#  Equity Protection Swaps: A New Type of Investment Insurance
for Holders of Superannuation Accounts

Huansang Xu$\,{}^{a}$, Ruyi Liu$\,{}^{a}$ and Marek Rutkowski$\,{}^{a,b}\,$

${}^{a\,}$School of Mathematics and Statistics, University of Sydney
Sydney, NSW 2006, Australia

${}^{b\,}$Faculty of Mathematics and Information Science, Warsaw University of
Technology
00-661 Warszawa, Poland

###### Abstract

We propose to develop a new class of investment insurance products for holders
of superannuation accounts in Australia, which we tentatively call equity
protection swaps (EPSs). An EPS is a standalone financial derivative, which is
reminiscent of a total return swap but also shares some features with the
variable annuity known as the registered index-linked annuity (RILA). The
buyer of an EPS obtains a partial protection against losses on a reference
portfolio and, in exchange, agrees to share portfolio gains with the insurance
provider if the realised return on a reference portfolio is above a
predetermined threshold. Formally, a generic EPS consists of protection and
fee legs with participation rates agreed upon by the provider and holder. A
general fair pricing formula for an EPS is obtained by considering a static
hedging strategy based on traded European options. It is argued that to make
the contract appealing to holders, the provider should select an appropriate
protection and fee rates that make the fair premium at contract’s inception
equal to zero. A numerical study based on the Black-Scholes model and
empirical tests based on market data for S&P 500 and S&P/ASX 200 indices for
2020-2022 demonstrate the benefits of an EPS as an efficient investment
insurance tool for superannuation accounts.

Keywords: equity protection swap, portfolio insurance, superannuation account,
fair premium, static hedging.

##  1 Introduction

Since the 1990s, we observe the global trend of shifting pension plans from
defined benefit (DB) schemes to defined contribution (DC) structures. Within a
defined contribution type pension scheme, the ultimate retirement benefit
hinges upon the accumulated contributions and the effectiveness of their
investment over time. Generally speaking, under a DC plan a proportion of
employee’s earnings is deposited into an investment fund so that the DC
account holder may lawfully access the money at their retirement. In contrast
to defined benefit pension plans, DC pension funds provide greater control
over retirement savings, albeit introducing market risks to participants.
Nowadays, a growing number of DC pension account holders are undergoing
transition from the accumulation phase toward pension phase. Given the
substantial balances at hand, there is a growing interest in managing the
market risks and keeping a proportion of the balance on risky assets with a
limited potential loss. For concreteness, we will refer to the pension system
in Australia but analogous schemes are available in all major economies.

The Superannuation Guarantee (SG) is a comprehensive retirement savings
scheme, which was introduced by the Australian Government on 1 July 1992.
Under the SG scheme (also known as superannuation or, simply, super) employers
are required to contribute mandatory sums to individual superannuation
accounts of their employees. As of July 2023, the minimum employer’s
contribution charge is 11% of each eligible employee’s ordinary time earnings
and is scheduled to progressively grow to its maximum level 12% from 1 July
2027 onwards. Employer’s SG contributions are supplemented by personal
contributions of super fund members representing around 20% of total
contributions. Since the introduction of mandatory superannuation scheme, the
proportion of Australian population with superannuation account has reached
78%, which is one of the highest levels of coverage worldwide. According to
the recent data provided by ASFA [1] in September 2023, around 17 million
Australians are holders of superannuation account and their total
superannuation assets amount to around 140% of GDP in 2022.

In contrast to other pension systems, Australian super funds distinguish
themselves by establishing personal accounts for their members with two main
types of super funds: defined benefit super funds with shared investment risks
and limited risk-taking capacity ([10]), which are now mostly closed to new
members, and nowadays prevailing accumulation super funds, which grow with
contributions and idiosyncratic investment returns. We henceforth focus on the
accumulation funds since they enable the provision of highly tailored
investment options, accommodating diverse individual beliefs and investment
preferences. Accumulation account holders have the flexibility to allocate
their investments across various asset classes, including equities, cash
investments, property, infrastructure, as well as other assets such as hedge
funds and commodities, which also means that their risk-taking capacity is
virtually unlimited.

As of September 2023, Australians have collectively accumulated 3.56 trillion
AUD, out of which 2.47 trillion AUD held in 1368 APRA-regulated super funds
and 0.89 trillion AUD in ATO-regulated self-managed super funds; this makes
Australia the fourth largest holder of pension fund assets worldwide. In
particular, 1.00 trillion AUD (around 40% of the total) was invested in
MySuper, which is a default option offered by large APRA-regulated super
funds. From the perspective of asset classes, 53.3% of the total was invested
in equities, with a breakdown of 21.9% in Australian listed equities, 26.4% in
international listed equities, and 5.1% in unlisted equities. Fixed income and
cash investments constituted 28.8% of total investments, distributed as 20.3%
in fixed income securities (bonds) and 8.5% in cash (bank bills). Property and
infrastructure accounted for 15.6% of the total, while other asset classes,
encompassing hedge funds and commodities, represented 2.2% of the overall
investment portfolio. This nuanced investment allocation strategy underscores
the unique nature of Australian superannuation scheme, catering to members’
varied expectations and preferences.

Over the past 30 years, the average nominal return of Australian super funds
was 7.2% and the real return versus CPI was 4.5%. Despite this overall
positive trend, two noteworthy instances of negative returns occurred, first
negative 12.9% return in 2008 during the subprime mortgage crisis of 2007-2010
and then negative 3.3% return in 2022 amidst the post Covid-19 pandemic high
inflation crisis and the ensuing market debacle in the first half of 2022 when
the S&P 500 and ASX 200 (more precisely, the S&P/ASX 200 index) indices have
fallen by 20% and 12%, respectively. Not surprisingly, superannuation returns
exhibit a high correlation with major market indices due to the predominant
inclusion of these indices in investment options offered to members.

Figure 1: Superannuation nominal returns by year

One important issue for members of super funds that has not received deserved
attention reads: how to manage the risk of a global market downturn? While the
account holders often prioritise risk mitigation over pursuing higher returns
during the ultimate stage of their accumulation period, they meet a great
challenge in finding a desirable investment strategy. This is partially due to
the lack of user-friendly financial instruments available to protect their
investment returns against an unpredictable risk of a market downturn,
especially when in transition to pension phase. This shortage of investment
protection products underscores a critical gap in the existing framework for
superannuation risk management. We thus believe that it is important to
develop new widely available products serving as an efficient protection
against market downturns. By profiting from such standalone financial
instruments, account holders would be able choose more aggressive investment
strategies, even during the final stage of accumulation phase or in pension
phase. From a broader financial market perspective, the flow of additional
funds from super funds into the market would contribute to sustainable growth
of the economy.

Before delving into the Australian superannuation context and introducing a
new type of investment insurance, let us first take a brief look at other
insurance products offered on pension markets worldwide, especially in the
United States where the market for variable annuities has been well
established since 1950s. In general terms, a variable annuity is a financial
contract between a policyholder and an insurance company that promises a
stream of periodic payments linked to the investment performance of an
underlying portfolio. A typical variable annuity is a unit-linked or managed
fund vehicle, which also offers optional guarantee benefits, commonly referred
to as ‘riders,’ as a choice for the customer. Insurance companies charge
periodic fees for managing these annuities and primarily invest in publicly
traded securities to create a diversified reference portfolio. They have also
responded to the need to protect investors’ future income by introducing
various additional covenants (riders) to variable annuities. These riders are
designed not only to manage risks, such as market and longevity risks, but
also to allow for optimal withdrawal strategies in the face of future
uncertainty. Two major types of riders are: the Guaranteed Minimum Death
Benefits (GMDB) and the Guaranteed Minimum Living Benefits (GMLB). GMLB riders
are further categorised into several classes, including Guaranteed Minimum
Income Benefits (GMIB), Guaranteed Minimum Withdrawal Benefits (GMWB), and
Guaranteed Lifetime Withdrawal Benefits (GLWB) but, typically, a single rider
combines various features tailored to individual needs. As reported in Drexler
et al. [9], there was a significant increase in the sale of variable annuities
after the late 1990s when guaranteed minimum benefit riders began to be
included in variable annuities and, in 2016, approximately 76% of
policyholders opted to purchase a variable annuity with guarantees, when this
option was available.

Regarding the GMDB rider, if the policyholder’s death occurs before the
contract matures, the beneficiary of the contract will receive the higher of
the investment account value or the death benefit. The pricing approach to
GMDB riders was first proposed and analysed by Milevsky and Posner [16].
Milevsky and Salisbury [17] examined a variety of static and dynamic methods
for pricing GMWB riders. Dai et al. [7] developed an efficient finite
difference algorithm with the penalty approximation to solve the singular
stochastic control problem under the optimal withdrawal strategy. Later on,
several papers discussed combinations of various riders. In particular, Bauer
et al. [4] dealt with the valuation of variable annuities with multiple
guarantees, with a holder’s strategy not only consisting of the decision of
surrender option but also possible withdrawals at a set of predetermined
dates. Luo and Shevchenko[14] formulated the pricing problem for GMWDB rider
(i.e., GMDB combined with GMWB) as a stochastic control problem by using the
conditional expectation of dynamic GMWDB riders for a given death time.
Moreover, Yang and Dai [21] and Dai et al. [8] considered stochastic mortality
in GMWB riders and concluded that ignoring mortality risk would lead to an
overpricing of the rider. From the existing literature, it is clear that the
fair pricing of variable annuities with riders is a complex optimisation
problem, which can be solved by sophisticated providers, but a sound
assessment of their advantages and drawbacks is beyond reach of members of
super funds.

As a supplement to compulsory annuities, a new type of insurance product
called Registered Index-Linked Annuity (RILA) has been introduced in the U.S.,
also known as ‘index-linked annuities’ or ‘structured variable annuities.’
Moenig [18] provided a timely first academic study on RILAs. A RILA is a
complex annuity with insurance properties that offer policyholders the
flexibility to prioritise their growth opportunities while limiting potential
losses. In a generic RILA contract, the policyholder can choose from several
index options, protection levels, and other available options to achieve their
preferred profile of potential growth. Typically, holders of a RILA can choose
to tie their wealth account with the performance of a particular market index,
such as the S$\&$P 500 for U.S. large-cap stocks, NASDAQ Composite for all
American and foreign common stocks, and the MSCI EAFE for stocks from
developed economies outside of North America. In essence, if the reference
index performs poorly, the credited loss is lessened by either a floor (i.e.,
a maximum loss percentage), a buffer (i.e., only index losses over a specific
threshold are credited), or a downside participation rate (e.g., the loss
credited to the account corresponds to only 50$\%$ of index losses). As
pointed out in [18], the insurer can benefit from the imbalance between the
guarantee’s downside protection and its upside cap, the ability to invest RILA
funds into other products (e.g., corporate bonds) and hence earn a credit risk
premium and, in some cases, from using a price-based index not accounting for
dividend payments.

It is striking that in today’s Australian superannuation system no widely
available and functional risk management instruments similar to RILAs are
offered. It should be acknowledged that some kinds of variable annuities with
riders are available to members of super funds but they are mostly retirement
income products (see, for instance, Shen et al. [20] and the references
therein) whereas insurance products for both accumulation and pension phase
are missing. Furthermore, as emphasised in [20], the valuation and assessment
of existing products rely on extremely complex numerical techniques and hence
their pricing is subject to model risk. This should be contrasted with a
robust market-based approach presented here where fair pricing does not
require to make use of any stochastic model, provided that suitable options on
indices are traded. Most importantly, we propose to completely separate the
superannuation account from insurance product, as opposed to the case variable
annuities where insurance is embedded in a pension product, for instance, a
lifetime pension such a Group Self-Annuity (GSA), which was proposed by
Piggott et al. [19] and recently reexamined by Shen et al. [20]. Although a
proportion of holders of superannuation accounts are likely to invest in
variable annuities to support their living expenses in retirement phase, it is
also reasonable to expect that not only holders of accumulation accounts but
also a group of holders of pensions account may show interest in investment
protection products.

In response to the increasing demand for investment insurance by members of
super funds, we propose to completely depart from variable annuities with
riders and to introduce of a new class of standalone financial products, which
we propose to call Equity Protection Swaps (EPS). Recall that a traditional
equity swap involves the exchange of future cash flows between two parties,
allowing each party to diversify their income for a predetermined period while
retaining their original assets. An EPS shares some similarities with a common
type of equity swaps known as a Total Return Swap (TRS), which is a financial
contract transferring both the credit and market risks of an underlying asset.
Due to its design, an EPS can offer effective protection in case the value of
a reference portfolio declines, and it can be easily tailored to meet the
specific needs of super fund members.

It is important to note that the proposed approach is completely independent
of the issue of optimal investment strategies since we take for granted that
the majority of pension account holders lack the necessary financial or
mathematical expertise to construct an optimal portfolio and hence they are
choosing the default option, such as MySuper. Moreover, the current
superannuation system in Australia is characterised by a relatively limited
selection of investment options offered by super funds, restricting investors
from completely freely choosing their desired investments. Given the limited
understanding of investment principles among most super funds members,
pursuing an optimal investing approach to superannuation products would be
clearly unrealistic and of no practical importance. At the same time, there
remains a significant, and intuitively clear, desire among investors to
mitigate their exposure to investment risk. This underscores the relevance of
Equity Protection Swaps, which are tailored to provide an effective and
relatively simple way of risk reduction (investment insurance) to holders of
superannuation account.

When compared to the existing insurance products for variable annuities, an
EPS offer several advantages. Firstly, it has a more transparent structure
than a typical variable annuity rider, making it easier for super funds
members to assess and appreciate. Second, it does not require from its holder
to make any decisions during the contract’s lifetime. In particular, there is
no need to consider an embedded optimal stopping problem, which is a feature
commonly encountered in most riders. Thirdly, the fair pricing of an EPS
relies on static perfect hedging using liquidly traded options. Hence it is
expected that an EPS would be a straightforward and cost-effective tool for
insurance providers and holders of superannuation accounts. This should be
contrasted with opaque and extremely complex utility-based pricing methods for
riders such as GMWB, GMIB or GSA, typically with no reference to effective
hedging for providers, although a martingale measure for equities model is
sometimes formally utilised. Most importantly, the fair pricing of a basic EPS
can be considered to be model-free since market data for traded European
options can be used to compute its fair premium.

Finally, an EPS is not inherently tied to any annuity and hence it can be
issued by a third party, which is here referred to as a provider of investment
insurance. In fact, the key distinction between a RILA and an EPS is that the
latter is a standalone financial derivative, which can be added to a
superannuation account at any phase (not only after retirement), whereas RILA
is a genuine variable annuity, which can be purchased by retirees using funds
from their pension accounts. In the context of the Australian superannuation
system, it should be stressed that members are not allowed to withdraw funds
from their superannuation accounts during the accumulation phase to purchase
an investment guarantee such as, e.g., a RILA. Instead, members are compelled
to select a combination of investment options, which are offered by a super
fund, and thus they are entirely responsible for managing market risks. An EPS
is tailored to serve as a convenient risk mitigation tool for super funds
members by offering a highly desirable and transparently designed protection
against potential losses at a minimal cost, which is only incurred when their
investment portfolio yields substantial gains.

The paper is organised as follows. We first present in Section 2 the
definition and main features of a generic EPS. We propose a classification of
the fee and protection legs of an EPS with special attention to buffer and
floor legs. Section 3 is devoted to an analysis of hedging strategies for
equity protection swaps. We show in Proposition 3.1 that an EPS admits a
static hedge composed of European call and put options and thus its fair price
can be computed in any arbitrage-free model of the financial market. Even more
importantly, the fair price of an EPS can also be obtained directly from
market data for traded European call and put options if such options are
available for the reference portfolio (usually a market index). Then the
static hedge for an EPS can be easily constructed by investing in traded
options and hence the hedge is independent of a model. If, however, European
options needed for hedging are not available, then their synthetic
counterparts can be produced through a dynamic trading strategy using a
preferred dynamical model for the reference portfolio (e.g., the classical
Black-Scholes [5] model, the Merton [15] jump-diffusion model, the Heston [11]
stochastic volatility model, or the Bates [3] model).

In Section 4, we present forward performance tests of buffer and floor EPSs
within the framework of the classical Black-Scholes model using the pricing
formula for a generic EPS obtained in Proposition 4.1. In the first part of
model-based numerical studies, we use buffer and floor EPSs introduced before
as examples of a generic EPS. By specifying interest rates, volatility, and
other relevant market parameters, we compute the fair initial premium by using
static hedging strategies for an EPS. We can see the impact of different
characteristics of an EPS, such as participation rate, leg setting, and
maturity, on fair premiums. It is widely acknowledged that a typical investor
would not be willing to pay an upfront premium, especially when it is
substantial. Therefore, we propose to focus on EPS products with a null fair
premium at inception. An EPS with a null fair premium still has the
participation rate for the protection leg desired by the holder, whereas the
participation rate for the fee leg is chosen to ensure that the provider can
implement a hedging strategy based on long and short positions in European
options at no additional cost.

Unlike Section 4 where all results are model-based, Section 5 is devoted to
backtesting of real-world benefits for holders of EPSs using historical data
for S&P 500 and S&P/ASX 200 indices and hence results and conclusions are
based on model-free market data. We ponder over two situations during a period
under study: a market crash generating substantial losses and a general market
characterised by moderate gains. An analysis of the first situation shows that
an EPS can provide highly effective protection against large losses, whereas
the second situation illustrates the tradeoff between loss protection and
profit sharing. The initial premia for EPS products used in our analysis can
be obtained directly from the market data for European call and put options,
as opposed to calculations based on a hypothetical model. The conclusion from
backtesting is that an EPS can serve as an efficient tool to mitigate
financial risks associated with a superannuation account.

Finally, in Section 6 we briefly summarise our findings and we give an outline
possible directions for future research. In particular, we emphasise the
importance of a cross-currency variant of an equity protection swap for
holders of superannuation accounts in Australia.

##  2 Properties and Classification of Equity Protection Swaps

In this section, we will build a general pricing model for adjusted return of
the underlying reference portfolio for an equity protection swap (EPS). By an
EPS, we mean a financial contract in which a single terminal payoff occurs at
maturity date $T$ and is a predetermined function of the realised simple rate
of return on a reference portfolio during the lifetime of an EPS. For
instance, a reference portfolio may represent the value of a particular market
index, a combination of domestic and foreign indices or a bespoke portfolio of
domestic and foreign equities.

###  2.1 Specification of a Generic Equity Protection Swap

Let $S_{t}$ stand for the value at time $t\in[0,T]$ of a given reference
portfolio of assets, which is denoted as $S$. The simple rate of return on $S$
over the time period $[0,T]$ is denoted as $R_{S}(0,T)$ but, for brevity, we
will also write $R_{T}$ instead of $R_{S}(0,T)$ so that
$R_{T}=R_{S}(0,T):=(S_{T}-S_{0})/S_{0}$ and the inequality $R_{T}\geq-1$ holds
under our standing assumptions that $S_{0}$ is a strictly positive number and
$S_{T}$ is a nonnegative random variable. Formally, the stochastic process $S$
is defined on a probability space
$(\Omega,\mathcal{F},{\mathbb{F}},{\mathbb{P}})$ endowed with the filtration
${\mathbb{F}}$ where ${\mathbb{P}}$ is the statistical probability and thus
the rate of return $R_{T}$ is an $\mathcal{F}_{T}$-measurable random variable.

Let us consider a generic equity protection swap so that an explicit
specification of a reference portfolio $S$ is not important at this stage.
Suppose that a holder of a superannuation account decides at time 0 to invest
some of their funds in $S$ and let $N_{p}$ denote the nominal principal of
their holdings that are invested (either directly or indirectly) in $S$ at
time 0. Then the dynamics of their funds that are invested in $S$ will follow
the fluctuations of the value of a portfolio $S$ and thus the current market
value of their funds invested in $S$ will grow (if $R_{T}>0$) or decline (if
$R_{T}<0$) to the terminal value $N_{p}R_{T}$ at time $T$.

As was already mentioned, the goal of an EPS is to reduce the holder’s risk
exposure by providing either a full or partial protection against a
substantial decline of holder’s wealth invested in $S$ by entering into a
customised contract offered by a financial company. The two counterparties of
an EPS are henceforth referred to as the protection provider who has the short
position in an EPS and the holder of an EPS (the protection buyer or, simply,
the investor) who has the long position in the contract. It should be stressed
that aims of the two counterparties are completely different: the holder is
reducing their risk exposure by partially ‘selling’ it to the protection
provider, whereas the financial company accept the holder’s risk and their
goal is to completely hedge their risk exposure using traded financial
derivatives (European options on $S$). By analysing available hedging
strategies and market prices of options, the company is also able to provide a
fair valuation of an EPS in the form of the fee participation rate in holder’s
gains.

From the viewpoint of the protection provider, the cash flow at time $T$ is
given by $N_{p}\psi(R_{T})$ where $N_{p}$ is the nominal principal of an EPS
and a function $\psi:[-1,\infty)\to{\mathbb{R}}$ is used to encode the
structure of an EPS. By the adjusted return we mean a specific transformation
$\psi(R_{T})$ of the realised return $R_{T}$, as given by the covenants of
each particular EPS. Before offering a product, the provider of an EPS needs
to examine the pricing and hedging problem although, formally, a solution to
the pricing and hedging problem is symmetric if the bid-ask spread for traded
options is neglected and thus, at least in principle, an investor can also
derive that solution since it does not require to use sophisticated
mathematical tools.

From the holder’s perspective, it is important to notice that the net payoff
to the buyer of an EPS is given by $N_{p}(R_{T}-\psi(R_{T}))$ or,
equivalently, it equals $R_{T}-\psi(R_{T})$ per one unit of the nominal
principal. The benefit from entering an EPS for investors lies in reduction of
losses and thus the transformation $\psi$ should be specified in accordance
with their individual investment strategies and risk preferences. Therefore,
several variants of an EPS are likely to be offered to investors and some of
them are studied in the present work. However, the main features shared by all
EPSs are: a desirable reduction of the holder’s loss, the null initial cost,
and a fair protection fee in the form of partial sharing of the holder’s gain
from their investment in $S$. It is clear that an EPS is a standalone
financial derivative written on $S$ and the values of $N_{p}$ and $S_{0}$ are
unrelated.

We are in a position to formalise the definition of a generic EPS. Unless
explicitly stated otherwise, we henceforth assume, without loss of generality,
that the notional principal of an EPS equals 1, that is, $N_{p}=1$. In fact,
due to specification of a generic EPS, one could also assume that $S_{0}=1$
since only the return $R_{T}$ is relevant but for clarity of presentation we
are not going to make that assumption. Since it is natural to postulate that
an EPS can be hedge using plain-vanilla call and put options, we postulate
that the payoff profile $\psi:[-1,\infty)\to{\mathbb{R}}$ of a generic EPS is
a piecewise linear, non-decreasing, continuous function such that $\psi(0)=0$
where the last condition is natural.

###### Definition 2.1

In an equity protection swaps starting at time $0$ with the maturity date $T$
and the reference portfolio $S$, the cash flow for the EPS provider is
specified by the adjusted return $\psi(R_{T})$ where the derivative of the
adjusted return function $\psi:[-1,\infty)\to\mathbb{R}$ equals

$\psi^{\prime}(R)=\sum_{k=-(n+1)}^{m}\alpha_{k+1}\mathds{1}_{(\beta_{k},\beta_{k+1})}(R)$
(1)

where the sequence of endpoints of return ranges, $(\beta_{k})_{k=-n}^{m}$, is
a strictly increasing finite sequence of real numbers, $\beta_{-(n+1)}=-1$ and
$\beta_{m+1}=\infty$. The sequence $(\alpha_{k})_{k=-n}^{m+1}$ of marginal
participation rates is an arbitrary sequence of numbers from $[0,1]$.
Furthermore, we assume that $\beta_{0}=0$ and thus
$-1=\beta_{-(n+1)}<\beta_{-n}<\cdots<\beta_{-2}<\beta_{-1}<\beta_{0}=0<\beta_{1}<\cdots<\beta_{m}<\beta_{m+1}=\infty$.

Consistently with its financial interpretation, we expect that the number
$\alpha_{k}$ will belong to the interval $[0,1]$ since it represents the
provider’s marginal participation rate in gains or losses of the reference
portfolio when the realised return falls within the range of $\beta_{k}$ to
$\beta_{k+1}$ (though in some exceptional cases the fee participation rate can
be greater than 1). In particular, $\alpha_{k}=1$ corresponds to the full
participation of the provider in gains or losses, whereas $\alpha_{k}\in(0,1)$
represents a partial participation in gains (upside participation rate) or
losses (downside participation rate). Of course, $\alpha_{k}=0$ means that the
provider does not participate in incremental returns belonging to the interval
$[\beta_{k},\beta_{k+1}]$. Since $\alpha_{k}$ represents the provider’s
participation rate, it is clear that the corresponding buyer’s participation
rate equals $1-\alpha_{k}$ for every $k$.

When analyzing particular instances of fee and protection legs of an EPS, it
will be convenient to modify the notation by independently labeling the
negative and positive values of $\beta$. Specifically, we write
$-1=l_{n+1}<l_{n}<\cdots<l_{2}<l_{1}<l_{0}=0$ for non-positive values of
$\beta$ and $0=g_{0}<g_{1}<\cdots<g_{m}<g_{m+1}=\infty$ for non-negative
values of $\beta$. For each $k$, the participation rate over the loss interval
$[l_{n+1},l_{n})$ is denoted as $p_{n+1}$ whereas the participation rate over
the gain interval $(g_{k},g_{k+1}]$ is denoted as $f_{k+1}$.

In a simple example where $n=m=1$, we have
$-1=l_{2}<l_{1}<l_{0}=0=g_{0}<g_{1}<g_{2}=\infty$ and we need to specify four
values of marginal participation rates: $p_{1}$ and $p_{2}$ for the protection
leg and $f_{1}$ and $f_{2}$ for the fee leg. It is thus clear that we deal
with six parameters, which fully specify the structure of an EPS when $n=m=1$.
The following lemma is an easy consequence of (1) and the postulated
equalities $\psi(0)=\psi(g_{0})=\psi(l_{0})=0$.

###### Lemma 2.1

We have that $\psi(R)\leq 0$ for $R\in[-1,0]$ and $\psi(R)\geq 0$ for
$R\in[0,\infty)$. The protection leg and the fee leg of an EPS are given by

$\psi^{p}(R):=\psi(R)\mathds{1}_{\\{R<0\\}},\quad\psi^{f}(R):=\psi(R)\mathds{1}_{\\{R>0\\}}$

so that $\psi(R_{T})=\psi^{p}(R_{T})+\psi^{f}(R_{T})$ where the non-positive
payoff $\psi^{p}(R_{T})$ and the non-negative payoff $\psi^{f}(R_{T})$
represent the provider’s loss (protection payout) and gain (fee income) from
an EPS, respectively.

Proof.  Let us define $\Delta l_{i}=l_{i+1}-l_{i}<0$ and $\Delta
g_{j}=g_{j+1}-g_{j}>0$ and recall that $g_{0}=l_{0}=0$. Then the protection
and fee legs satisfy, for $i=0,1,\dots,n$ and $j=0,1,\dots,m$,

$\displaystyle\psi^{p}(R)$ $\displaystyle=\Big{(}\sum_{k=0}^{i-1}p_{k+1}\Delta
l_{k}+p_{i+1}(R-l_{i})\Big{)}\mathds{1}_{\\{R\in(l_{i+1},l_{i})\\}},$
$\displaystyle\psi^{f}(R)$ $\displaystyle=\Big{(}\sum_{k=0}^{j-1}f_{k+1}\Delta
g_{k}+f_{j+1}(R-g_{j})\Big{)}\mathds{1}_{\\{R\in(g_{j},g_{j+1})\\}},$

and it is clear that the adjusted return function of an EPS satisfies
$\psi(R)=\psi^{p}(R)+\psi^{f}(R)$. $\Box$

Lemma 2.1 leads to the following definition of fairness of a generic EPS,
which explains why the term ‘swap’ is suitable for a generic EPS. Although, by
convention, the inception date of an EPS is chosen to be 0, it can be
substituted with any date $t<T$.

###### Definition 2.2

We say that an EPS initiated at time 0 with a payoff structure $\psi$ is fair
if the payoff $\psi(R_{T})$ with settlement date $T$ has null arbitrage-free
price at its inception time 0.

It will be shown in Section 3 that the issue of finding a fair structure for
an EPS is closely related to the pricing problem for European options with the
underlying asset $S$ and maturity date $T$.

Let us make some comments on Definition 2.1 and Lemma 2.1:

  * •

Since the main goal of an EPS is to protect against substantial losses, the
loss leg should be chosen by the holder from several available options. As
soon as the holder’s choice is completed, the issue of ‘fair pricing’ of an
EPS hinges on finding parameters for the fee leg that make the contract
worthless at its inception.

  * •

In a typical structure, the fee leg $\psi^{f}$ (resp., the protection leg
$\psi^{p}$) can be formally identified with the payoff of a particular
portfolio of European call options (resp., European put options) written on
the underlying static portfolio (e.g., a market index).

  * •

Suppose now that the underlying portfolio is dynamic, that is, the portfolio
is actively managed by a holder of a superannuation account. If a holder of an
EPS on a given portfolio wishes to switch at some moment before $T$ from one
investment option to another, then an EPS can be terminated and settled at its
market price and a new EPS can be initiated at no cost if this is desired by a
holder. This means that a flexible roll-over strategy in EPSs is available
since at any time the market value of an EPS is readily available due to the
fact that the hedging portfolio is composed of traded options.

  * •

One could also consider an EPS with a discontinuous adjusted reward function
$\psi$ but it is not clear if this would be beneficial for practical purposes
and, obviously, the hedging portfolio for an EPS would require options with
discontinuous payoff profiles.

Despite some similarities, there are several crucial differences between an
EPS and a RILA. Firstly, the main difference is that a RILA is special case of
a variable annuity while an EPS is a standalone financial derivative (a swap),
which is designed as a complement to a superannuation account. The provider of
RILA receives the nominal principal from the investor at the start of
contract, whereas there should be no cash flow at the inception of an EPS, of
course, provided that it is set up and priced correctly. Although
representations of the provider’s gains and losses in terms of European
options are analogous to those obtained for RILAs in Section 3.2 of Moenig
[18], their analysis is different since the provider of an EPS does not need
to manage the nominal principal and thus has lower expenses and risks.

Secondly, an EPS has typically a more flexible structure than a RILA, for
which one can only consider an existing variable annuity, which is provided by
an insurance company and thus always has a fixed structure with either floor,
buffer or cap and some predetermined participation rates. In contrast, an EPS
is a bespoke financial derivative so its structure can be brokered by the two
parties of the contract. Thirdly, the underlying portfolio of the RILA is also
fixed (a particular market index), whereas the reference portfolio of an EPS
can be chosen arbitrarily, for instance: a market index, an bespoke equity
portfolio, or an investment portfolio involving assets from several economies
and, as was explained above, if the reference portfolio is changed before the
nominal maturity $T$ then an EPS can be terminated by its holder and settled
in reference to its current market value.

###  2.2 Basic EPS with Proportional Fee and Protection Legs

Let us assume $n=m=1$ and thus $-1<l_{1}<l_{0}=0=g_{0}<g_{1}<\infty$. The most
basic fee leg relies upon setting $f:=f_{1}=f_{2}\in(0,1]$, which means that
the provider is awarded proportional participation in all positive returns on
a reference portfolio $S$. Similarly, the simplest protection leg is obtained
by taking $p:=p_{1}=p_{2}\in(0,1]$, which means the provider has a
proportional participation in all negative returns. Then the cash flow
$\psi(R_{T})$ equals

$\displaystyle\psi(R_{T})=pR_{T}\mathds{1}_{\\{R_{T}\in[-1,0)\\}}+fR_{T}\mathds{1}_{\\{R_{T}\in(0,\infty)\\}}=-p(-R_{T})^{+}+f(R_{T})^{+}$

and hence it can be represented by a short position in $p/S_{0}$ units of a
European put combined with a long position in $f/S_{0}$ units of a European
call written on $S$ with strike $S_{0}$ and maturity $T$ since

$\displaystyle-p(-R_{T})^{+}=-\frac{p}{S_{0}}\,(S_{0}-S_{T})^{+},\quad
f(R_{T})^{+}=\frac{f}{S_{0}}\,(S_{T}-S_{0})^{+}.$

We will now show that the basic EPS can be fairly priced in relation to market
prices of European call and put options. Let $\text{Call}_{t}(K,T)$ and
$\text{Put}_{t}(K,T)$ denote the price at time $t$ of a European call and put
on the reference portfolio with strike $K$ and maturity $T$, respectively.
Then

$\displaystyle\psi(R_{T})=\frac{f}{S_{0}}\,\text{Call}_{T}(S_{0},T)-\frac{p}{S_{0}}\,\text{Put}_{T}(S_{0},T)$

and thus the fair value of the EPS at time 0 for its provider equals

$\displaystyle\frac{f}{S_{0}}\,\text{Call}_{0}(S_{0},T)-\frac{p}{S_{0}}\,\text{Put}_{0}(S_{0},T).$

An application of the put-call parity, which reads
$\text{Call}_{0}(K,T)-\text{Put}_{0}(K,T)=S_{0}-Ke^{-rT}$, gives the equality
$\text{Call}_{0}(S_{0},T)-\text{Put}_{0}(S_{0},T)=S_{0}(1-e^{-rT})$ and thus
yields the following representation for the fair value for the provider of the
basic EPS

$\displaystyle\frac{f}{S_{0}}\,\text{Call}_{0}(S_{0},T)-\frac{p}{S_{0}}\,\text{Put}_{0}(S_{0},T)=\frac{f}{S_{0}}\,\text{Call}_{0}(S_{0},T)-\frac{p}{S_{0}}\Big{(}\text{Call}_{0}(S_{0},T)-S_{0}\big{(}1-e^{-rT}\big{)}\Big{)}$
$\displaystyle=\frac{f-p}{S_{0}}\,\text{Call}_{0}(S_{0},T)+p\big{(}1-e^{-rT}\big{)}=0.$
(2)

Notice that the last equality in (2.2) is not trivially satisfied for any
choice of $f$ and $p$ but is postulated here for consistency with the swap
requirement that there should be no cash flow in the basic EPS at time 0 and
thus its fair value at time 0 should vanish. It is clear that if the risk-free
rate $r$ and the market price $\text{Call}_{0}(S_{0},T)$ are known, then it is
possible to find a relationship between the parameters $f$ and $p$ that makes
the basic EPS worthless at time 0. Finally, the theoretical equality
$\text{Call}_{0}(S_{0},T)=S_{0}\,\text{Call}_{0}(1,T)$ holds and thus the
relationship between the fair values of $f$ and $p$ reduces to
$(f-p)\,\text{Call}_{0}(1,T)+p(1-e^{-rT})=0$. It is clear that the equality
$f=p$ holds if $r=0$.

In contrast, if $r>0$ then the basic EPS with $f=p$ is not fair since an EPS
allows the provider to participate in returns on a portfolio $S$ without
actually investing in $S$ his wealth, presumably borrowed at a positive
interest rate $r$. Of course, it becomes a fair EPS for some values $f>p$,
which can be computed from (2.2). An analogous analysis can be conducted for
the basic EPS in the less commonly encountered case where $r<0$.

###  2.3 Classification of Fee and Protection Legs

In this subsection, we will usually assume that $n=m=1$ (except for the
buffer-cap and buffer-floor legs) and thus
$-1<l_{1}<l_{0}=0=g_{0}<g_{1}<\infty$. Then each particular structure of an
EPS depends on a specification of four participation rates, namely, $p_{1}$
and $p_{2}$ for the protection leg and $f_{1}$ and $f_{2}$ for the fee leg.
For brevity, the parameters $p_{1}$ and $p_{2}$ (resp. $f_{1}$ and $f_{2}$)
are henceforth referred as protection rates (resp. fee rates).

Let us first describe the most typical and practically appealing instances of
a protection leg. Although all four cases can be considered, we contend that
the buffer (and perhaps also buffer-cap) case would be the most suitable,
especially when combined with an analogous structure for the protection leg.
Generally speaking, the goal of an EPS is to give protection against
substantial losses so it is natural to expect that there will be no cash flow
at settlement if the realised return falls within the range $(l_{1},g_{1})$.
Only when realised losses are below the trigger $l_{1}<0$ a buffer (or a
buffer-cap) EPS provides a full or partial protection to its holder who, in
exchange, agrees to share with the provider portfolio’s gains if they fall
above a predetermined level $g_{1}>0$.

###### Definition 2.3

The proportional fee leg is given by $f_{1}=f_{2}\in(0,1]$ so that

$\displaystyle\psi^{f,1}(R)=f_{1}R\mathds{1}_{\\{R\in(0,\infty)\\}}.$

The buffer fee leg is given by $f_{1}=0$ and $f_{2}\in(0,1]$ so that

$\displaystyle\psi^{f,2}(R)=f_{2}(R-g_{1})\mathds{1}_{\\{R\in(g_{1},\infty)\\}}.$

The cap fee leg is given by $f_{1}\in(0,1]$ and $f_{2}=0$ so that

$\displaystyle\psi^{f,3}(R)=f_{1}R\mathds{1}_{\\{R\in(0,g_{1}]\\}}+f_{1}g_{1}\mathds{1}_{\\{R\in(g_{1},\infty)\\}}.$

The buffer-cap fee leg is given by $f_{1}=0,f_{2}\in(0,1]$ and $f_{3}=0$ so
that

$\displaystyle\psi^{f,4}(R)=f_{2}(R-g_{1})\mathds{1}_{\\{R\in(g_{1},g_{2}]\\}}+f_{2}(g_{2}-g_{1})\mathds{1}_{\\{R\in(g_{2},\infty)\\}}.$

The most typical specifications of the protection leg are analogous to those
of the fee leg. A selection of a particular protection leg depends on the
buyer’s preferences and thus it would be natural to expect that a broad
spectrum of products should be offered by EPS providers. Assuming that a
perfect hedge of an EPS is feasible, the provider would be indifferent with
respect to the buyer’s choice of the structure of the protection leg. However,
in reality only a partial hedging can be attained for more complex cross-
currency products and thus some forms of the protection leg are likely to be
preferred by providers. For instance, the presence of a floor clause is
expected to be appreciated by providers since it provides a lower bound on
their downside risk exposure even when the market experiences a catastrophic
downturn.

###### Definition 2.4

The proportional protection leg is given by $p_{1}=p_{2}\in(0,1]$ so that

$\displaystyle\psi^{p,1}(R)=p_{1}R\mathds{1}_{\\{R\in[-1,0)\\}}.$

The buffer protection leg is given by $p_{1}=0$ and $p_{2}\in(0,1]$ so that

$\displaystyle\psi^{p,2}(R)=p_{2}(R-l_{1})\mathds{1}_{\\{R\in[-1,l_{1})\\}}.$

The floor protection leg is given by $p_{1}\in(0,1]$ and $p_{2}=0$ so that

$\displaystyle\psi^{p,3}(R)=p_{1}R\mathds{1}_{\\{R\in[l_{1},0)\\}}+p_{1}l_{1}\mathds{1}_{\\{R\in[-1,l_{1})\\}}.$

The buffer-floor protection leg is given by $p_{1}=0,p_{2}\in(0,1]$ and
$p_{3}=0$ so that

$\displaystyle\psi^{p,4}(R)=p_{2}(R-l_{1})\mathds{1}_{\\{R\in[l_{2},l_{1})\\}}+p_{2}(l_{2}-l_{1})\mathds{1}_{\\{R\in[-1,l_{2})\\}}.$

####  2.3.1 Buffer EPS

Let us introduce two most practically relevant forms of an EPS, which are
called the buffer EPS and the floor EPS. Notice that the proposed terminology
for a generic EPS is referring directly to the protection leg, rather than the
fee leg for which the choice of a buffer structure seems to be the most
appealing for holders. It should be observed that the net payoff to the holder
of a portfolio $S$ and an EPS is given by $R_{T}-\psi(R_{T})$ per one unit of
the nominal principal and thus it can be easily computed for any choice of
$\psi$.

We first define the buffer EPS where, incidentally, the concept of a buffer
leg is applied to both the protection and fee legs and thus it could also be
called a double-buffer EPS or a buffer/buffer EPS.

###### Definition 2.5

A buffer EPS is obtained by taking $p_{1}=f_{1}=0$ and any values for
$p_{2}\in(0,1]$ and $f_{2}\in(0,1]$. Hence $\psi_{B}(R)=0$ for all
$R\in[l_{1},g_{1}]$ where $l_{1}<0$ and $g_{1}>0$ and the payoff at maturity
$T$ for the provider of a buffer EPS equals

$\displaystyle\psi_{B}(R_{T})$
$\displaystyle:=\psi^{p,2}(R_{T})+\psi^{f,2}(R_{T})=p_{2}(R_{T}-l_{1})\mathds{1}_{\\{R_{T}\in[-1,l_{1})\\}}+f_{2}(R_{T}-g_{1})\mathds{1}_{\\{R_{T}\in(g_{1},\infty)\\}}$
$\displaystyle=-p_{2}(l_{1}-R_{T})^{+}+f_{2}(R_{T}-g_{1})^{+}.$

It is clear that the provider of the buffer EPS only participates in negative
returns below the buffer threshold $l_{1}$ and the positive return above the
cap threshold $g_{1}$ with respective participation rates $p_{2}$ and $f_{2}$.
The extremal case where $p_{2}=1$ and $f_{2}=1$ corresponds to the provider’s
full participation in portfolio’s losses below the buffer threshold $l_{1}<0$
combined with his full participation in gains above the cap $g_{1}>0$.

The adjusted return function $\psi_{B}$ for the buffer EPS that the provider
receives at maturity $T$ is presented in Figure 2 where the horizontal axis
represents the return of the underlying reference portfolio at maturity and
the vertical axis represents the provider’s gain or loss. Notice that the
protection rate $p_{2}$ and the fee rate $f_{2}$ give the slopes of the
adjusted return on a reference portfolio.

We will argue that the fair price of the buffer EPS can be easily expressed in
terms of prices of European put and call options on a reference portfolio
(e.g., a particular investment choice or a market index) with expiry $T$ and
suitable nominal values and strikes. If $l_{1}$ and $g_{1}$ are given, then
for any value of the protection rate $p_{2}$ chosen by the buyer, one can
compute a unique level of the fee rate $f_{2}$ for which the fair value of the
buffer EPS at its inception is null.

adjusted realised return $\psi_{B}(R_{T})$realised return
$R_{T}$$-1$$l_{1}$$g_{1}$$p_{1}=0$$p_{2}$$f_{1}=0$$f_{2}$ Figure 2: Provider’s
adjusted return for a buffer EPS

####  2.3.2 Floor EPS

The so-called floor EPS, which can also be called floor/buffer EPS, is
obtained by setting $p_{2}=0,p_{1}\in(0,1],f_{1}=0$ and $f_{2}\in(0,1]$, which
means that the provider fully covers the buyer’s losses above the level
$l_{1}$ if $p_{1}=1$ but the buyer’s losses below $l_{1}$ are only partially
covered since $p_{1}l_{1}$ acts as a floor for the adjusted return. By
convention, the buffer specification is applied to the fee leg and thus the
fee leg is exactly the same as in the buffer EPS of Definition 2.5. This
choice for the fee leg is motivated by our practical arguments that the buffer
fee leg is likely to be appreciated by a buyer of an EPS since there will be
no fee payment at all, unless the portfolio’s gains during the lifetime of an
EPS are substantial. The adjusted return function $\psi_{F}(R_{T})$ for the
floor EPS is shown in Figure 3.

###### Definition 2.6

A floor EPS is specified by parameters $p_{2}=f_{1}=0$ and arbitrary values of
$p_{1}\in(0,1]$ and $f_{2}\in(0,1]$. Hence $\psi_{B}(R)=0$ for all
$R\in[0,g_{1}]$ where $g_{1}>0$ and the payoff for the provider equals

$\displaystyle\psi_{F}(R_{T})$
$\displaystyle:=\psi^{p,3}(R_{T})+\psi^{f,2}(R_{T})=p_{1}l_{1}\mathds{1}_{\\{R_{T}\in[-1,l_{1}]\\}}-p_{1}(-R_{T})^{+}\mathds{1}_{\\{R_{T}\in(l_{1},0]\\}}+f_{2}(R_{T}-g_{1})^{+}$
$\displaystyle=-p_{1}(-R_{T})^{+}+p_{1}(l_{1}-R_{T})^{+}+f_{2}(R_{T}-g_{1})^{+}.$

adjusted realised return $\psi_{F}(R_{T})$realised return
$R_{T}$$-1$$l_{1}$$g_{1}$$p_{1}$$p_{2}=0$$f_{1}=0$$f_{2}$ Figure 3: Provider’s
adjusted return for a floor EPS

####  2.3.3 Examples of a Buffer and Floor EPS

In Section 4, we present selected numerical results for some EPSs with
different payoff structures. We refer, in particular, to Table 1 in Section 4
for the values of fair premia for EPSs considered in Examples 2.1 and 2.2. It
should be noted that an EPS should be designed in such a way that its initial
fair premium is null and this is clearly not satisfied in basic examples given
below (see, however, Section 4.2).

###### Example 2.1

Consider a buffer EPS with the nominal principal one million AUD ($1 mm), the
buffer threshold $l_{1}=-5\%$ with the protection rate $p_{2}=80\%$, the fee
threshold $g_{1}=10\%$ with the fee rate $f_{2}=80\%$, and maturity one year.
Its fair premium within the Black-Scholes setup of Section 4 amounts to $1,168
(see the buffer EPS No. 9 in Table 1) and, in three sample scenarios, the
buffer EPS is settled at time $T$ in the following way:

  * •

If the realised rate of return $R_{T}$ is above $10\%$, say $12\%$, then the
buyer pays $(12\%-10\%)\times 80\%\times\$1\ \mbox{\rm mm}=\$16$ k to the
provider.

  * •

If $R_{T}$ is between $-5\%$ and $10\%$, then there is no cash exchange
between the buyer and provider.

  * •

If $R_{T}$ is below $-5\%$, say $-8\%$, then the buyer receives
$(-8\%+5\%)\times 80\%\times\$1\ \mbox{\rm mm}=\$24$ k from the provider.

###### Example 2.2

Consider a floor EPS with the nominal principal $1 mm, the floor threshold
$l_{1}=-15\%$ with the protection rate $p_{1}=80\%$, the fee threshold
$g_{1}=10\%$ with the fee rate $f_{2}=80\%$, and maturity one year. Its fair
premium within the Black-Scholes setup equals $4,276 (see the floor EPS No. 13
in Table 1) and the floor EPS is settled at time $T$ as follows:

  * •

If the realised rate of return $R_{T}$ is above $10\%$, say $12\%$, then the
buyer pays $(12\%-10\%)\times 80\%\times\$1\ \mbox{\rm mm}=\$16$ k to the
provider.

  * •

If $R_{T}$ is between $0\%$ and $10\%$, then there is no cash exchange between
the buyer and provider.

  * •

If $R_{T}$ is between $-15\%$ and $0\%$, say $-8\%$, then the buyer gets
$8\%\times 80\%\times\$1\ \mbox{\rm mm}=\$64$ k from the provider.

  * •

If $R_{T}$ is below $-15\%$, say $-20\%$, then the buyer receives $15\%\times
80\%\times\$1\ \mbox{\rm mm}=\$120$ k from the provider. This is the maximum
amount that the buyer may receive, no matter how large is her actual loss
above $15\%$ cap.

###  2.4 Classification of Equity Protection Swaps

Let us make an attempt to classify Equity Protection Swaps according to
underlying portfolios. One can distinguish the following cases: (i) an index
EPS, (ii) a bespoke portfolio EPS, (iii) a cross-currency index EPS, and (iv)
a cross-currency bespoke portfolio EPS. A classification is important since
only for some instances of an EPS a robust (i.e., model-free) pricing and
static hedging can be developed, provided that the associated index and equity
options are liquidly traded.

Since EPSs are just a specific class of financial derivatives, they can be
analysed using well-established methods from the arbitrage pricing theory
based on the concept of continuous-time dynamic trading but then, obviously,
the issue of a model choice arises even if the theoretical market completeness
is postulated so that the replication argument can be used. In contrast, if
the market completeness is not assumed, then one needs to refer to less
effective pricing tools developed by academics to deal with incomplete models
but it should be acknowledged that these methods are less appreciated by the
finance industry. In addition, any stochastic model needs to be calibrated to
market data and, in practice, it is also periodically recalibrated in order to
mitigate the model risk.

Finally, one can also apply the most recent robust methodology where a wide
class of models is simultaneously covered but, once again, a large amount of
market data are required to obtain reliable pricing results with a narrow bid-
ask spread. To sum up, depending on a class of an EPS and availability of
traded options one needs to develop the most effective approach to obtain the
fair price interval and hedging strategies.

####  2.4.1 Index EPS

As expected, the easiest to handle is an index EPS where the process $S$
represents a particular stock index in the domestic economy, such as the
S&P/ASX 200 index, which measures the performance of the 200 largest index-
eligible stocks listed on the Australian Securities Exchange (ASX). The
related exchange-traded index options are cash-settled calls and puts of
European style so they can only be exercised on their expiry date. As will be
shown in Section 3, due to the fact that options on S&P/ASX 200 are actively
traded, the problem of pricing and static hedging of a generic EPS referencing
S&P/ASX 200 does not require to use any stochastic model for the index
dynamics. Also, it will be argued that the fair value of an index EPS can be
easily computed at any date in reference to market data for index options so
that an index EPS can be made puttable by the holder at its market value. In
principle, they can also be made callable by the provider but such a clause
would not suit the main goal of an EPS, which is to give protection to holders
of a superannuation account.

Let $\text{Call}_{\,t}(K,U)$ and $\text{Put}_{\,t}(K,U)$ denote the price at
time $t$ of a European call and put option written on the index $S$, with
strike $K\geq 0$ and maturity $U$. For future reference, we observe that the
following equalities hold, for any $-1\leq l\leq 0$,

$\displaystyle\big{(}l-R_{S}(0,T)\big{)}^{+}=\frac{(S_{0}(1+l)-S_{T})^{+}}{S_{0}}=\frac{1}{S_{0}}\,\text{Put}_{T}(S_{0}(1+l),T)$

and, similarly, for any nonnegative $g$,

$\displaystyle\big{(}R_{S}(0,T)-g\big{)}^{+}=\frac{(S_{T}-S_{0}(1+g))^{+}}{S_{0}}=\frac{1}{S_{0}}\,\text{Call}_{T}(S_{0}(1+g),T).$

Notice also that $S_{0}$ is the initial value of the index and the notional
principal of an index EPS is any quantity $N_{p}>0$ denominated in domestic
currency. Hence we may and do take $N_{p}=1$ since $N_{p}$ is independent of
the value of $S_{0}$ and our pricing and hedging approach through replication
is linear.

####  2.4.2 Bespoke Portfolio EPS

A bespoke portfolio EPS is referencing an arbitrary static portfolio composed
of investments in major companies traded on ASX. European options on
individual stocks are traded on ASX for around 20 top companies. Tailor-made
options of both European and American style on a larger selection of
Australian companies are offered on an over-the-counter basis by the major
banks: ANZ, CBA, NAB, and Westpac and thus equity options are readily
available. However, an EPS on a bespoke portfolio of Australian companies is
related to a basket option and thus its pricing and hedging is necessarily
relying on a multidimensional stochastic model for equities, which will
typically require to calibrate volatilities and correlations).

Let $S^{i}$ represent the market price of the $i$th domestic stock from the
reference portfolio so that $R^{i}(0,T):=(S^{i}_{T}-S^{i}_{0})/S^{i}_{0}$ is
the realised return on the $i$th domestic stock. Then the portfolio’s value
process equals $S_{t}:=\sum_{i=1}^{k}n_{i}S^{i}_{t}$ and thus the realised
return can be represented as follows

$R_{S}(0,T)=(S_{T}-S_{0})/S_{0}=\frac{\sum_{i=1}^{k}n_{i}S^{i}_{T}-\sum_{i=1}^{k}n_{i}S^{i}_{0}}{\sum_{i=1}^{k}n_{i}S^{i}_{0}}=\frac{\sum_{i=1}^{k}n_{i}R^{i}(0,T)S^{i}_{0}}{\sum_{i=1}^{k}N^{i}_{p}}$

where $n_{i},\,i=1,2,\dots,k$ is the number of shares of the $i$th stock held
at time 0 and thus the wealth invested at time 0 in the $i$th stock equals
$N^{i}_{p}:=n_{i}S^{i}_{0}$ and the notional principal equals
$N_{p}=\sum_{i=1}^{k}N^{i}_{p}$.

As in the case of an index EPS, it is convenient to assume, without loss of
generality, that the notional principal of an EPS equals $N_{p}=1$ so that
$\sum_{i=1}^{k}N^{i}_{p}=1$ where $N^{i}_{p}>0$ for every $i$. Equivalently,
we denote by $\pi^{i}_{0}:=n_{i}S^{i}_{0}/\sum_{j=1}^{k}n_{j}S^{j}_{0}$ a
(constant) proportion of the nominal principal invested at time 0 in shares of
the $i$th stock so that $\pi^{i}_{0}>0$ for every $i$ and
$\sum_{i=1}^{k}\pi^{i}_{0}=1$. Under this convention, for the realised return
on a bespoke portfolio we obtain

$R_{S}(0,T)=\sum_{i=1}^{k}N^{i}_{p}R^{i}(0,T)=\sum_{i=1}^{k}\pi^{i}_{0}R^{i}(0,T).$

Consequently, for any number $-1\leq l\leq 0$ we obtain the following
equalities

$\displaystyle(l-R_{S}(0,T))^{+}$
$\displaystyle=\frac{1}{S_{0}}\,(S_{0}(1+l)-S_{T})^{+}=\bigg{(}l-\sum_{i=1}^{k}\pi^{i}_{0}R^{i}(0,T)\bigg{)}^{+}=\bigg{(}(1+l)-\sum_{i=1}^{k}\pi^{i}_{0}\frac{S^{i}_{T}}{S^{i}_{0}}\bigg{)}^{+}$

where the last equality shows that in the case of a bespoke portfolio we deal
with a particular basket put option. Similarly, for any number $g\geq 0$ we
have that

$\displaystyle(R_{S}(0,T)-g)^{+}$
$\displaystyle=\frac{1}{S_{0}}\,(S_{T}-S_{0}(1+g))^{+}=\bigg{(}\sum_{i=1}^{k}\pi^{i}_{0}R^{i}(0,T)-g\bigg{)}^{+}=\bigg{(}\sum_{i=1}^{k}\pi^{i}_{0}\frac{S^{i}_{T}}{S^{i}_{0}}-(1+g)\bigg{)}^{+},$

which is the payoff of a basket call option. Since options on a bespoke
portfolio are not traded, it can be useful to observe that the following
inequalities hold

$\displaystyle\bigg{(}l-\sum_{i=1}^{k}\pi^{i}_{0}R^{i}(0,T)\bigg{)}^{+}\leq\sum_{i=1}^{k}\pi^{i}_{0}\big{(}l-R^{i}(0,T)\big{)}^{+}=\sum_{i=1}^{k}\frac{\pi^{i}_{0}}{S^{i}_{0}}\,\text{Put}_{T}(S^{i}_{0}(1+l),T)$

and

$\bigg{(}\sum_{i=1}^{k}\pi^{i}_{0}R^{i}(0,T)-g\bigg{)}^{+}\leq\sum_{i=1}^{k}\pi^{i}_{0}\big{(}R^{i}(0,T)-g\big{)}^{+}=\sum_{i=1}^{k}\frac{\pi^{i}_{0}}{S^{i}_{0}}\,\text{Call}_{T}(S^{i}_{0}(1+g),T).$

In general, one can use results pertaining to arbitrage-free pricing of basket
options of European style, for instance in a multidimensional version of the
Black-Scholes model.

####  2.4.3 Cross-Currency EPS

A cross-currency market index EPS references a combination of stock indices in
foreign securities markets and possibly also the domestic one. Similarly, a
cross-currency bespoke portfolio EPS is based on the realised return on a
bespoke portfolio of domestic and foreign equities. Understandably, the
pricing and hedging of a cross-currency EPS is a rather complex exercise but
it falls within the well-studied area of arbitrage pricing of cross-currency
derivatives. Since, by construction, all payoffs from an EPS are expressed in
domestic currency, an analysis of a cross-currency EPS requires stochastic
models not only for relevant indices (or stocks) but also applicable exchange
rates. It is expected that not only domestic and foreign equity options, but
also currency options or forward contracts can be used for hedging of cross-
currency EPSs although, typically, a static hedging portfolio is unlikely to
be available. For instance, exchange-traded options on U.S. indices and stocks
are available for investors in Australia but they are regular options
denominated in U.S. dollar and thus the exchange rate risk is not accounted
for in their payoff profile, which explains why they are not suitable for
construction of a static hedging portfolio for a cross-currency AUD/USD EPS.

For concreteness, we assume that U.S. equities market plays the role of a
generic foreign market. Then the exchange rate $Q$ is an
${\mathbb{F}}$-adapted stochastic process, which at any date $t$ is quoted as

$Q_{t}=\frac{\text{Number of units of the domestic currency (AUD)}}{\text{One
unit of the foreign currency (USD)}}.$

If $S^{i}$ represents the market price of the $i$th foreign stock from the
reference portfolio, then the realised return on $S^{i}$ expressed in domestic
currency is given by

$R^{Q,i}(0,T):=(Q_{T}S^{i}_{T}-Q_{0}S^{i}_{0})/Q_{0}S^{i}_{0}.$

Therefore, if a cross-currency EPS references a bespoke portfolio, which is
composed of domestic stocks (or indices) $S^{i}$ for $i=1,2,\dots,m$ and
foreign stocks (or indices) $S^{i}$ for $i=m+1,m+2,\dots,k$, then

$S_{t}=\sum_{i=1}^{m}n_{i}S^{i}_{t}+\sum_{i=m+1}^{k}n_{i}Q_{t}S^{i}_{t}$

and thus if we assume that $N_{p}=1$ so that $\sum_{i=1}^{k}N^{i}_{p}=1$ where
$N^{i}_{p}=n_{i}S^{i}_{0}$ for $i=1,2,\dots,m$ and
$N^{i}_{p}=n_{i}Q_{0}S^{i}_{0}$ for $i=m+1,m+2,\dots,k$, then

$R_{S}(0,T)=\sum_{i=1}^{m}N^{i}_{p}R^{i}(0,T)+\sum_{i=m+1}^{k}N^{i}_{p}R^{Q,i}(0,T)=\sum_{i=1}^{m}\pi^{i}_{0}R^{i}(0,T)+\sum_{i=m+1}^{k}\pi^{i}_{0}R^{Q,i}(0,T)$

where, as before, $\pi^{i}_{0}$ is a proportion of the nominal principal
invested at time 0 in shares of the $i$th stock.

Consequently, for any $l\leq 0$,

$\displaystyle(l-R_{T})^{+}$
$\displaystyle=\frac{1}{S_{0}}\,(S_{0}(1+l)-S_{T})^{+}=\bigg{(}l-\sum_{i=1}^{m}\pi^{i}_{0}R^{i}(0,T)-\sum_{i=m+1}^{k}\pi^{i}_{0}R^{Q,i}(0,T)\bigg{)}^{+}$
$\displaystyle=\bigg{(}(1+l)-\sum_{i=1}^{m}\pi^{i}_{0}\frac{S^{i}_{T}}{S^{i}_{0}}-\sum_{i=m+1}^{k}\pi^{i}_{0}\frac{Q_{T}S^{i}_{T}}{Q_{0}S^{i}_{0}}\bigg{)}^{+}$

and, for any $g\geq 0$,

$\displaystyle(R_{T}-g)^{+}$
$\displaystyle=\frac{1}{S_{0}}\,(S_{T}-S_{0}(1+g))^{+}=\bigg{(}\sum_{i=1}^{k}\pi^{i}_{0}R^{i}(0,T)+\sum_{i=m+1}^{k}\pi^{i}_{0}R^{Q,i}(0,T)-g\bigg{)}^{+}$
$\displaystyle=\bigg{(}\sum_{i=1}^{m}\pi^{i}_{0}\frac{S^{i}_{T}}{S^{i}_{0}}-\sum_{i=m+1}^{k}\pi^{i}_{0}\frac{Q_{T}S^{i}_{T}}{Q_{0}S^{i}_{0}}-(1+g)\bigg{)}^{+}.$

It is readily seen that extensions of these expressions to the case of several
foreign markets can be obtained. Although we do not elaborate in pricing of
hedging of cross-currency EPSs, it should be observed that this can be done in
a cross-currency multidimensional variant of the Black-Scholes model
complemented by a stochastic model for exchange rates, e.g., the classical
Garman-Kohlhagen model.

##  3 Hedging Strategies for Equity Protection Swaps

Although an EPS can be based on a virtually arbitrary portfolio of traded
domestic and foreign securities, we find it convenient to focus on an EPS
referencing a domestic stock index for which there exists a liquid market of
related exchange-traded options. The foregoing analysis formally covers also
some other EPSs provided that call and put options involved in a static
hedging strategy of Proposition 3.1 are actively traded either on organised
exchanges or OTC market. However, if an EPS involves several domestic indices
or a bespoke portfolio of domestic assets, then it represents a multi-asset
financial derivative and thus its static hedging using only index options and
single-name stock options cannot be achieved.

After representing the payoff of a generic EPS in terms of a combination of
European calls and puts, our next goal is to build a hedging portfolio for two
particular cases of an EPS and, subsequently, to extend it to a generic EPS.
Let us temporarily assume that the provider receives at time 0 from the buyer
of an EPS a negative or positive premium $c$ per one unit of the nominal
principal. According to our convention, the provider receives at time $T$ the
cash flow $\psi(R_{T})$ from the buyer.

Since the buyer’s goal is to use an EPS as a tool for the reduction of the
risk exposure associated with their investment in $S$, so the holder is
concerned with the net position $R_{T}-\psi(R_{T})$. In contrast, the provider
is not holding the reference portfolio and thus he is only concerned with
elimination of the risk associated with the cash flow $\psi(R_{T})$ through
replication (if feasible) of the opposite cash flow, that is, the payoff
$-\psi(R_{T})$ at time $T$. The price of an EPS computed by the provider
(seller) of an EPS will also be fair for its holder (buyer), under the
assumption that the provider and holder have access to the same set of traded
options and the bid-ask spread for traded options is disregarded, e.g., by
focusing on the mid-price.

We henceforth take the view of the provider who establishes at time 0 an
appropriate portfolio of plain-vanilla call and put options written on the
underlying portfolio $S$. We assume that the hedging portfolio, denoted as
$H$, costs $H_{0}$ at time 0 and yields a random payoff $H_{T}$ at maturity
date $T$. Unlike a RILA, the expected expenses of the holder’s portfolio are
not relevant for the provider since it is not exchanged at time 0 between the
two parties. Hence the management fee charge, which is considered when
studying RILAs and other similar variable annuities, is also immaterial
because the provider does not manage the holder’s portfolio. Note also that if
a static hedge of an EPS using traded European options is feasible, then it
suffices to independently investigate the hedging portfolios for the two legs
of an EPS and combine the respective hedges in order to obtain a complete
static hedge for the whole EPS. For simplicity of presentation, we henceforth
assume that the continuously compounded annualised risk-free rate equals $r$
although this assumption is not essential.

###### Definition 3.1

For an EPS introduced in Definition 2.1, the provider’s hedged cash flow at
time $T$ per one unit of the nominal principal $N_{p}$, denoted by $\mbox{\rm
CF}_{T}(c,H)$, equals

$\mbox{\rm CF}_{T}(c,H)=(c-H_{0})e^{rT}+H_{T}+\psi(R_{T})$

where the provider’s payoff $\psi(R_{T})$ from an EPS is given by (1). We say
that an EPS can be statically hedged by the provider if there exists a static
portfolio $\widehat{H}$ established at time 0 and an initial premium
$\widehat{c}\in\mathbb{R}$ such the equality $\mbox{\rm
CF}_{T}(\widehat{c},\widehat{H})=0$ holds almost surely. Then the number
$\widehat{c}$ is called the fair premium for an EPS per one unit of the
nominal principal.

###  3.1 Static Hedge for a Buffer Index EPS

In the next step, we establish the existence of a static hedge, and hence the
initial premium, for the most typical examples of an index EPS. Furthermore,
we analyse particular structures of an EPS for which the fair premium is null
and hence there is no cash flow at inception. In Sections 3.1, 3.2 and 3.3, we
assume that European options written on a reference index $S$ are liquidly
traded and we will show that a static hedge for an index EPS is feasible.
Therefore, the pricing and hedging results are independent of a stochastic
model for the portfolio’s value process $S$.

To justify this claim, it suffices to show that the terminal payoff of an EPS
can be expressed through a combination of European call and put options
written on $S$. We work here under the postulate that European options with
$S$ as the underlying asset are actively traded and we denote by
$\text{Call}_{\,t}(K,T)$ and $\text{Put}_{\,t}(K,T)$ denote the price at time
$t$ of a European call and put option written on $S$, with strike $K\geq 0$
and maturity $T$.

The provider’s cash flow $\mbox{\rm CF}_{T}(c,H)$ for the hedged buffer EPS
equals

$\displaystyle\mbox{\rm CF}_{T}(c,H)$
$\displaystyle=(c-H_{0})e^{rT}+H_{T}+\psi_{B}(R_{T})$
$\displaystyle=(c-H_{0})e^{rT}+H_{T}-p_{2}(l_{1}-R_{T})^{+}+f_{2}(R_{T}-g_{1})^{+}$
(3)
$\displaystyle=(c-H_{0})e^{rT}+H_{T}-\frac{p_{2}}{S_{0}}\,\text{Put}_{T}(S_{0}(1+l_{1}),T)+\frac{f_{2}}{S_{0}}\,\text{Call}_{T}(S_{0}(1+g_{1}),T)$

where to obtain the last equality, it suffices to observe (as in Section
2.4.1) that

$\displaystyle(l_{1}-R_{T})^{+}=\frac{(S_{0}(1+l_{1})-S_{T})^{+}}{S_{0}}=\frac{1}{S_{0}}\,\text{Put}_{T}(S_{0}(1+l_{1}),T)$

and, similarly,

$\displaystyle(R_{T}-g_{1})^{+}=\frac{(S_{T}-S_{0}(1+g_{1}))^{+}}{S_{0}}=\frac{1}{S_{0}}\,\text{Call}_{T}(S_{0}(1+g_{1}),T).$

From the above equations, it is readily seen that the buffer EPS can be
statically hedged by investing in European options written on $S$.
Specifically, we deduce from (3.1) that the provider should take the following
positions at time $0$:

  * •

Long $\frac{p_{2}}{S_{0}}$ units of a put with strike
$K^{l}_{1}:=S_{0}(1+l_{1})$,

  * •

Short $\frac{f_{2}}{S_{0}}$ units of a call with strike
$K^{g}_{1}:=S_{0}(1+g_{1})$,

since this portfolio, denoted as $\widehat{H}$, provides at time $T$ the
payoff

$\displaystyle\widehat{H}_{T}=\frac{p_{2}}{S_{0}}\,\text{Put}_{T}(K^{l}_{1},T)-\frac{f_{2}}{S_{0}}\,\text{Call}_{T}(K^{g}_{1},T)=p_{2}(l_{1}-R_{T})^{+}-f_{2}(R_{T}-g_{1})^{+}=-\psi_{B}(R_{T}),$

as was required. It is clear that its initial value equals

$\displaystyle\widehat{H}_{0}=\frac{p_{2}}{S_{0}}\,\text{Put}_{0}(K^{l}_{1},T)-\frac{f_{2}}{S_{0}}\,\text{Call}_{0}(K^{g}_{1},T)$

and for the above choice of the static hedge $\widehat{H}$ we have that
$\mbox{\rm
CF}_{T}(\widehat{c},\widehat{H})=(\widehat{c}-\widehat{H}_{0})e^{rT}=0$, which
shows that the fair premium for a buffer EPS with the nominal principal
$N_{p}=1$ satisfies $\widehat{c}=\widehat{H}_{0}$. Notice also that
portfolio’s wealth $\widehat{H}_{T}$ can be represented explicitly in terms of
the realised value of the index $S_{T}$ (see Figure 4)

$\displaystyle\widehat{H}_{T}=\frac{p_{2}}{S_{0}}\,(K^{l}_{1}-S_{T})\mathds{1}_{\\{S_{T}\in[0,K^{l}_{1}]\\}}-\frac{f_{2}}{S_{0}}\,(S_{T}-K^{g}_{1})\mathds{1}_{\\{S_{T}\in[K^{g}_{1},\infty[\\}}.$

portfolio’s wealth $\widehat{H}_{T}$realised index
$S_{T}$$p_{2}(1+l_{1})$$K_{1}^{l}$$S_{0}$$K_{1}^{g}$$-p_{2}/S_{0}$$-f_{2}/S_{0}$
Figure 4: Provider’s static hedge for a buffer EPS

###  3.2 Static Hedge for a Floor Index EPS

An analysis of a static hedge for a floor EPS is analogous to a buffer EPS
with some modifications of the payoff. The provider’s cash flow $\mbox{\rm
CF}_{T}(c,H)$ for the hedged floor EPS equals, per one unit of the nominal
principal,

$\displaystyle\mbox{\rm CF}_{T}(c,H)=(c-H_{0})e^{rT}+H_{T}+\psi_{F}(R_{T})$
$\displaystyle=(c-H_{0})e^{rT}+H_{T}-p_{1}(-R_{T})^{+}+p_{1}(l_{1}-R_{T})^{+}+f_{2}(R_{T}-g_{1})^{+}.$

We claim that the provider of the floor EPS can statically hedge a floor EPS
by investing at time 0 in three European options on $S$:

  * •

Short $\frac{p_{1}}{S_{0}}$ units of a put with strike
$K^{l}_{1}:=S_{0}(1+l_{1})$,

  * •

Long $\frac{p_{1}}{S_{0}}$ units of a put with strike $S_{0}$,

  * •

Short $\frac{f_{2}}{S_{0}}$ units of a call with strike
$K^{g}_{1}:=S_{0}(1+g_{1})$.

This portfolio $\widehat{H}$ of European options yields at time $T$ the payoff
given by

$\displaystyle\widehat{H}_{T}$
$\displaystyle=-\frac{p_{1}}{S_{0}}\,\text{Put}_{T}(K^{l}_{1},T)+\frac{p_{1}}{S_{0}}\,\text{Put}_{T}(S_{0},T)-\frac{f_{2}}{S_{0}}\,\text{Call}_{T}(K^{g}_{1},T)$
$\displaystyle=-p_{1}(l_{1}-R_{T})^{+}+p_{1}(-R_{T})^{+}-f_{2}(R_{T}-g_{1})^{+}$
$\displaystyle=-p_{1}l_{1}\mathds{1}_{\\{R_{T}\in[-1,l_{1}]\\}}+p_{1}(-R_{T})^{+}\mathds{1}_{\\{R_{T}\in(l_{1},0]\\}}-f_{2}(R_{T}-g_{1})^{+}=-\psi_{F}(R_{T})$

and thus the fair premium for the floor EPS with the nominal principal
$N_{p}=1$ equals

$\widehat{c}=\widehat{H}_{0}=-\frac{p_{1}}{S_{0}}\,\text{Put}_{0}(K^{l}_{1},T)+\frac{p_{1}}{S_{0}}\,\text{Put}_{0}(S_{0},T)-\frac{f_{2}}{S_{0}}\,\text{Call}_{0}(K^{g}_{1},T).$

It can also be checked that $\widehat{H}_{T}$ can be represented as follows
(see Figure 5)

$\displaystyle\widehat{H}_{T}=\frac{p_{1}}{S_{0}}\,(S_{0}-K^{l}_{1})\mathds{1}_{\\{S_{T}\in[0,K^{l}_{1}]\\}}+\frac{p_{1}}{S_{0}}(S_{0}-S_{T})\mathds{1}_{\\{S_{T}\in[K^{l}_{1},S_{0}]\\}}-\frac{f_{2}}{S_{0}}\,(S_{T}-K^{g}_{1})\mathds{1}_{\\{S_{T}\in[K^{g}_{1},\infty[\\}}$

where $\frac{p_{1}}{S_{0}}\,(S_{0}-K^{l}_{1})=-p_{1}l_{1}$ is the (positive)
difference between the prices at time 0 of put options with strikes $S_{0}$
and $K^{l}_{1}$.

portfolio’s wealth $\widehat{H}_{T}$realised index
$S_{T}$$K_{1}^{l}$$-p_{1}l_{1}$$S_{0}$$K_{1}^{g}$$-p_{1}/S_{0}$$-f_{2}/S_{0}$
Figure 5: Provider’s static hedge for a floor EPS

###  3.3 Static Hedge for a Generic Index EPS

We do not present an explicit solution for the provider’s static hedge of an
EPS with the buffer-floor protection leg and buffer fee leg since it is
covered by a result for a generic EPS of Definition 2.1 (see Example 3.1).
Generally speaking, if the parameters $\alpha_{k}$ and $\beta_{k}$ in
Definition 2.1 are given, then the question of fair valuation of an EPS
reduces to the computation of the fair premium $\widehat{c}=\widehat{H}_{0}$.
However, a more interesting question arises if it is not assumed a priori that
$\widehat{c}=0$ but at least one of parameters $\alpha_{k}$ or $\beta_{k}$ is
left unspecified. Then the issue of fair pricing hinges on finding the value
of that parameter (typically, the provider’s fee rate) for which the initial
value $\widehat{H}_{0}$ is null so that the fair premium $\widehat{c}$
vanishes as well. By definition, a static portfolio in European put and call
options is a vector
$h=(a_{0},a_{1},a_{2},\dots,a_{n},b_{0},b_{1},b_{2},\dots,b_{m})\in\mathbb{R}^{n+m}$
and the value process $H_{t},\,t\in[0,T]$, is given by the equality, for every
$t\in[0,T]$,

$H_{t}=\sum_{i=0}^{n}a_{i}\,\text{\rm
Put}_{t}(K^{l}_{i},T)+\sum_{j=0}^{m}b_{j}\,\text{\rm Call}_{t}(K^{g}_{j},T).$

The above arguments lead to the following hedging result for a generic index
EPS with virtually arbitrary structure of the two legs in which the fair
premium is computed in relation to a family of call and put options on $S$
with maturity $T$ and various strikes. Recall that $l_{0}=g_{0}=0$. For
convenience, we also adopt the convention that $f_{0}=p_{0}=0$.

###### Proposition 3.1

The static hedge $(\widehat{c},\widehat{H})$ composed of call and put options
for an index EPS with the nominal principal $N_{p}=1$ is obtained by setting
$a_{j}=(p_{i+1}-p_{i})/S_{0}$ and $b_{j}=-(f_{j+1}-f_{j})/S_{0}$ so that

$\widehat{H}_{T}=\sum_{i=0}^{n}\frac{p_{i+1}-p_{i}}{S_{0}}\,\text{\rm
Put}_{T}(K^{l}_{i},T)-\sum_{j=0}^{m}\frac{f_{j+1}-f_{j}}{S_{0}}\,\text{\rm
Call}_{T}(K^{g}_{j},T)=-\psi(R_{T})$ (4)

where $K^{l}_{i}=S_{0}(1+l_{i})$ and $K^{g}_{j}=S_{0}(1+g_{j})$ for every
$i=0,1,\dots,n$ and $j=0,1,\dots,m$ and the fair premium for an EPS satisfies
$\widehat{c}=\widehat{H}_{0}$, that is,

$\widehat{c}=\sum_{i=0}^{n}\frac{p_{i+1}-p_{i}}{S_{0}}\,\text{\rm
Put}_{0}(K^{l}_{i},T)-\sum_{j=0}^{m}\frac{f_{j+1}-f_{j}}{S_{0}}\,\text{\rm
Call}_{0}(K^{g}_{j},T).$ (5)

Proof.  Observe that, using the ideas introduced in Lemma 2.1, we can split an
EPS into two parts, the protection and fee legs, and then compute a static
hedge for each of them independently. Recall from Lemma 2.1 that
$\psi(R_{T})=\psi^{f}(R_{T})-\psi^{p}(R_{T})$ where for every $i=0,1,\dots,n$
and $j=0,1,\dots,m$

$\displaystyle\psi^{p}(R_{T})$
$\displaystyle=\Big{(}\sum_{k=0}^{i-1}p_{k+1}\Delta
l_{k}+p_{i+1}(R_{T}-l_{i})\Big{)}\mathds{1}_{\\{R\in(l_{i+1},l_{i})\\}},$
$\displaystyle\psi^{f}(R_{T})$
$\displaystyle=\Big{(}\sum_{k=0}^{j-1}f_{k+1}\Delta
g_{k}+f_{j+1}(R_{T}-g_{j})\Big{)}\mathds{1}_{\\{R\in(g_{j},g_{j+1})\\}}.$

According to Definition 3.1, it suffices to find a particular static portfolio
of options, denoted as $\widehat{h}$, such that
$\widehat{H}_{T}+\psi(R_{T})=0$ and set $\widehat{c}=\widehat{H}_{0}$ so that
$\mbox{\rm CF}_{T}(\widehat{c},\widehat{H})=0$.

We first consider the fee leg of an EPS. Observe that either $f_{1}>0$ or
$f_{1}=0$ and $f_{2}>0$. We now start with the first nonzero fee rate
$f_{j+1}$ so that $f_{j+1}>f_{j}=0$ and thus either $f_{j+1}=f_{1}$ or
$f_{j+1}=f_{2}$. In order to hedge the fee leg, the provider needs to short
$\frac{f_{j+1}}{S_{0}}$ units of European call option with strike
$K^{g}_{j}=S_{0}(1+g_{j})$ at time 0. Then, if the inequality $f_{k+1}>f_{k}$
holds for some $k=j+1,j+2,\dots,m$, the provider needs to short further
$\frac{f_{k+1}-f_{k}}{S_{0}}$ units of a European call option with strike
$K^{g}_{k}=S_{0}(1+g_{k})$. If, on the contrary, we have that $f_{k+1}<f_{k}$,
then the provider needs to long $\frac{f_{k}-f_{k+1}}{S_{0}}$ units of a
European call option with strike $K^{g}_{k}=S_{0}(1+g_{k})$. This portfolio of
call options constitutes the first part of a static hedge.

It now remains to find a hedge for the protection leg of an EPS. Notice that
we have that either $p_{1}>0$ or $p_{1}=0$ and $p_{2}>0$. We start with the
first nonzero protection rate $p_{i+1}$ so that $p_{i+1}>p_{i}=0$ where, by
convention, $p_{0}=0$ so that either $p_{i+1}=p_{1}$ or $p_{i+1}=p_{2}$. In
order to hedge the protection leg of an EPS, the provider needs to long
$\frac{p_{i+1}}{S_{0}}$ units of European put option with strike
$K^{l}_{i}=S_{0}(1+l_{i})$ at the inception date of an EPS. Then, for any $k$
such that $p_{k+1}>p_{k}$ with $k=i+1,i+2,\dots,n$, the provider should long
$\frac{p_{k+1}-p_{k}}{S_{0}}$ units of a European put option with strike
$K^{l}_{k}=S_{0}(1+l_{k})$. If, however, the inequality $p_{k+1}<p_{k}$ holds,
then the provider need to short $\frac{p_{k}-p_{k+1}}{S_{0}}$ units of a
European put option with strike $K^{l}_{k}=S_{0}(1+l_{k})$. These put options
give the static hedge for the fee leg.

The equality $\widehat{c}=\widehat{H}_{0}$ and hence equality (5) for the fair
premium $\widehat{c}$ are now immediate consequences of Definition 3.1. $\Box$

In Proposition 3.1 we offer a solution to the hedging problem for an EPS but
the issue of uniqueness of a static hedge for an EPS is not examined and, as
usual when dealing with a combination of European options, it may fail to
hold, in general. However, the uniqueness of the fair premium for an EPS
relative to a set of market prices of traded options is always true since its
non-uniqueness would be inconsistent with the principle of arbitrage-free
pricing. In the context of Proposition 3.1, European options written on $S$
maturing at $T$ are assumed to play the role of primary traded assets and an
EPS is a derivative security. Hence the pricing formula (5) is robust, meaning
that it does not depend on a choice of a stochastic model for $S$, provided
that options prices are observed. However, Proposition 3.1 is also valid in
the framework of any arbitrage-free model for assets $B$ and $S$ where $B$
represents the money market account and $S$ is assumed to be a primary traded
asset so that European options are derivative securities and their arbitrage-
free prices are computed, not observed on the market. Finally, it is clear the
same arguments can be applied the case of an EPS written on a bespoke
portfolio of domestic and foreign indices or stocks although, for technical
reasons, the problem may become more difficult to solve.

###### Example 3.1

As a practical illustration of the static hedge $(\widehat{c},\widehat{H})$
obtained in Proposition 3.1, let us consider an index EPS with buffer-floor
protection leg and buffer-cap fee leg, as introduced in Definitions 2.3 and
2.4. Then the modified realised return equals
$\psi(R)=\psi^{p,4}(R)-\psi^{f,4}(R)$ where

$\displaystyle\psi^{p,4}(R)=p_{2}(R-l_{1})\mathds{1}_{\\{R\in[l_{2},l_{1})\\}}+p_{2}(l_{2}-l_{1})\mathds{1}_{\\{R\in[-1,l_{2})\\}}$

and

$\displaystyle\psi^{f,4}(R)=f_{2}(R-g_{1})\mathds{1}_{\\{R\in(g_{1},g_{2}]\\}}+f_{2}(g_{2}-g_{1})\mathds{1}_{\\{R\in(g_{2},\infty)\\}}.$

adjusted realised return $\psi_{B}(R_{T})$realised return
$R_{T}$$-1$$l_{1}$$l_{2}$$g_{1}$$g_{2}$$p_{1}=0$$p_{2}$$p_{3}=0$$f_{1}=0$$f_{2}$$f_{3}=0$
Figure 6: Provider’s adjusted return for a buffer-floor/buffer-cap EPS

Since $f_{1}=f_{3}=0,f_{2}\in(0,1]$ and $p_{1}=p_{3}=0,p_{2}\in(0,1]$ a direct
application of equality (4) with $n=m=2$ yields

$\displaystyle\widehat{H}_{T}$
$\displaystyle=\frac{p_{2}}{S_{0}}\,\Big{(}\text{\rm
Put}_{T}(K^{l}_{1},T)-\text{\rm
Put}_{T}(K^{l}_{2},T)\Big{)}-\frac{f_{2}}{S_{0}}\Big{(}\text{\rm
Call}_{T}(K^{g}_{1},T)-\text{\rm Call}_{T}(K^{g}_{2},T)\Big{)}$

where $K^{l}_{i}=S_{0}(1+l_{i})<S_{0}$ and $K^{l}_{i}=S_{0}(1+g_{i})>S_{0}$
for $i=1,2$ and, obviously, $\widehat{c}=\widehat{H}_{0}$.

portfolio’s wealth $\widehat{H}_{T}$realised index
$S_{T}$$K^{l}_{2}$$p_{2}(l_{1}-l_{2})$$K^{l}_{1}$$S_{0}$$K_{1}^{g}$$K_{2}^{g}$$-p_{2}/S_{0}$$-f_{2}/S_{0}$
Figure 7: Provider’s static hedge for a buffer-floor/buffer-cap EPS

##  4 Forward Performance Testing of Buffer and Floor EPSs

To provide the reader with a preliminary insight into the properties of an
EPS, we first report results of forward performance tests for some classes of
EPSs. For simplicity and concreteness, we employ the Black-Scholes options
pricing formula to compute the fair premium for an EPS, as specified through
its static hedging strategy of Definition 3.1.

As in the classical Black-Scholes model, we assume that the dynamics of the
process $S$ under the martingale probability measure ${\mathbb{Q}}$ are given
by the SDE

$dS_{t}=(r-\kappa)S_{t}\,dt+\sigma S_{t}\,dW_{t},\quad S_{0}>0,$ (6)

where $W$ is a one-dimensional standard Brownian motion, $\sigma>0$ is a
constant volatility parameter, $r$ is a constant risk-free interest rate, and
$\kappa$ is a constant dividend yield. Obviously, this also means that the
process $Z_{t}=S_{t}/S_{0}$ satisfies

$dZ_{t}=(r-\kappa)Z_{t}\,dt+\sigma Z_{t}\,dW_{t},\quad Z_{0}=1.$ (7)

Under the postulated dynamics of the portfolio’s value process, we have the
following well-known result from [5], which gives the arbitrage-free prices of
European call and put options on $S$ with a fixed strike $K>0$ and maturity
date $T$.

###### Theorem 4.1

The arbitrage-free price of a European call option with the payoff
$(S_{T}-K)^{+}$ at maturity date $T$ equals, for every $t\in[0,T)$,

$C(S_{t},t)=e^{-\kappa(T-t)}S_{t}N\big{(}d_{+}(S_{t},T-t)\big{)}-e^{-r(T-t)}KN\big{(}d_{-}(S_{t},T-t)\big{)}$

where

$d_{\pm}(S_{t},T-t)=\frac{1}{\sigma\sqrt{T-t}}\bigg{[}\ln\frac{S_{t}}{K}+\bigg{(}r-\kappa\pm\frac{1}{2}\sigma^{2}\bigg{)}(T-t)\bigg{]}$

and $N$ is the cumulative density function of the standard normal
distribution. Similarly, the arbitrage-free price of a European put option
with the payoff $(K-S_{T})^{+}$ at maturity date $T$ is given by, for every
$t\in[0,T)$,

$P(S_{t},t)=e^{-r(T-t)}KN\big{(}-d_{-}(S_{t},T-t)\big{)}-e^{-\kappa(T-t)}S_{t}N\big{(}-d_{+}(S_{t},T-t)\big{)}.$

By combining Proposition 3.1 with Theorem 4.1, we immediately obtain the
following result.

###### Proposition 4.1

The arbitrage-free price at time 0 in the Black-Scholes model of a generic
index EPS with the nominal principal $N_{p}=1$ equals

$\displaystyle\widehat{c}$
$\displaystyle=\sum_{i=0}^{n}(p_{i+1}-p_{i})\Big{(}e^{-rT}(1+l_{i})N\big{(}-h_{-}(l_{i},T)\big{)}-e^{-\kappa
T}N\big{(}-h_{+}(l_{i},T)\big{)}\Big{)}$
$\displaystyle-\sum_{j=0}^{m}(f_{j+1}-f_{j})\,\Big{(}e^{-\kappa
T}N\big{(}h_{+}(g_{j},T)\big{)}-e^{-rT}(1+g_{j})N\big{(}h_{-}(g_{j},T)\big{)}\Big{)}$

where we denote, for every $x>-1$,

$h_{\pm}(x,T)=\frac{1}{\sigma\sqrt{T}}\bigg{[}-\ln(1+x)+\bigg{(}r-\kappa\pm\frac{1}{2}\sigma^{2}\bigg{)}T\bigg{]}.$

Notice that the price $\widehat{c}$ of an EPS does not depend on the initial
value $S_{0}$ of the underlying portfolio (e.g., a reference index). This
remarkable property was expected since the arbitrage-free price of an EPS
should depend on the interest rate $r$, the dividend yield $\kappa$, the
notional principal $N_{p}$ (here chosen to be equal to 1) and the volatility
$\sigma$ of the random return rate $R_{T}$ on portfolio $S$, but it should be
independent on the actual size of a reference portfolio $S$. This is due to
the fact that the future returns on investor’s holdings in a portfolio $S$
coincide with the future returns on the reference portfolio $S$ (e.g., the S&P
500 index). Of course, to obtain the pricing formula for an EPS, the dynamics
of the process $S$ was specified under the martingale probability measure
${\mathbb{Q}}$ but the real-world benefits from holding an EPS will be
examined in Section 5 through backtesting under the statistical probability
$\mathbb{P}$.

We stress that the Black-Scholes prices of European options are only used here
for illustrative purposes. It is well known that the Black-Scholes model has
some limitations and it is not fully consistent with the real-world options
market. In Section 4, we use this model in order to compare the theoretical
fair premia for EPSs with different parameters but in Section 5 we refer
instead to the market data for traded European options to obtain empirical
pricing results and conclusions in regard of the price dependence on a
structure of an EPS. We also give a preliminary analysis of sensitivities of
the price of an EPS with respect to the underlying asset $S$ and model
parameters.

Needless to say, the pricing of an EPS is dynamic and thus the arbitrage-free
price of a fair EPS at any date after its inception date is usually nonzero
and with random values that can be either positive or negative for each party.
It is thus natural to ask the following question: suppose that a holder
decided to enter into a one-year EPS with inception date 0: what should be the
arbitrage-free price at time 0 of an option to extend the EPS with the same
covenants for the second year at no additional cost at time $T=1$? More
generally, one can examine an option to extend an EPS at time $T=1$ for the
next $n$ years at no additional cost at time $T=1$. It is rather clear that
the answer to the question of fair valuation of options to extend an EPS
hinges on arbitrage-free pricing of a particular compound option on $S$ in
either the domestic market or a cross-currency framework so the usual pricing
techniques based on dynamic hedging can be applied.

###  4.1 Theoretical Black-Scholes Pricing of an EPS

We first compute the fair premia for typical instances of an EPS from the
perspective of their provider whose aim is to offer an index EPS to members of
super funds. As was explained, in practice the pricing of an EPS should rely
on observed market prices of traded options but, for clarity of presentation,
we assume in Section 4 that arbitrage-free option prices are obtained using
Theorem 4.1. It is well known that the Black-Scholes model does not yield
option prices that perfectly match the market data across various strikes and
maturities. Nevertheless, it is widely accepted by practitioners for market
quotes for options via their implied volatilities and is largely sufficient
for illustration of the model-free pricing method established in Proposition
3.1. Regarding the parameters of the Black-Scholes dynamics, we assume
throughout that the interest rate $r=1.5\%$, the dividend yield $\kappa=0$
(consistently with the market convention for stock indices), and the
volatility parameter $\sigma=20\%$ (as a proxy for the one-year volatility
index).

Table 1: Theoretical fair premia of buffer and floor equity protection swaps No. | EPS class | $T$ | $l_{1}$ | $g_{1}$ | $p_{1}$ | $p_{2}$ | $f_{1}$ | $f_{2}$ | Fair premium
---|---|---|---|---|---|---|---|---|---
[1] | Buffer | 1 year | -5% | 5% | 0 | 0.5 | 0 | 0.5 | -0.007890
[2] | Buffer | 1 year | -5% | 5% | 0 | 0.8 | 0 | 0.5 | 0.006873
[3] | Buffer | 1 year | -5% | 5% | 0 | 0.8 | 0 | 0.8 | -0.012623
[4] | Buffer | 1 year | -5% | 10% | 0 | 0.5 | 0 | 0.5 | 0.000730
[5] | Buffer | 1 year | -5% | 10% | 0 | 0.6 | 0 | 0.5 | 0.005651
[6] | Buffer | 1 year | -5% | 10% | 0 | 0.7 | 0 | 0.5 | 0.010572
[7] | Buffer | 1 year | -5% | 10% | 0 | 0.8 | 0 | 0.5 | 0.015493
[8] | Buffer | 1 year | -5% | 10% | 0 | 0.9 | 0 | 0.5 | 0.020414
[9] | Buffer | 1 year | -5% | 10% | 0 | 0.8 | 0 | 0.8 | 0.001168
[10] | Buffer | 1 year | -10% | 10% | 0 | 0.5 | 0 | 0.5 | -0.008082
[11] | Buffer | 1 year | -10% | 10% | 0 | 0.8 | 0 | 0.5 | 0.001393
[12] | Buffer | 1 year | -10% | 10% | 0 | 0.8 | 0 | 0.8 | -0.012931
[13] | Buffer | 2 year | -5% | 5% | 0 | 0.8 | 0 | 0.5 | 0.006601
[14] | Buffer | 2 year | -5% | 10% | 0 | 0.8 | 0 | 0.5 | 0.015960
[15] | Buffer | 2 year | -10% | 10% | 0 | 0.8 | 0 | 0.5 | 0.000237
[1] | Floor | 1 year | -5% | 5% | 0.5 | 0 | 0 | 0.5 | -0.021180
[2] | Floor | 1 year | -5% | 5% | 0.8 | 0 | 0 | 0.5 | -0.014391
[3] | Floor | 1 year | -5% | 10% | 0.5 | 0 | 0 | 0.5 | -0.012560
[4] | Floor | 1 year | -5% | 10% | 0.8 | 0 | 0 | 0.5 | -0.005771
[5] | Floor | 1 year | -5% | 10% | 0.8 | 0 | 0 | 0.8 | -0.049895
[6] | Floor | 1 year | -10% | 10% | 0.5 | 0 | 0 | 0.5 | -0.003748
[7] | Floor | 1 year | -10% | 10% | 0.8 | 0 | 0 | 0.5 | 0.008328
[8] | Floor | 1 year | -15% | 10% | 0.5 | 0 | 0 | 0.5 | 0.002672
[9] | Floor | 1 year | -15% | 10% | 0.6 | 0 | 0 | 0.5 | 0.007982
[10] | Floor | 1 year | -15% | 10% | 0.7 | 0 | 0 | 0.5 | 0.013291
[11] | Floor | 1 year | -15% | 10% | 0.8 | 0 | 0 | 0.5 | 0.018601
[12] | Floor | 1 year | -15% | 10% | 0.9 | 0 | 0 | 0.5 | 0.023910
[13] | Floor | 1 year | -15% | 10% | 0.8 | 0 | 0 | 0.8 | 0.004276
[14] | Floor | 2 year | -5% | 5% | 0.8 | 0 | 0 | 0.5 | -0.033581
[15] | Floor | 2 year | -5% | 10% | 0.8 | 0 | 0 | 0.5 | -0.024222
[16] | Floor | 2 year | -10% | 10% | 0.8 | 0 | 0 | 0.5 | -0.008500
[17] | Floor | 2 year | -15% | 10% | 0.8 | 0 | 0 | 0.5 | 0.004358
[18] | Floor | 2 year | -15% | 10% | 0.8 | 0 | 0 | 0.8 | -0.021315

In Table 1, we report the fair premia $\widehat{c}$ for selected buffer and
floor EPSs. Recall from Proposition 3.1 that the fair premium satisfies
$\widehat{c}=\widehat{H}_{0}$ and thus its value can be found using the Black-
Scholes formula, as was shown in Proposition 4.1. We give the fair premia for
fifteen buffer EPSs and eighteen floor EPSs with different sets of parameters.

Let us analyse some basic features of hedging costs for an EPS. First, we can
conclude that for a buffer EPS, the maturity of either one or two years does
not make a big difference on the fair premium since its static hedge involves
only two options with opposite directions (a long put and a short call). In
contrast, the fair premium for a floor EPS is markedly different when the
maturity increases. Thus for an EPS with same types of options and different
directions, the maturity should not make a big influence on fair premium with
all else variables equal. We need to mention that, similar to RILA examined in
[18], we find it natural to assume that one-year or two-year EPSs should be
offered by a typical provider and some providers may also offer contracts with
longer maturities.

Recall from Definition 2.5 of the buffer EPS that $l_{1}$ is the level of
negative returns that triggers protection and $g_{1}$ is the level of positive
returns such that the provider begins to collect fees. From rows 2, 7, 11 in
the top half of Table 1 we can see that, with other variables being fixed, if
either $l_{1}$ or $g_{1}$ increases, then the fair premium grows. According to
Definition 2.6, the parameter $l_{1}$ in the floor EPS is the level of
negative returns such that the provider does not offer any additional
compensation beyond that level. As can be seen from cases 4, 7, 11 in the
bottom half of Table 1 the fair premium for the floor EPS increases when the
level $l_{1}$ decreases, as was expected.

Recall that the parameters $p_{2}$ and $f_{2}$ of the buffer EPS represent the
provider’s participation rates in portfolio’s losses and gains, respectively.
Obviously, the EPS buyer would appreciate if the provider bears more losses
and is entitled to less gains. This explains why the buyer is required to pay
a higher fair premium when the protection rate $p_{2}$ is increased, as can be
seen from the top half of Table 2. Similarly, the buyer still should pay a
higher fair premium when the fee rate $f_{2}$ is lower, as shown in rows 2, 3,
7, 9, 11, and 12 in the buffer EPS of Table 1.

Analogous properties can be observed for the floor EPS, The fair premium is
higher when the protection rate $p_{1}$ is increased, as can be seen in rows
8-12 in the bottom half of Table 3, and is lower when the fee rate $f_{2}$ is
increased, as can be observed in rows 11, 13, 17, 18 in the bottom half of
Table 3. We stress that all these properties (except for the one regarding
maturity) can be formally established by considering the dependence of the
payoff profile of an EPS on its parameters and hence they are universally
true.

Table 2: Fair premium dependence on parameters $p_{1}$ (floor) and $p_{2}$ (buffer) No. | EPS class | $T$ | $l_{1}$ | $g_{1}$ | $p_{1}$ | $p_{2}$ | $f_{1}$ | $f_{2}$ | Fair premium
---|---|---|---|---|---|---|---|---|---
[4] | Buffer | 1 year | -5% | 10% | 0 | 0.5 | 0 | 0.5 | 0.000730
[5] | Buffer | 1 year | -5% | 10% | 0 | 0.6 | 0 | 0.5 | 0.005651
[6] | Buffer | 1 year | -5% | 10% | 0 | 0.7 | 0 | 0.5 | 0.010572
[7] | Buffer | 1 year | -5% | 10% | 0 | 0.8 | 0 | 0.5 | 0.015493
[8] | Buffer | 1 year | -5% | 10% | 0 | 0.9 | 0 | 0.5 | 0.020414
[8] | Floor | 1 year | -15% | 10% | 0.5 | 0 | 0 | 0.5 | 0.002672
[9] | Floor | 1 year | -15% | 10% | 0.6 | 0 | 0 | 0.5 | 0.007982
[10] | Floor | 1 year | -15% | 10% | 0.7 | 0 | 0 | 0.5 | 0.013291
[11] | Floor | 1 year | -15% | 10% | 0.8 | 0 | 0 | 0.5 | 0.018601
[12] | Floor | 1 year | -15% | 10% | 0.9 | 0 | 0 | 0.5 | 0.023910
Table 3: Fair premium dependence on fee rate $f_{2}$ No. | EPS class | $T$ | $l_{1}$ | $g_{1}$ | $p_{1}$ | $p_{2}$ | $f_{1}$ | $f_{2}$ | Fair premium
---|---|---|---|---|---|---|---|---|---
[2] | Buffer | 1 year | -5% | 5% | 0 | 0.8 | 0 | 0.5 | 0.006873
[3] | Buffer | 1 year | -5% | 5% | 0 | 0.8 | 0 | 0.8 | -0.012623
[7] | Buffer | 1 year | -5% | 10% | 0 | 0.8 | 0 | 0.5 | 0.015493
[9] | Buffer | 1 year | -5% | 10% | 0 | 0.8 | 0 | 0.8 | 0.001168
[11] | Buffer | 1 year | -10% | 10% | 0 | 0.8 | 0 | 0.5 | 0.001393
[12] | Buffer | 1 year | -10% | 10% | 0 | 0.8 | 0 | 0.8 | -0.012931
[11] | Floor | 1 year | -15% | 10% | 0.8 | 0 | 0 | 0.5 | 0.018601
[13] | Floor | 1 year | -15% | 10% | 0.8 | 0 | 0 | 0.8 | 0.004276
[17] | Floor | 2 year | -15% | 10% | 0.8 | 0 | 0 | 0.5 | 0.004358
[18] | Floor | 2 year | -15% | 10% | 0.8 | 0 | 0 | 0.8 | -0.021315

When comparing buffer EPSs with floor EPSs, we observe that the hedging costs
for buffer EPSs are higher than for floor EPSs. In order to explain this
feature, we focus on their respective structures and we note that the fee legs
for the buffer EPS and floor EPS shown in Table 1 are identical but their
protection legs differ. From the profile of cash flow function for a buffer
EPS (Figure 2) and a floor EPS (Figure 3), we can see that in a floor EPS the
protection payment for large losses is capped, while it is unlimited in a
buffer EPS. Hence the protection provided by a buffer EPS is more effective
and, consequently, the hedging cost (i.e., the fair premium) for a buffer EPS
is typically higher than for a floor EPS with identical fee leg.

###  4.2 Numerical Study of a Fair EPS

In order to make EPS products more attractive to holders of superannuation
accounts, we propose to set the fair premium to zero so that the buyer of an
EPS is not required to pay an initial premium to the provider. Furthermore,
since investors usually pay more attention to limit their losses, rather than
to maximise their gains, we will set the protection rate together with the
null fair premium, and then attempt to find a proper fee rate $f_{2}$. In
Table 4, we give the fee rate $f_{2}$ when all other parameters are fixed and
the fair premium is null. We continue studying some examples of EPSs from the
Table 4.

Table 4: Buffer and floor EPSs with null fair premium No. | EPS class | $T$ | $l_{1}$ | $g_{1}$ | $p_{1}$ | $p_{2}$ | $f_{1}$ | $f_{2}$
---|---|---|---|---|---|---|---|---
[1] | Buffer | 1 year | -5% | 5% | 0 | 0.5 | 0 | 0.38
[2] | Buffer | 1 year | -5% | 5% | 0 | 0.8 | 0 | 0.61
[3] | Buffer | 1 year | -5% | 10% | 0 | 0.5 | 0 | 0.52
[4] | Buffer | 1 year | -5% | 10% | 0 | 0.6 | 0 | 0.62
[5] | Buffer | 1 year | -5% | 10% | 0 | 0.7 | 0 | 0.72
[6] | Buffer | 1 year | -5% | 10% | 0 | 0.8 | 0 | 0.82
[7] | Buffer | 1 year | -5% | 10% | 0 | 0.9 | 0 | 0.93
[8] | Buffer | 1 year | -10% | 10% | 0 | 0.5 | 0 | 0.33
[10] | Buffer | 1 year | -10% | 10% | 0 | 0.8 | 0 | 0.53
[11] | Buffer | 2 year | -5% | 5% | 0 | 0.8 | 0 | 0.56
[12] | Buffer | 2 year | -5% | 10% | 0 | 0.8 | 0 | 0.69
[13] | Buffer | 2 year | -10% | 10% | 0 | 0.8 | 0 | 0.50
[1] | Floor | 1 year | -5% | 5% | 0.5 | 0 | 0 | 0.17
[2] | Floor | 1 year | -5% | 5% | 0.8 | 0 | 0 | 0.28
[3] | Floor | 1 year | -5% | 10% | 0.5 | 0 | 0 | 0.24
[4] | Floor | 1 year | -5% | 10% | 0.8 | 0 | 0 | 0.38
[5] | Floor | 1 year | -10% | 10% | 0.5 | 0 | 0 | 0.42
[6] | Floor | 1 year | -10% | 10% | 0.8 | 0 | 0 | 0.67
[7] | Floor | 1 year | -15% | 10% | 0.5 | 0 | 0 | 0.56
[8] | Floor | 1 year | -15% | 10% | 0.6 | 0 | 0 | 0.67
[9] | Floor | 1 year | -15% | 10% | 0.7 | 0 | 0 | 0.78
[10] | Floor | 1 year | -15% | 10% | 0.8 | 0 | 0 | 0.89
[11] | Floor | 1 year | -15% | 10% | 0.9 | 0 | 0 | 1.0
[12] | Floor | 2 year | -5% | 5% | 0.8 | 0 | 0 | 0.18
[13] | Floor | 2 year | -5% | 10% | 0.8 | 0 | 0 | 0.22
[14] | Floor | 2 year | -10% | 10% | 0.8 | 0 | 0 | 0.40
[15] | Floor | 2 year | -15% | 10% | 0.8 | 0 | 0 | 0.55

###### Example 4.1

We take the buffer EPS No. 6 from Table 4 as an example. Assume that a holder
uses $1 mm as the nominal principal, the EPS product has the buffer threshold
$l_{1}=-5\%$ with the protection rate $p_{2}=80\%$, the cap threshold
$g_{1}=10\%$, and maturity one year. The fair premium for this product is
null, which means that there no cost for EPS buyer at the inception of the
EPS, if the fee rate equals $f_{2}=82\%$.

  * •

If the realised rate of return $R_{T}$ is above $10\%$, say $12\%$, then the
EPS buyer pays $(12\%-10\%)\times 82\%\times\$1\ \mbox{\rm mm}=\$16.4$ k

  * •

If $R_{T}$ is between $-5\%$ and $10\%$, there is no cash flow between the
buyer and provider.

  * •

If $R_{T}$ is below $-5\%$, say $-8\%$, then the buyer receives
$(-8\%+5\%)\times 80\%\times\$1\ \mbox{\rm mm}=\$24$ k.

If a high loss arises due to a negative rate of realised return $R_{T}$, the
provider offers the same level of protection to a holder in both Examples 2.1
and 4.1. In contrast, the EPS buyer is required to share more gains from
positive returns with the provider in fair EPS. However, we contend that a
typical holder would pay more attention to the protection level from potential
losses than the retention rate of potential gains and hence a fair EPS would
be preferred by most holders. The the protection rate is the same in both EPSs
but the EPS from Example 4.1 has null fair premium and hence the latter EPS is
likely to have a greater appeal to investors.

###### Example 4.2

We now examine the floor EPS No. 10 from Table 4 and we assume that the
nominal principal equals $1 mm. The EPS has the buffer threshold $l_{1}=-15\%$
with the protection rate $p_{1}=80\%$, the cap threshold $g_{1}=10\%$ with the
fee rate $f_{2}=89\%$, and one year maturity. Then the holder does not need to
pay an initial premium to the provider, since the fair premium is null.

  * •

If the realised rate of return $R_{T}$ is above $10\%$, say $12\%$, then the
buyer pays $(12\%-10\%)\times 89\%\times\$1\ \mbox{\rm mm}=\$19.8$ k to the
EPS provider.

  * •

If $R_{T}$ is between $0\%$ and $10\%$, there is no cash flow between the
buyer and provider.

  * •

If $R_{T}$ is between $-15\%$ and $0\%$, such that $-8\%$, then the provider
pays $8\%\times 80\%\times\$1\ \mbox{\rm mm}=\$64$ k to the buyer.

  * •

If $R_{T}$ is below $-15\%$, say $-20\%$, then the provider pays $15\%\times
80\%\times\$1\ \mbox{\rm mm}=\$120$ k to the buyer. This is the fixed amount
that the buyer receives when $R_{T}$ is less than $-15\%$.

From Table 4, we can conclude that if the protection rate ($p_{2}$ for the
buffer EPS and $p_{1}$ for the floor EPS) increases, then the proper level of
fee rate $f_{2}$ increases as well. Rows 3-7 for the buffer EPS in the top
half of Table 4 and rows 7-11 in the bottom half of Table 4 justify this
observation. For the reader’s convenience, these EPSs are also presented in
Table 5.

Table 5: Buffer and floor EPSs with null fair premium No. | EPS class | $T$ | $l_{1}$ | $g_{1}$ | $p_{1}$ | $p_{2}$ | $f_{1}$ | $f_{2}$
---|---|---|---|---|---|---|---|---
[3] | Buffer | 1 year | -5% | 10% | 0 | 0.5 | 0 | 0.52
[4] | Buffer | 1 year | -5% | 10% | 0 | 0.6 | 0 | 0.62
[5] | Buffer | 1 year | -5% | 10% | 0 | 0.7 | 0 | 0.72
[6] | Buffer | 1 year | -5% | 10% | 0 | 0.8 | 0 | 0.82
[7] | Buffer | 1 year | -5% | 10% | 0 | 0.9 | 0 | 0.93
[7] | Floor | 1 year | -15% | 10% | 0.5 | 0 | 0 | 0.56
[8] | Floor | 1 year | -15% | 10% | 0.6 | 0 | 0 | 0.67
[9] | Floor | 1 year | -15% | 10% | 0.7 | 0 | 0 | 0.78
[10] | Floor | 1 year | -15% | 10% | 0.8 | 0 | 0 | 0.89
[11] | Floor | 1 year | -15% | 10% | 0.9 | 0 | 0 | 1.0

Similarly, if the protection leg parameter $l_{1}$ for the buffer EPS drops,
which means the starting point for a loss protection is lowered, then the fee
rate $f_{2}$ for the EPS provider decreases, with all else equal. For the
floor EPS, if $l_{1}$ decreases, then the protection interval for losses
becomes wider. We conclude that the fee rate $f_{2}$ for an EPS with null fair
premium is increasing when the protection leg level $l_{1}$ decreases (this
observation is illustrated in Table 6). By combining results in Tables 5 and
6, we can draw the following conclusion: if the protection strength for losses
raises, then the fee rate for all EPSs should increase when the fair premium
is postulated to vanish.

Table 6: Buffer and floor EPSs with null fair premium No. | EPS class | $T$ | $l_{1}$ | $g_{1}$ | $p_{1}$ | $p_{2}$ | $f_{1}$ | $f_{2}$
---|---|---|---|---|---|---|---|---
[6] | Buffer | 1 year | -5% | 10% | 0 | 0.8 | 0 | 0.82
[10] | Buffer | 1 year | -10% | 10% | 0 | 0.8 | 0 | 0.53
[12] | Buffer | 2 year | -5% | 10% | 0 | 0.8 | 0 | 0.69
[13] | Buffer | 2 year | -10% | 10% | 0 | 0.8 | 0 | 0.50
[4] | Floor | 1 year | -5% | 10% | 0.8 | 0 | 0 | 0.38
[6] | Floor | 1 year | -10% | 10% | 0.8 | 0 | 0 | 0.67
[10] | Floor | 1 year | -15% | 10% | 0.8 | 0 | 0 | 0.89
[13] | Floor | 2 year | -5% | 10% | 0.8 | 0 | 0 | 0.22
[14] | Floor | 2 year | -10% | 10% | 0.8 | 0 | 0 | 0.40
[15] | Floor | 2 year | -15% | 10% | 0.8 | 0 | 0 | 0.55

Lastly, we briefly discuss the influence of different values of the fee leg
parameter $g_{1}$. If $g_{1}$ increases, the starting point of rate of return
for which the EPS provider can participate in realised gains is higher. Hence
if $g_{1}$ increases, then the fee rate $f_{2}$ should increase as well for
both the buffer and floor EPS in order to make the fair premium vanish. Rows
2, 6, 11, 12 for the buffer EPS in the top half and rows 2, 4, 12, 13 for the
floor EPS in the bottom half of Table 7 are consistent with that finding.

Table 7: Buffer and floor EPSs with null fair premium No. | EPS class | $T$ | $l_{1}$ | $g_{1}$ | $p_{1}$ | $p_{2}$ | $f_{1}$ | $f_{2}$
---|---|---|---|---|---|---|---|---
[2] | Buffer | 1 year | -5% | 5% | 0 | 0.8 | 0 | 0.61
[6] | Buffer | 1 year | -5% | 10% | 0 | 0.8 | 0 | 0.82
[11] | Buffer | 2 year | -5% | 5% | 0 | 0.8 | 0 | 0.56
[12] | Buffer | 2 year | -5% | 10% | 0 | 0.8 | 0 | 0.69
[2] | Floor | 1 year | -5% | 5% | 0.8 | 0 | 0 | 0.28
[4] | Floor | 1 year | -5% | 10% | 0.8 | 0 | 0 | 0.38
[12] | Floor | 2 year | -5% | 5% | 0.8 | 0 | 0 | 0.18
[13] | Floor | 2 year | -5% | 10% | 0.8 | 0 | 0 | 0.22

###  4.3 Forward Performance of an EPS

We now take the view of the holder and we examine the theoretical forward
performance of an EPS during market downturns. Consistently with the primary
purpose of an EPS, its buyer is assumed to be a holder of a superannuation
account and thus we assume that she will receive the return $R_{T}$ on a
reference portfolio at maturity date $T$. Recall that the buyer an EPS is
required to deliver to its provider the adjusted return $\psi(R_{T})$ on the
reference portfolio where $\psi(R_{T})$ is nonnegative (resp. nonpositive) if
the realised return $R_{T}$ is nonnegative (resp. nonpositive). Therefore, in
a study of performance of an EPS we will focus on the realised net return for
the holder of an superannuation account supplemented by an EPS. Formally, the
net return is equal to the difference $R_{T}-\psi(R_{T})$ where $R_{T}$ is the
original return and $\psi(R_{T})$ is the adjusted return. In the following
example, we consider an investor holding a fair one-year index EPS for five
consecutive years, meaning that he/she immediately enters into a new one-year
contract upon termination of the previously held one-year index EPS. The
simulated performance of the index in the Black-Scholes model and the net
returns for the holder of an EPS are presented in Figure 8.

We acknowledge that the Black-Scholes model does not offer the most reliable
description of market crises but we contend that in a jump-diffusion model the
performance of an EPS would be even more visible. To explicitly illustrate the
latter conjecture by means of an example, we artificially introduce two
negative jumps at deterministic dates in weeks 80 and 250. As before, we take
the interest rate $r=1.5\%$, the dividend yield $\kappa=0$, the volatility
parameter $\sigma=20\%$, and the initial value $S_{0}=100$. We perform 200
simulations of sample paths of asset returns over 5 years with the dynamics of
the asset price given by (6). The original and net returns are plotted with
the $x$-axis representing the number of trading weeks. Three typical sample
paths of returns are shown in black.

In addition, to account for the possibility of market crises, we introduce two
down-ticks in weeks 80 and 250, which are indicated by red dashed lines. We
then apply five back-to-back one-year fair buffer EPSs with parameters
$p_{1}=0.8,\,l_{1}=-10\%,\,f_{2}=0.53$ and $g_{1}=10\%$. The distributions of
the asset’s original returns (green line) and net returns for a buffer EPS
(blue line) are plotted year by year. Furthermore, we mark the extreme and
mean values in the distributions of original and net returns. To enhance
clarity, we only show the statistics for the first and fifth years.

Figure 8: Simulation of original and net returns with a buffer EPS

In Figure 8, we make the first key observation that the distribution curve of
net returns is still reminding a normal curve when the buffer EPS is used. It
is notable that the expected net returns are visibly higher while the spread
narrows toward the mean. This indicates that the net returns have a smaller
standard deviation compared to original returns, resulting in a more
concentrated distribution with a higher peak. If no unpredictable events
happen in the first year, holders of the EPS are protected from a loss less
than $l_{1}$, and, as a trade-off, she/he needs to share any benefits greater
than the threshold $g_{1}$.

Furthermore, the impact of the EPS on benefits sharing and loss protection is
exactly the same in relation to the market factors and the participation rates
$p_{1}$ and $f_{2}$. This is because the considered EPS is fair to both
parties. The extreme positive values of the original and net returns with the
buffer EPS at the end of the first year are $56.6\%$ and $38.5\%$,
respectively, while the extreme negative values are $-50.0\%$ and $-32.0\%$,
respectively. Additionally, for risk-averse investors, the buffer EPS can
offer even more protection against losses if an insurance premium is paid to
the provider. At the end of the fifth year, after experiencing two
unpredictable market crises, the expected value of original returns drops from
$0.033\%$ to $-0.324\%$ which represents the worse case scenarios for
investors in real life. However, with the help of five one-year buffer EPSs,
the mean of net returns was lifted to $-17.4\%$, which is much higher than the
mean of original returns at the end of the fifth year ($-32.4\%$). The entire
net returns curve is shifted upward towards a more concentrated distribution,
which results in the higher mean and significantly elevated negative returns.

On the gains side, the extreme positive value of net returns decreased only
from $93.6\%$ (original returns) to $77.2\%$ (net returns), which is a
difference of $16.4\%$. In contrast, compared to the extreme negative value of
the original return ( $-158.3\%$), the net returns rose to $-113\%$, an
increase of $45.3\%$, nearly three times the difference of the extreme
positive returns. The occurrence of market downturns accentuates the
prominence of the protection leg over the fee leg, leading to asymmetric
contraction levels in the two legs. These results underscore the role of EPSs
as an investment insurance, demonstrating their enhanced performance in
periods of increased market risks, particularly in the presence of market
crises.

##  5 Backtesting of Performance of Buffer and Floor EPSs

In our numerical analysis of some classes of equity protection swaps, we
computed the theoretical fair premia for several EPSs with predetermined
participation rates, as outlined in Section 4.1. Subsequently, we have
examined the level of the fee rate for which the fair premium
$\widehat{c}=\widehat{H}_{0}$ vanishes. In the final step, we analyse the
impact of an EPS on holder’s investment through backtesting using the real-
world data for two major stock indices: S&P 500 and S&P/ASX 200 in the U.S.
and Australia, respectively. It should be acknowledged that we do not consider
the currency risk and thus backtesting is done independently for the domestic
and foreign investments, denominated in AUD and USD, respectively. It is
important to stress that throughout this section, all of our discussions and
pricing results are model-free, in the sense that we do not make any
assumptions about the valuation models or the future dynamics of indices.

We argued that, unlike other insurance products for variable annuities, the
fair pricing of basic EPS products can be considered to be model-free since
their valuation can be done directly in reference to the market data for
traded options without using any stochastic model for future dynamics of the
underlying index. Specifically, the fair premium for an EPS can be calculated
using directly the market prices of the European call and put options with
weights given by a static hedging strategy. Needless to say, since the fair
premia (or, equivalently, the fair fee rates) are implicitly given by the real
market data, they necessarily vary when market conditions change. In our
empirical study of the real-world benefits of an EPS for its holder, we will
focus on the following six specifications for buffer and floor EPSs:

  * •

Buffer 1: $p_{2}=0.5,\ f_{2}=0.63,\ l_{1}=-5\%,\ g_{1}=5\%$;

  * •

Buffer 2: $p_{2}=0.7,\ f_{2}=1.51,\ l_{1}=-5\%,\ g_{1}=10\%$;

  * •

Buffer 3: $p_{2}=0.7,\ f_{2}=1.21,\ l_{1}=-10\%,\ g_{1}=10\%$;

  * •

Floor 1: $p_{1}=0.5,\ f_{2}=0.52,\ l_{1}=-10\%,\ g_{1}=10\%$;

  * •

Floor 2: $p_{1}=0.7,\ f_{2}=0.73,\ l_{1}=-10\%,\ g_{1}=10\%$;

  * •

Floor 3: $p_{1}=0.7,\ f_{2}=0.98,\ l_{1}=-15\%,\ g_{1}=10\%$.

These six products were postulated to be fair EPSs, meaning that their initial
premia at inception should vanish. To this end, we first selected the
protection rate ($p_{1}$ or $p_{2}$), as well as the protection and fee
thresholds ($l_{1}$ and $g_{1}$) and then we used the market data for European
options to identify a unique level of the fee rate $f_{2}$, which makes each
EPS a fair swap. As was mentioned, the initial fair fee rate $f_{2}$ varies
from day to day but here, for the sake of simplicity, we have only used one
day market data for European options on S&P 500 index to compute the fair fee
rate $f_{2}$. The date (2022.02.02) was chosen as the middle date of the
period from 3 May 2021 to 23 December 2022 when the market downturn has
occurred. The closing prices for in-the-money and at-the-money European put
and call options on S&P 500 index with a one-year maturity are reported in
Table 8. We give there the quote and expiration dates, the strike price and
the spot level of S&P 500 index, and bid and ask prices for relevant options.
Notice that the values of the moneyness, here defined by convention as the
ratio strike/spot are aligned with the value $1+l_{1}$ for put options and
$1+g_{1}$ for call options (recall that $K^{l}_{1}=S_{0}(1+l_{1})$ and
$K^{g}_{1}=S_{0}(1+g_{1})$).

Table 8: Market data for European options on S&P 500 index No. | Quote Date | Expiration | Strike | Type | Bid | Ask | Spot | Moneyness
---|---|---|---|---|---|---|---|---
[1] | 2022/02/02 | 2023/02/17 | 3900 | Put | 184.6 | 187.8 | 4576.8 | 85.2%
[2] | 2022/02/02 | 2023/02/17 | 4125 | Put | 235.8 | 239.8 | 4576.8 | 90.1%
[3] | 2022/02/02 | 2023/02/17 | 4350 | Put | 298.3 | 302.7 | 4576.8 | 95.0%
[4] | 2022/02/02 | 2023/02/17 | 4575 | Put | 375.2 | 379.8 | 4576.8 | 100%
[5] | 2022/02/02 | 2023/02/17 | 4575 | Call | 366.1 | 370.4 | 4576.8 | 100%
[6] | 2022/02/02 | 2023/02/17 | 4800 | Call | 239.0 | 243.4 | 4576.8 | 104.9%
[7] | 2022/02/02 | 2023/02/17 | 5025 | Call | 139.0 | 142.8 | 4576.8 | 109.8%

We use the same six EPS products in empirical studies for both S&P 500 index
and S&P/ASX 200 index, although historical market data for the European option
on S&P/ASX 200 index were not available. Hence it was assumed that the market
prices of European options for both indices will be close. In practice, our
approach can be easily extended to the fully dynamic fair pricing of EPSs but
we contend that the conclusions from our study would not change substantially.

###  5.1 Backtesting During Market Downturn

To assess the benefits from holding an EPS, we initially focus on scenarios in
which the market experiences a downturn, which means that the underlying
assets have negative returns. We assume that a cohort of investors engage on
any given trading day in one-year EPS contracts of a given type and the EPSs
commenced on any given day have the same combined notional principal, the size
of which is in fact irrelevant for our conclusions.

Regarding the market data, we use the daily closing prices of S&P 500 and
S&P/ASX 200 indices from 2 January 2020 to 23 December 2022 to determine the
realised trailing returns on the index serving as a reference portfolio for an
EPS under study. In order to analyse the impact of predominantly negative
returns, we collected 164 one-year trailing returns of the S&P 500 index and
165 one-year trailing returns of the S&P/ASX 200 index from 3 May 2021 to 23
December 2022. The histograms in Figure 9 present empirical frequencies of
realised one-year trailing returns on S&P 500 and S&P/ASX 200 indices during
the considered period.

---
(a) S&P 500 one year trailing return

---
(b) S&P/ASX 200 one year trailing return

Figure 9: Histograms of trailing returns on S&P 500 and S&P/ASX 200 indices Figure 10: Empirical densities of original and net trailing returns on S&P 500 under market downturn Table 9: Empirical quantiles of original and net trailing returns on S&P 500 under market downturn Case | Min | $5\%$ | $10\%$ | $25\%$ | $50\%$ | $75\%$ | $90\%$ | Max
---|---|---|---|---|---|---|---|---
Original | -0.2027 | -0.1822 | -0.1777 | -0.1543 | -0.1180 | -0.0649 | -0.0365 | 0.0325
Buffer1 | -0.1264 | -0.1161 | -0.1139 | -0.1021 | -0.0840 | -0.0575 | -0.0365 | 0.0325
Buffer2 | -0.0958 | -0.0897 | -0.0883 | -0.0813 | -0.0704 | -0.0545 | -0.0365 | 0.0325
Buffer3 | -0.1308 | -0.1247 | -0.1233 | -0.1163 | -0.1054 | -0.0649 | -0.0365 | 0.0325
Floor1 | -0.1527 | -0.1322 | -0.1277 | -0.1043 | -0.0680 | -0.0325 | -0.0183 | 0.0325
Floor2 | -0.1327 | -0.1122 | -0.1077 | -0.0843 | -0.0480 | -0.0195 | -0.0110 | 0.0325
Floor3 | -0.0977 | -0.0772 | -0.0727 | -0.0493 | -0.0354 | -0.0195 | -0.0110 | 0.0325

From Figure 9 it transpires that the realised one-year trailing returns in the
considered period are characterised by a negative mean value. By combining
either a buffer or a floor EPS with the original returns, we obtain the sample
of net trailing returns on S&P 500 index. To compare the original and net
trailing returns, we display in Figure 10 their respective empirical densities
and in Table 9 we give the values of empirical quantiles. Figure 10 and Table
9 reveal that, on the negative side, the net trailing returns dominate the
original trailing returns, have a less negative empirical mean, and exhibit an
empirical distribution not far away from the normal distribution. It is worth
noting that the buffer EPS No. 2 in Table 9 offers the best protection among
three buffer contracts, which is due to the fact that it has a higher
protection rate ($p_{2}=0.7$) than the buffer EPS No. 1 ($p_{2}=0.5$), and a
higher protection threshold ($l_{1}=-5\%$) than the buffer EPS No. 3
($l_{1}=-10\%$). Similarly, the floor EPS No. 3 offers the highest protection
among three floor contracts, has a higher protection rate ($p_{1}=0.7$) than
the floor No. 1 ($p_{1}=0.5$), and a lower floor ($l_{1}=-15\%$) than the
floor EPS No. 2 ($l_{1}=-10\%$). In Figure 10, we observe that floor EPSs
shift negative original trailing returns to higher levels than buffer EPSs
when considering the market during a downturn period. This feature was in fact
expected, since a buffer EPS only protects against sizeable negative returns,
while a floor EPS provides protection against any negative return.

We have also examined the impact of EPSs on the S&P/ASX 200 index using 165
one-year realised trailing returns on S&P/ASX 200. Figure 11 displays the
empirical densities of one-year trailing returns on the S&P/ASX 200 index and
net returns for various EPS products. Additionally, in Table 10 we report the
empirical quantiles of S&P/ASX 200 index returns for the period under study.

Figure 11: Empirical densities of original and net trailing returns on S&P/ASX 200 under market downturn Table 10: Empirical quantiles of original and net trailing returns on S&P/ASX 200 under market downturn Case | Min | $5\%$ | $10\%$ | $25\%$ | $50\%$ | $75\%$ | $90\%$ | Max
---|---|---|---|---|---|---|---|---
Original | -0.1290 | -0.1092 | -0.0988 | -0.0858 | -0.0654 | -0.0248 | 0.0076 | 0.0478
Buffer1 | -0.0895 | -0.0796 | -0.0744 | -0.0679 | -0.0577 | -0.0248 | 0.0076 | 0.0478
Buffer2 | -0.0737 | -0.0678 | -0.0646 | -0.0607 | -0.0546 | -0.0248 | 0.0076 | 0.0478
Buffer3 | -0.1087 | -0.1028 | -0.0988 | -0.0858 | -0.0654 | -0.0248 | 0.0076 | 0.0478
Floor1 | -0.0790 | -0.0592 | -0.0494 | -0.0429 | -0.0327 | -0.0124 | 0.0076 | 0.0478
Floor2 | -0.0590 | -0.0392 | -0.0297 | -0.0257 | -0.0196 | -0.0074 | 0.0076 | 0.0478
Floor3 | -0.0387 | -0.0328 | -0.0296 | -0.0257 | -0.0196 | -0.0074 | 0.0076 | 0.0478

It is no surprising to see in Figure 11 and Table 10 that the relationship
between different EPS contracts is similar to that observed in Figure 10 and
Table 9. We conclude that, regardless of the data used, the benefits of EPS
contracts for portfolio’s protection are consistent.

It can also be observed that a floor EPS has a stronger impact on S&P/ASX 200
index returns yielding relatively more concentrated net returns, while a
buffer EPS typically shows a more pronounced influence on S&P 500 index
returns. This phenomenon is due to the fact that a floor EPS protects against
any losses of a limited size, whereas a buffer EPS can only protect against
relatively large losses but is unlimited. A buffer EPS would be preferred by a
holder characterised by a high level of risk aversion to large losses, whereas
a floor EPS would be more appropriate for a holder expecting only relatively
low losses during the holding period of an EPS. Finally, since original
trailing returns on S&P/ASX 200 index were broadly less negative compared to
those on S&P 500 index, a floor EPS would be more suitable when the S&P/ASX
200 index is chosen as the reference portfolio.

###  5.2 Backtesting During Market Downturn and Upturn

Next, we study the impact of EPS contracts on a general scenario where the
real-world market has experienced both a slump and an upsurge. To this end, we
analyse the daily closing prices of S&P 500 and S&P/ASX 200 between 2 January
2020 and 23 December 2022, resulting in 499 one-year trailing returns for the
S&P 500 index and 502 one-year trailing returns for the S&P/ASX 200 index.
Figure 12 shows the empirical histograms of one-year trailing returns on S&P
500 and S&P/ASX 200 indices. We observe that the average trailing return on
the S&P 500 index is higher than that on the S&P/ASX 200 index and,
furthermore, the realised trailing returns on S&P 500 index are characterised
by a higher volatility.

---
(a) S&P 500 one year trailing return

---
(b) S&P/ASX 200 one year trailing return

Figure 12: Histograms of trailing returns for S&P 500 and S&P/ASX 200 indices

To examine the effect of an EPS on both positive and negative returns, we
continue to examine six EPSs (three buffers and three floors) introduced at
the beginning of this section to compute the net returns for the holder of an
EPS. Figure 13 illustrates the empirical densities for original and net
returns on the S&P 500 index and Table 11 gives the respective empirical
quantiles.

Figure 13: Empirical densities of trailing returns on S&P 500 under market downturn and upturn Table 11: Empirical quantiles of original and net trailing returns on S&P 500 under market downturn and upturn Case | Min | $5\%$ | $10\%$ | $25\%$ | $50\%$ | $75\%$ | $90\%$ | Max
---|---|---|---|---|---|---|---|---
Original | -0.2027 | -0.1654 | -0.1444 | -0.0617 | 0.1641 | 0.3236 | 0.4172 | 0.7382
Buffer1 | -0.1264 | -0.1077 | -0.0972 | -0.0559 | 0.0922 | 0.1512 | 0.1859 | 0.3046
Buffer2 | -0.2255 | -0.0969 | -0.0880 | -0.0704 | -0.0269 | 0.0219 | 0.0726 | 0.0997
Buffer3 | -0.1308 | -0.1196 | -0.1133 | -0.0617 | 0.0427 | 0.0655 | 0.0882 | 0.0997
Floor1 | -0.1527 | -0.1154 | -0.0944 | -0.0309 | 0.1308 | 0.2073 | 0.2523 | 0.4063
Floor2 | -0.1327 | -0.0954 | -0.0744 | -0.0185 | 0.1173 | 0.1604 | 0.1857 | 0.2723
Floor3 | -0.0977 | -0.0604 | -0.0433 | -0.0185 | 0.1013 | 0.1045 | 0.1063 | 0.1128

In addition, Figure 14 presents the empirical densities for the original and
net trailing returns on the S&P/ASX 200 index, accompanied by their empirical
quantiles given in Table 12. Observe that the empirical densities of returns
on the S&P/ASX 200 index exhibit similar main characteristics as those
obtained for the S&P 500 index. While we previously examined the impact of
EPSs on negative returns in the benefit testing for investors, we now focus on
their influence on positive returns.

Figure 14: Empirical densities of trailing returns on S&P/ASX 200 under market downturn and upturn Table 12: Empirical quantiles of original and net trailing returns on S&P/ASX 200 during market downturn and upturn Case | Min | $5\%$ | $10\%$ | $25\%$ | $50\%$ | $75\%$ | $90\%$ | Max
---|---|---|---|---|---|---|---|---
Original | -0.1290 | -0.0939 | -0.0804 | -0.0386 | 0.0630 | 0.2239 | 0.2690 | 0.4912
Buffer1 | -0.0895 | -0.0719 | -0.0652 | -0.0386 | 0.0548 | 0.1144 | 0.1310 | 0.2132
Buffer2 | -0.0995 | -0.0635 | -0.0593 | -0.0418 | 0.0165 | 0.0475 | 0.0837 | 0.0999
Buffer3 | -0.1087 | -0.0939 | -0.0804 | -0.0386 | 0.0509 | 0.0730 | 0.0915 | 0.0999
Floor1 | -0.0790 | -0.0469 | -0.0402 | -0.0193 | 0.0630 | 0.1595 | 0.1811 | 0.2877
Floor2 | -0.0590 | -0.0282 | -0.0241 | -0.0116 | 0.0630 | 0.1335 | 0.1456 | 0.2056
Floor3 | -0.0387 | -0.0282 | -0.0241 | -0.0116 | 0.0630 | 0.1025 | 0.1034 | 0.1078

When comparing the empirical densities of original and net trailing returns on
S&P 500 and S&P/ASX 200 indices, we can observe that the net returns are more
tightly distributed and exhibit a smaller variance. Furthermore, the
frequencies of positive and negative values for both the original and net
trailing returns are similar, with positive (resp., negative) net returns
being systematically lower (resp., higher) for the net returns. It is common
in the economic literature to assume that investors are risk-averse and pay
more attention to losses, rather than gains. Although the net trailing returns
have compressed positive parts, there are ample reasons to contend that EPS
products would prove advantageous for holders of superannuation accounts since
their protection leg is an effective tool to mitigate portfolio’s losses.

To summarise, the empirical densities of net returns for the holder of an EPS
presented in Figures 13 and 14 exhibit sharper peaks than the original
returns, which indicates that an EPS offers effective protection against
substantial losses at a cost of an insurance fee in the form of a minor
reduction of large gains. As expected, a holder of an EPS receives a higher
return in the case of portfolio’s loss combined with a lower return when gain
occurs, which helps to mitigate the uncertainty in portfolio’s future values.

##  6 Concluding Remarks

This work examines the performance of a new class of financial derivatives,
tentatively called equity protection swaps (EPS), whose aim is to provide
tailored-made protection for individual superannuation accounts. A generic EPS
is a special kind of equity swap that combines the features of a total return
swap with a customised contract that allows both the holder and provider to
select levels of the protection and fee legs, as well as the corresponding
participation rates.

An EPS is purposely designed to have a relatively simple structure in
comparison to other existing guarantee products for variable annuities, while
it still retains the important attribute of providing at least a partial
guarantee for the value of a reference portfolio at no initial cost.
Therefore, we contend that they are likely to have a huge potential for
holders of superannuation accounts. In essence, the provider of an EPS
receives potential gains when the value of the reference portfolio increases
between the contract’s inception date and its maturity and offers partial
protection to the holder when the value of the reference portfolio drops
during that period. Since an EPS can be easily customized, it should be
attractive to portfolio holders as a tool to mitigate losses at a desired
level of protection based on their individual preferences.

Additionally, an EPS offers a unique structure that allows for the fair
premium to be easily calculated for varying maturities, protection thresholds
and rates, as well as fee thresholds and rates. The existence of an explicit
formula for the fair premium is beneficial for both the EPS provider and
buyer. According to behavioral finance, investors tend be unwilling to pay a
large amount upfront in exchange for insurance of a potential loss in the
future. To address this issue, the provider has also the option to adjust the
fee rate so that the fair initial premium is null. In that case the only cost
for investors is a predetermined share of potential gains at maturity, which
should make an EPS more attractive to them. The flexibility and simplicity of
an EPS make it an ideal product for the pension market, which is characterised
by unsophisticated investors whose livelihoods depend on income stream drawn
from their superannuation accounts.

From the provider’s perspective, we argued that the use of European call and
put options is an effective method for the static (and hence model-free)
hedging strategy for an EPS. By using call options to hedge potential gains in
the fee leg and put options to hedge potential losses in the protection leg,
providers can create a well-structured static hedging portfolio. Two specific
structures of an EPS, the buffer EPS and the floor EPS, were used throughout
to explicitly illustrate the hedging portfolios but the method works for other
cases as well. Of these two structures, the floor EPS was found to have a
superior hedging structure, due to its cap on potential provider’s losses in
case of counterparty credit risk of issuers of put options. While the present
work focuses on these two specific structures, other specification, such as
buffer-floor/buffer-cap EPS of Example 2.1, will be explored in future
research.

Another important advantage of an EPS is the fact that it can be offered by a
third-party provider, as this completely eliminates the pressure on managing
the underlying reference portfolio. However, in such a situation, it would be