# When Do Decompositions Help for Machine Reading?

Kangda Wei1, Dawn Lawrie2, Benjamin Van Durme2, Yunmo Chen2*, Orion Weller2*
1University of North Carolina Chapel Hill
2Johns Hopkins University
kangda<EMAIL_ADDRESS>
###### Abstract

Answering complex questions often requires multi-step reasoning in order to
obtain the final answer. Most research into decompositions of complex
questions involves open-domain systems, which have shown success in using
these decompositions for improved retrieval. In the machine reading setting,
however, work to understand when decompositions are helpful is understudied.
We conduct experiments on decompositions in machine reading to unify recent
work in this space, using a range of models and datasets. We find that
decompositions can be helpful in the few-shot case, giving several points of
improvement in exact match. However, we also show that when models are given
access to around a few hundred or more examples, decompositions are not
helpful (and can actually be detrimental). Thus, our analysis implies that
models can learn decompositions implicitly even with limited data.

##  1 Introduction

00footnotetext: * Joint advising

Much recent work has examined and improved models’ ability to answer complex
questions that require multiple steps of reasoning Wolfson et al. (2020); Dua
et al. (2019a); Yang et al. (2018); Welbl et al. (2018); Talmor and Berant
(2018a). A consistent theme in these works is to break the main complex
question down into a series of sub-questions to be solved, which is referred
to as question decomposition. These works generally represent decompositions
as a human would, with explicit natural language sub-questions that build
together to the final answer. Research has consistently shown that open-domain
question answering (QA) models perform better on multi-step questions when
they use decomposed sub-questions rather than answering with the main question
alone Wolfson et al. (2020); Perez et al. (2020); Geva et al. (2021a).

Figure 1: An instance of HotpotQA in BREAK Wolfson et al. (2020), showing
three different decomposition settings: (1) No Decomposition, i.e. regular
question answering, (2) Explicit Decompositions that use iterative sub-
questions, and (3) Implicit Decompositions that prepend the reasoning steps as
special tokens.

Figure 2: Main results showing the effect of various decomposition strategies.
Left shows results on DROP, right shows HotpotQA. Runs were done over three
random seeds and report the mean; error bars indicate 1 std. dev.

Despite a large amount of research in decompositions for multi-step question
answering, the majority of it has focused on using question decomposition for
both information retrieval and question answering Wolfson et al. (2020); Perez
et al. (2020). The few works that have used decompositions for machine reading
generally evaluate in limited settings Guo et al. (2022); Patel et al. (2022).

Therefore, we seek to shed light on if and when decompositions are helpful for
machine reading. To do so, we analyze decomposition methods for seq2seq models
across two multi-step QA datasets. Our results show that decompositions are
only helpful in the low data setting, where there are less than a few hundred
examples. Using decompositions in anything other than that setting performs
the same (or much worse, depending on the strategy) than simply training the
model end-to-end.

Thus, overall, decompositions are helpful for question answering when they are
used in two main settings: (1) information retrieval, where the decompositions
help isolate distinct aspects of the question, or (2) in zero to low resource
settings, where there isn’t enough data to implicitly learn the multi-step
process through end-to-end training.

##  2 Experiment Setup

###  2.1 Data

We use the BREAK Wolfson et al. (2020) resource which contains annotated
decompositions for 10 different benchmarks, including three reading
comprehension benchmarks. We use two of these three datasets (HotpotQA and
DROP) as the third, ComplexWebQuestions Talmor and Berant (2018b), does not
currently provide a training set.111Note that although the ComplexWebQuestions
(CWQ) paper initially released the training set the authors have since removed
it from the official Dropbox. Furthermore, even if the dataset was available
CWQ does not verify that the questions are answered by the returned Google
search. Following BREAK and other follow-up work Geva et al. (2021b), we train
and test our models using the high-level decompositions (also known as QDMRs).

Thus, we use HotpotQA Yang et al. (2018) and DROP Dua et al. (2019b) in our
experiments, using the portions annotated from the train and validation sets.
HotpotQA was created from workers on Mechanical Turk who annotated
compositional questions from Wikipedia by using the page links (an example can
be found in Figure 1). DROP was created to test discrete multi-hop reasoning
and was also annotated by Mechanical Turk workers who used Wikipedia articles
as the context.

The BREAK annotations include the list of decomposed questions that eventually
yield the same answer as the original question, along with an operator
assigned to each decomposition that represents the type of reasoning for that
step (e.g. Boolean, Comparison, etc.). Note that there are no gold labels for
intermediate decomposition steps; the only ground truth label is for the main
question.

###  2.2 Models

To explore the effect of decompositions on various types of common NLP models,
we employ three different models: BART Lewis et al. (2019), vanilla T5 Raffel
et al. (2019), and UnifiedQA-v2 Khashabi et al. (2020, 2022) that uses a t5
backbone and has been fine-tuned on other QA datasets but not on HotpotQA
Raffel et al. (2019). This allows us to demonstrate the effect of additional
fine-tuning (UnifiedQA vs vanilla T5) as well as between different
architectures (BART vs T5). Note that because UnifiedQA-v2 was multi-task
trained on DROP, its scores are noticeably higher than the other models
(Figure 2). However, our purpose is not to compare scores between models, but
rather to compare scores between different decomposition strategies. Thus, the
inclusion of this model on DROP shows us that our results hold even if the
model was pretrained on it. For more hyperparameter and compute details, see
Appendix A.

###  2.3 Decomposition Strategies

There are many possible ways to combine decomposition with model fine-tuning.
We try a wide variety of techniques (including novel ones) that we group into
three categories: (1) no decomposition e.g. the baseline QA format, (2)
explicit decomposition, and (3) implicit decomposition.

#### Explicit Decomposition

Explicit decompositions are the most common approach in the decomposition
literature, generating the answer iteratively through sub-questions: the model
answers the first decomposition step, then replaces placeholders in future
decomposition steps with that predicted answer, then predicts the second
decomposition step, and so forth. Note that using this method (Explicit)
naively presents issues with backpropagation, as the model can only
backpropagate through the last decomposition step. Variations of strategies in
this category include giving the model all previous decomposition steps as
context (Explicit w/Decomp) or including all decomposition steps and all
predicted answers as context (Explicit Everything).

#### Implicit Decomposition

Another way to do decompositions could be to add them implicitly. To do so, we
utilize the operators provided in the BREAK annotations which describe the
type of reasoning needed, removing duplicate operators and keeping them in
their given order. For example, in Figure 1 the model uses select twice and
then intersection in the explicit decomposition reasoning steps (but we remove
the duplicate select). We implement this in practice by adding a new special
token for each operator and prepending them to the original question. Although
this approach is novel in the context of decompositions, it bears similarity
to work in the prompting literature, such as soft prompts Qin and Eisner
(2021); Liu et al. (2022). Variations to this approach in the same category
include randomizing the order of the special tokens (Implicit Unordered),
leaving in duplicate special tokens (Implicit w/Dupl.), or even prepending
these operators to the explicit sub-questions in the Explicit Decomposition
approach to combine the two strategies (Explicit + Implicit).

##  3 Results

#### Full-Data Experiments

We see the main results in Figure 2 with results for DROP on the left and
HotpotQA on the right. Bars are colored-coded according to their method. All
bars are the mean of three random seeds and error bars indicate the standard
deviation. We see that most methods perform nearly the same, except for two
that underperform: Explicit and Explicit + Implicit. Note that both of these
have issues with training end-to-end, as an Explicit decomposition is not
differentiable through all decomposition steps. Thus, we only end up
differentiating through the last step of the explicit decomposition steps,
leaving the model unable to learn as effectively. All other approaches to
decomposition perform comparably, given random seed variance (e.g. t5-base
DROP Implicit Decomp. is 33.7% exact match ± 0.9% vs No Decomp 34.0% ± 0.4%).
In fact, in this full data setting, the No Decomp method performs better or
statistically similar to every other method according to two-sample t-tests
with the Bonferroni correction Weisstein (2004), across all datasets and
models.

Question |  Decompositions |  Content (shortened) |  Intermediate Predictions |  Answer |  Error
---|---|---|---|---|---
Who was born first, Kwok Kin Pong or Edison Chen? |  #1: when was Kwok Kin Pong born?
#2: when was Edison Chen born?
#3: which is the lowest of #1,#2? |  Edison Koon-hei Chen (born 7 October 1980) .. Kwok Kin Pong (born 30 March 1987 in Hong Kong) .. |  #1. 7 October 1980
#2. 30 March 1987
#3: Kwok |  Edison Chen |  Wrong Prediction at Last Step
(18%)
Are both Deerhunter and Nine Lashes American Christian rock bands? |  #1. is Deerhunter a American Christian rock band?
#2. is Nine Lashes a American Christian rock band?
#3: if both #1 and #2 are true |  .. Nine lashes is an American Christian rock band .. Deerhunter is an American rock band from Atlanta .. |  #1. Yes
#2. Yes
#3. Yes |  No |  Error propagation
(40%)
What actor from “Willow" also starred in “The Usual Suspects"? |  #1: who is the actor that starred in The Usual Suspects?
#2: #1 that was a actor from Willow? |  .. Kevin Elliot Pollak .. a role in "Willow" .. the Usual Suspects stars Kevin Pollak .. |  #1. Kevin Pollak
#2. Kevin Pollak |  Kevin Elliot Pollak |  Invalid or Missing Annotation
(42%)

Table 1: Error analysis for standard decomposition (i.e. Explicit Decomp with
no fine-tuning, a la Patel et al. (2022)) on HotpotQA. Percentages calculated
from annotation of 50 instances with representative examples shown.

(a) Results from UnifiedQA-v2

(b) Results from T5

Figure 3: Size Experiments on HotpotQA. The red line indicates the performance
of zero-shot decompositions while the blue line is the No Decomposition
method.

#### Size Experiment

How much data is needed for the No Decomposition method with fine-tuning to
perform comparably to decomposition methods without fine-tuning (e.g. the
setting in Patel et al. (2022))? We answer this question in Figure 3. As we
need a model fine-tuned on QA to evaluate in the zero-shot case, we use
UnifiedQA-v2 and a version of T5 fine-tuned on SQuAD Rajpurkar et al.
(2016).222Note that we do not evaluate on DROP as UnifiedQA-v2 was already
trained on DROP. Points indicate the mean score over three random seeds and
error bars indicate one standard deviation. We see that as the amount of data
increases, the fine-tuned model performs better; it is better than the zero-
shot method between 100-250 examples for UnifiedQA-v2 and 250-500 for T5.

#### Error Analysis

Why does the standard not fine-tuned decomposition method (e.g. Patel et al.
(2022)) perform worse than the fine-tuned No Decomp method? In Table 1 we show
representative errors from the decomposition approach with how often they
occurred. We randomly sample 50 instances of errors and categorize them into
three groups: wrong predictions in the last step, error propagation from
intermediate steps, and invalid/missing annotations from BREAK (i.e. not the
model’s fault). We found that the biggest category was predicting an invalid
annotation (42%), i.e. an alias that the dataset did not contain, followed by
error propagation (40%) and then wrong predictions (18%). Thus, compared to
the non-iterative methods, the iterative process allows error propagation that
occurs in roughly 40% of errors, contributing to its lower comparative
performance.

##  4 Related Work

#### Decompositions in QA

Decompositions for QA have a long history in complex question answering Perez
et al. (2020); Geva et al. (2021a) with recent interest in using them for
large language models Wei et al. (2022); Dua et al. (2022). Two of the most
related works to ours include Patel et al. (2022) who show improvements when
using decompositions in the zero-shot setting and concurrent work Guo et al.
(2022) that uses decompositions on the full DROP dataset (beyond BREAK). Note
that our reported no-decomposition results on DROP show comparable or greater
performance to Guo et al. (2022) when using only the BREAK dataset. Our work
complements these papers by showing when decompositions help w.r.t. data size.

#### Decompositions in other fields

Our understanding of decompositions in textual question answering helps to
unify results across machine learning fields, as similar results have been
shown in Computer Vision Hudson and Manning (2019); Amizadeh et al. (2020); Li
et al. (2021) and through results on similar visual QA leaderboards.

#### Decomposition Strategies and Prompting

Decompositions methods are also related to prompting, where the explicit
decompositions can be seen as a hard prompt Liu et al. (2021); Su et al.
(2022) and the implicit decompositions are similar to soft prompts Qin and
Eisner (2021); Liu et al. (2022).

##  5 Conclusion

Our work explored when decompositions are helpful for machine reading. We
showed that decompositions are helpful when there is limited data available,
or when parameters cannot be tuned. However, when enough data exists
(empirically around a few hundred instances) and parameters can be fine-tuned,
it is best to let the model learn the decompositions implicitly through end-
to-end training. We hope that our work will help to inform readers as they
create new datasets and select methods to use for complex question answering.

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##  Appendix A Hyperparameter and Compute Details

Models were trained with a default learning rate of 1e-5 for 5 epochs, using
early stopping on a holdout of the training set (5%) to determine the best
saved model. The best models were typically ones trained for around 2 epochs.

For the non-fine-tuned decomposition UnifiedQA-v2 approach (perhaps due to its
multi-task pre-training on other QA datasets), we found that using the
question plus the iterative decompositions added a solid boost to final
performance and hence we use those results. Each non-fine-tuned decomposition
run takes approximately 15-30 minutes to evaluate.

Compute time ranged from 1 hr for the shortest jobs with smaller data sizes
and non-iterative training to 18 hours for iterative decompositions methods
with UnifiedQA-v2 on 1 RTX 6000 GPU.

Data was prepared using the original BREAK authors’ code. Models were accessed
via the Huggingface repository Wolf et al. (2019).


# Attribution Methods in Asset Pricing: Do They Account for Risk?

Dangxing Chen 1 ∗ Corresponding author. Zu Chongzhi Center for Mathematics and
Computational Sciences
Duke Kunshan University
Kunshan, China
<EMAIL_ADDRESS>Yuan Gao Charlotte, U.S.
<EMAIL_ADDRESS>
###### Abstract

Over the past few decades, machine learning models have been extremely
successful. As a result of axiomatic attribution methods, feature
contributions have been explained more clearly and rigorously. There are,
however, few studies that have examined domain knowledge in conjunction with
the axioms. In this study, we examine asset pricing in finance, a field
closely related to risk management. Consequently, when applying machine
learning models, we must ensure that the attribution methods reflect the
underlying risks accurately. In this work, we present and study several axioms
derived from asset pricing domain knowledge. It is shown that while Shapley
value and Integrated Gradients preserve most axioms, neither can satisfy all
axioms. Using extensive analytical and empirical examples, we demonstrate how
attribution methods can reflect risks and when they should not be used.

###### Index Terms:

Attribution Methods, Risk Management, Explainability

##  I Introduction

Recent decades have seen a tremendous amount of success with machine learning
(ML). The use of ML increases the accuracy of traditional statistical models,
but often at the expense of transparency and explainability. In industries
with high stakes, such as the financial industry, explainability is essential.
In the model risk management
handbook111https://www.occ.treas.gov/publications-and-
resources/publications/comptrollers-handbook/files/model-risk-
management/index-model-risk-management.html recently released by the Office of
the Comptroller of the Currency, it is stressed that ML models must be
explained in a clear and concise manner when applied. This has led to
extensive research on explainable machine learning [18, 25, 15, 30].

Our work is focused on the attribution problem, i.e., how the ability to
ascribe a value to a base feature can be interpreted as its role in
predicting. A leading approach to attribution is based on the Shapley value by
[28], other popular methods include Integrated Gradients (IG) by [30]. An
elegant aspect of these approaches is the use of axioms as a guide. By
utilizing a few axioms, attribution methods can be uniquely determined.

In spite of the success of the axiomatic approach, previous studies focused
solely on the axioms of general models with no prior domain expertise.
However, domain knowledge has been extensively studied in science throughout
history. In recent studies, it has been demonstrated that when knowledge is
incorporated, ML models become more reasonable and could potentially achieve
higher accuracy [13, 8]. In this regard, we should not neglect the domain
knowledge implied in ML models when explaining them.

The knowledge required for different domains varies. This study focuses on
finance, a high-risk industry. Specifically, we focus on the issue of asset
pricing. In finance, risk management is crucial to making appropriate
decisions and avoiding significant financial losses. Inadequate risk
management could lead to catastrophic consequences. For example, we just
witnessed the collapse of Silicon Valley Bank (SVB), which is considered to be
the second-largest bank failure in the United States. Upon the failure of SVB,
a ripple effect was felt throughout the financial system, causing the stock
market to go into a panic. The improper management of risk is one of the major
causes of its failure, as outlined in [4]. In particular, the interest rate
risk has been neglected by SVB. In the short version, SVB’s portfolio was
extremely sensitive to interest rates, and the interest rate continued to
increase, resulting in tremendous losses. A careful risk management strategy
could have prevented the tragedy of SVB. In the wake of the failure of SVB, we
have learned a painful lesson that risk management is an extremely important
component of finance, and we should be aware of various risks on a regular
basis. As a result, we ask the following question: When applying attribution
methods to financial models, can attribution methods accurately capture the
risks implicit in the model?

We explore attribution problems for asset pricing in order to answer the above
question. We propose a number of axioms to reflect risks. These axioms are
carefully analyzed in relation to Shapley value and IG. Fortunately, both
attribution methods are capable of preserving most of the axioms. IG, however,
cannot maintain demand monotonicity, as if the model is monotonic in a certain
feature, then the attributions for that feature increase as its value
increases; Shapley value may involve calculations outside of the training
domain, which could present difficulties such as times of financial crisis.
Using extensive analytical and empirical examples, we demonstrate how
attribution methods can reflect risks and when their application might be
detrimental.

Our work has made the following contributions:

  * •

Several axioms are proposed based on different aspects for reflecting risk in
asset pricing.

  * •

Our examples demonstrate the motivations and significance of these axioms.

  * •

We provide a thorough analysis of Shapley value and IG, outlining their
advantages and disadvantages.

##  II Preliminaries

For problem setup, assume we have $n$ features. We denote a class of functions
$f:\mathbb{R}^{n}\rightarrow\mathbb{R}$ by $\mathcal{F}$. We assume
$\mathcal{F}$ to be the set of real analytic functions. For simplicity, we
omit the discussion of nonanalytic functions that can be approximated to
arbitrary precision by analytic functions.

###  II-A Attribution Methods

Following [20], we call the point of interest $\overline{\mathbf{x}}$ to
explain as an explicand and $\mathbf{x}^{\prime}$ a baseline. Without loss of
generality (WLOG), we assume $\overline{\mathbf{x}}\geq\mathbf{x}^{\prime}$.
The Baseline Attribution Method that interprets features’ importance is
defined below.

######  Definition II.1 (Baseline Attribution Method (BAM)).

Given $\overline{\mathbf{x}},\mathbf{x}^{\prime}\in\mathbb{R}^{n}$,
$f\in\mathcal{F}$, a baseline attribution method is any function of the form
$\mathbf{A}:\mathbb{R}^{n}\times\mathbb{R}^{n}\times\mathcal{F}\rightarrow\mathbb{R}^{n}$.

The Shapley value [27] takes as input a set function
$v:2^{N}\rightarrow\mathbb{R}$, which produces attributions $s_{i}$ for each
player $i\in N$ that add up to $v(N)$, where $N=\\{1,\dots,n\\}$.

######  Definition II.2 (Shapley value).

The Shapley value of a player $i$ is given by:

$\displaystyle s_{i}=\sum_{S\subseteq N\backslash
i}\frac{|S|!(|N|-|S|-1)!}{|N|!}(v(S\cup i)-v(S)).$ (1)

We focus on the Baseline Shapley (BShap) [29], in which

$\displaystyle
v(S)=f(\overline{\mathbf{x}}_{S};\mathbf{x}^{\prime}_{N\backslash S}).$ (2)

That is, baseline values replace the feature’s absence. We denote BShap
attribution by $\text{BS}_{i}(\overline{\mathbf{x}},\mathbf{x}^{\prime},f)$
and $\text{BS}_{i}$ sometimes. For example, suppose
$f(x_{1},x_{2})=x_{1}+x_{2}$,
$\overline{\mathbf{x}}=(\overline{x}_{1},\overline{x}_{2})$,
$\mathbf{x}^{\prime}=(0,0)$, and $S=\\{1\\}$, then we have
$v(S)=f(\overline{x}_{1},0)$. We focus on the BShap since it has better
theoretical properties than SHAP in terms of preserving axioms, as discussed
in [29]. As a result, we are more confident about using it for sectors with
high stakes.

Another popular method is the Integrated Gradients [30].

######  Definition II.3 (Integrated Gradients (IG)).

Given $\overline{\mathbf{x}},\mathbf{x}^{\prime}\in\mathbb{R}^{n}$ and
$f\in\mathcal{F}$, the Integrated Gradients attribution of the $i$-th
component of $\overline{\mathbf{x}}$ is defined as

$\displaystyle\text{IG}_{i}(\overline{\mathbf{x}},\mathbf{x}^{\prime},f)=(\overline{x}_{i}-x_{i}^{\prime})\int_{0}^{1}\frac{\partial
f}{\partial
x_{i}}\left(\mathbf{x}^{\prime}+t(\overline{\mathbf{x}}-\mathbf{x}^{\prime})\right)\
dt.$ (3)

For simplicity, we often use $\text{IG}_{i}$ for
$\text{IG}_{i}(\overline{\mathbf{x}},\mathbf{x}^{\prime},f)$.

##  III Asset Pricing

Asset pricing examines how financial assets are valued, such as stocks,
financial derivatives, and bonds. Since the Nobel Prize work by [6, 22],
stochastic differential equations with risk-neutral pricing have been used as
primary tools for pricing [1, 14, 31]. In recent years, ML models have been
increasingly used as an approximation of pricing formulas [3, 16]. These
studies have demonstrated that ML models offer advantages over classical
stochastic methods in terms of more flexible approximation formulas,
computational efficiency, and the ability to be data-driven with fewer
assumptions.

Explainability is crucial in sectors such as finance where stakes are high.
The application of ML models must be accompanied by an explanation. Existing
explainable ML tools have, however, focused primarily on general problems
without domain knowledge. As a result, explanations may be insufficient. In
the finance community, a considerable amount of attention is paid to
mathematical derivatives of pricing formulas, both first-order and higher-
order. Therefore, asset pricing considers mathematical derivatives to be
domain knowledge. Using mathematical derivatives, researchers and
practitioners have been able to gain a better understanding of financial
models. For this reason, we intend to incorporate mathematical derivatives
into our explanation of ML models. More specifically, BAMs must reflect risks
associated with sensitive features.

We provide two examples with discussions on their mathematical derivatives and
corresponding models under simplified assumptions, which will be used in the
remainder of the manuscript. By using these examples, we demonstrate how
researchers and practitioners interpret information derived from mathematical
derivatives.

###  III-A Coupon Bonds

A coupon is an interest payment received by a bondholder from the date of
issuance until the maturity date of the bond. Zero coupon bonds are the
simplest form of coupons. A zero coupon bond with a principal amount of $\$c$
and a maturity time $T$ will pay $\$c$ at $T$.

######  Example III.1.

Using continuous compounding, assume the interest rate is constant $r$, the
present value ($t=0$) of a zero coupon bond is calculated as follows:

$\displaystyle B(r,c,T)=ce^{-rT}.$ (4)

The first derivative of a bond based on its interest rate, $\frac{\partial
B}{\partial r}$, is known in finance as related to duration, introduced by
[21]. In general, the longer the duration, the more sensitive the bond price
is to changes in interest rates. A second-order derivative,
$\frac{\partial^{2}B}{\partial r^{2}}$ is known in finance as related to
convexity by [9]. Convexity is usually preferred as it reduces sensitivity to
interest rates.

###  III-B Option Pricing

Option contracts convey to their owners, the holders, the right, but not the
obligation, to purchase or sell a specified quantity of an underlying asset or
instrument at a specified strike price on or before a prescribed date.
European call options are a classic example.

######  Example III.2.

A call option is a contract between the buyer and the seller to exchange a
security at a strike price $K$ at a maturity time $T$. At time $T$, if the
stock price $S_{T}$ exceeds the strike price, then the option will be
exercised and the payoff will be $S_{T}-K$; otherwise, the option will not be
exercised and the payoff will be $0$. In summary, the value of the call option
at time $T$ is equal to $C(S_{T},K,T)=(S_{T}-K)^{+}$. In option pricing, the
key question is: What is the present value of an option at time $t=0$?

Based on a couple of assumptions, Black, Scholes, and Merton [6, 22] developed
a pricing formula $C(S_{0},K,T,\sigma,r)$, where $\sigma$ represents constant
volatility and $r$ represents the constant risk-free interest rate.

In option pricing, mathematical derivatives are referred to as Greeks [23].
Financial institutions typically set (risk) limits for each of the Greeks that
their traders are not permitted to exceed [17]. Suppose we denote $V$ as the
price of a general option, some common first-order Greeks include Delta, which
is calculated as $\frac{\partial V}{\partial S}$ to measure the sensitivity of
the asset price to the option price, and Vega, which is calculated as
$\frac{\partial V}{\partial\sigma}$ to measure the sensitivity of the
volatility to the option price. Greeks of higher order are also commonly
considered. Examples include Gamma $\frac{\partial^{2}V}{\partial S^{2}}$ and
Vanna $\frac{\partial^{2}V}{\partial S\partial\sigma}$. Based on the domain
knowledge, practitioners often are familiar with the implications of Greeks in
a wide range of situations, so it is crucial to incorporate this knowledge
into explanations. In the case of a call option, a positive Delta
$\left(\frac{\partial V}{\partial S}\right)$ implies that an increase in the
underlying asset price should result in an increase in the option price. As a
result, if BAMs are applied, they must provide consistent explanations. More
details about Greeks can be found in Appendix A-A.

##  IV Axioms From Domain Knowledge

###  IV-A Axioms on First-order Effects

Monotonicity can be used to reflect first-order effects in asset pricing.
WLOG, we assume that all monotonic features are monotonically increasing
throughout the paper. Suppose $\boldsymbol{\alpha}$ is the set of all
individual monotonic features and $\neg\boldsymbol{\alpha}$ its complement,
then the input $\mathbf{x}$ can be partitioned into
$\mathbf{x}=(\mathbf{x}_{\boldsymbol{\alpha}},\mathbf{x}_{\neg\boldsymbol{\alpha}})$.
Individual monotonicity is defined as follows.

######  Definition IV.1 (Individual Monotonicity).

$f$ is individually monotonic with respect to
$\mathbf{x}_{\boldsymbol{\alpha}}$ if $\forall\mathbf{x},\mathbf{x}^{*}\text{
s.t.
}\mathbf{x}_{\boldsymbol{\alpha}}\leq\mathbf{x}^{*}_{\boldsymbol{\alpha}}$

$\displaystyle
f(\mathbf{x})=f(\mathbf{x}_{\boldsymbol{\alpha}},\mathbf{x}_{\neg\boldsymbol{\alpha}})\leq
f(\mathbf{x}^{*}_{\boldsymbol{\alpha}},\mathbf{x}_{\neg\boldsymbol{\alpha}})=f(\mathbf{x}^{*}).$
(5)

where
$\mathbf{x}_{\boldsymbol{\alpha}}\leq\mathbf{x}_{\boldsymbol{\alpha}}^{*}$
means $x_{\alpha_{i}}\leq x_{\alpha_{i}}^{*},\forall{i}$.

As discussed by [7], applying BAMs to individually monotonic features should
result in positive attributions.

######  Definition IV.2 (Average Individual Monotonicity (AIM) Axiom).

Suppose $f$ is individually monotonic with respect to $x_{\alpha}$, then we
say a BAM preserves average individual monotonicity if
$\forall\overline{\mathbf{x}}\text{ s.t.
}\overline{\mathbf{x}}\geq\mathbf{x}^{\prime}$, we have

$\displaystyle A_{\alpha}(\overline{\mathbf{x}},\mathbf{x}^{\prime},f)\geq 0.$
(6)

######  Example IV.3.

In Example III.1, the interest rate should attribute negatively.

We expect in some situations that individual monotonicity will have a greater
impact in that, whenever a feature is increased, its attribution should be
increased accordingly. [11] introduced the concept of demand individual
monotonicity for this situation.

######  Definition IV.4 (Demand Individual Monotonicity (DIM) Axiom).

Consider two explicands
$\overline{\mathbf{x}}=(\overline{x}_{1},\dots,\overline{x}_{n})$ and
$\mathbf{x}^{*}=(\overline{x}_{1},\dots,\overline{x}_{\alpha}+c,\dots,\overline{x}_{n})$,
where $c>0$. Suppose $f$ is individually monotonic with respect to
$x_{\alpha}$. We say a BAM preserves demand individual monotonicity if
$\forall\overline{\mathbf{x}}\text{ s.t.
}\overline{\mathbf{x}}\geq\mathbf{x}^{\prime}$,

$\displaystyle A_{\alpha}(\mathbf{x}^{*},\mathbf{x}^{\prime},f)\geq
A_{\alpha}(\overline{\mathbf{x}},\mathbf{x}^{\prime},f).$ (7)

######  Example IV.5.

In Example III.1, any additional increase in the interest rate should always
result in the decrease of a zero-coupon bond, regardless of its principal.

###  IV-B Axioms on Second-order Main Effects

Second-order derivatives may provide additional insights into the model’s
behavior. The diminishing marginal effect is one of the most common phenomena
[12, 24].

######  Definition IV.6 (Diminishing Marginal Effect (DME)).

Suppose $\mathbf{x}=(x_{\alpha},\mathbf{x}_{\neg\alpha})$. We say $f$ has the
diminishing marginal effect with respect to $x_{\alpha}$ if
$\frac{\partial}{\partial x_{\alpha}}f(x_{\alpha},\mathbf{x}_{\neg\alpha})\geq
0$ and $\frac{\partial^{2}}{\partial
x_{\alpha}^{2}}f(x_{\alpha},\mathbf{x}_{\neg\alpha})\leq 0$. Similarly, we say
$f$ has the reverse DME (RDME) if $\frac{\partial}{\partial
x_{\alpha}}f(x_{\alpha},\mathbf{x}_{\neg\alpha})\leq 0$ and
$\frac{\partial^{2}}{\partial
x_{\alpha}^{2}}f(x_{\alpha},\mathbf{x}_{\neg\alpha})\geq 0$.

Basically, DME implies a slowing in the increment of the function. This
motivates us to propose the following axiom.

######  Definition IV.7 (Diminishing Marginal Axiom).

Suppose $f$ has the diminishing marginal effect with respect to $x_{\alpha}$,
then we say a BAM preserves DME if for $\overline{x}_{\alpha}\geq
x^{*}_{\alpha}>x_{\alpha}^{\prime}$, we have

$\displaystyle\frac{A_{\alpha}((\overline{x}_{\alpha},\overline{\mathbf{x}}_{\neg\alpha}),\mathbf{x}^{\prime},f)}{\overline{x}_{\alpha}-x_{\alpha}^{\prime}}\leq\frac{A_{\alpha}((x^{*}_{\alpha},\overline{\mathbf{x}}_{\neg\alpha}),\mathbf{x}^{\prime},f)}{x^{*}_{\alpha}-x_{\alpha}^{\prime}}.$
(8)

A BAM preserves reverse RDME if

$\displaystyle\frac{A_{\alpha}((\overline{x}_{\alpha},\overline{\mathbf{x}}_{\neg\alpha}),\mathbf{x}^{\prime},f)}{\overline{x}_{\alpha}-x_{\alpha}^{\prime}}\geq\frac{A_{\alpha}((x^{*}_{\alpha},\overline{\mathbf{x}}_{\neg\alpha}),\mathbf{x}^{\prime},f)}{x^{*}_{\alpha}-x_{\alpha}^{\prime}}.$
(9)

######  Example IV.8.

In Example III.1, the bond has the RDME with respect to the interest rate. The
bond benefits from such complexity, as it prevents huge losses when interest
rates rise significantly. RDME axiom can, therefore, be used to reflect
convexity in bonds.

Similarly, we define increasing marginal effects.

######  Definition IV.9 (Increasing Marginal Effect (IME)).

Suppose $\mathbf{x}=(x_{\alpha},\mathbf{x}_{\neg\alpha})$. We say $f$ has the
increasing marginal effect with respect to $x_{\alpha}$ if
$\frac{\partial}{\partial x_{\alpha}}f(x_{\alpha},\mathbf{x}_{\neg\alpha})\geq
0$ and $\frac{\partial^{2}}{\partial
x_{\alpha}^{2}}f(x_{\alpha},\mathbf{x}_{\neg\alpha})\geq 0$.

######  Definition IV.10 (Increasing Marginal Axiom).

Suppose $f$ has the increasing marginal effect with respect to $x_{\alpha}$,
then we say a BAM preserves IME if for
$\overline{x}_{\alpha}>x^{*}_{\alpha}>x_{\alpha}^{\prime}$, we have

$\displaystyle\frac{A_{\alpha}((\overline{x}_{\alpha},\overline{\mathbf{x}}_{\neg\alpha}),\mathbf{x}^{\prime},f)}{\overline{x}_{\alpha}-x_{\alpha}^{\prime}}\geq\frac{A_{\alpha}((x^{*}_{\alpha},\overline{\mathbf{x}}_{\neg\alpha}),\mathbf{x}^{\prime},f)}{x^{*}_{\alpha}-x_{\alpha}^{\prime}}.$
(10)

###  IV-C Axioms on Comparing Assets

The investor may wish to compare different assets before making a decision.
Bonds as well as options discussed in Section III for example, may be
considered together. As a result, different assets are involved, and their
features may differ. Nevertheless, they share many common features, such as
the market interest rate and volatility. If investors are concerned about the
potential increase in interest rates, they may find it useful to compare the
sensitivity of these assets. Therefore, we should be able to determine from
BAMs if a particular product is always more sensitive to interest rates than
another. This would allow investors to have a clear understanding of the risks
associated with different assets.

Consider
$\mathbf{x}=(x_{\alpha},\mathbf{x}_{\boldsymbol{\beta}},\mathbf{x}_{\neg})$
and
$\mathbf{y}=(y_{\alpha},\mathbf{y}_{\boldsymbol{\beta}},\mathbf{y}_{\neg})$.
That is, $\mathbf{x}$ and $\mathbf{y}$ have the same features of $\alpha$ and
$\boldsymbol{\beta}$, but not necessarily the others, and we are mostly
interested in the impact on $x_{\alpha}$. Note $\mathbf{x}_{\neg}$ and
$\mathbf{y}_{\neg}$ might have different dimensions.

######  Definition IV.11 (First-order Monotonic Dominance (FMD)).

Suppose we have two functions $f(\mathbf{x})$ and $g(\mathbf{y})$. We say $f$
dominates $g$ with respect to $x_{\alpha}$ for the first-order if $\forall
x_{\alpha}=y_{\alpha},\forall\mathbf{x}_{\boldsymbol{\beta}}=\mathbf{y}_{\boldsymbol{\beta}},\forall\mathbf{x}_{\neg},\mathbf{y}_{\neg}$,

$\displaystyle\frac{\partial}{\partial
x_{\alpha}}f(x_{\alpha},\mathbf{x}_{\boldsymbol{\beta}},\mathbf{x}_{\neg})\geq\frac{\partial}{\partial
y_{\alpha}}g(y_{\alpha},\mathbf{y}_{\boldsymbol{\beta}},\mathbf{y}_{\neg}).$
(11)

######  Definition IV.12 (First-order Monotonic Dominance Axiom).

Suppose $f$ dominates $g$ with respect to $x_{\alpha}$, then we say a BAM
preserves monotonic dominance for the first-order if for two explicands
$\overline{\mathbf{x}},\overline{\mathbf{y}}$ such that
$\overline{\mathbf{x}}_{\boldsymbol{\beta}}=\overline{\mathbf{y}}_{\boldsymbol{\beta}},\overline{\mathbf{x}}_{\boldsymbol{\beta}}^{\prime}=\overline{\mathbf{y}}_{\boldsymbol{\beta}}^{\prime},\overline{x}_{\alpha}=\overline{y}_{\alpha},\overline{x}_{\alpha}^{\prime}=\overline{y}_{\alpha}^{\prime}$,
we have

$\displaystyle A_{\alpha}(\overline{\mathbf{x}},\mathbf{x}^{\prime},f)\geq
A_{\alpha}(\overline{\mathbf{y}},\mathbf{y}^{\prime},g).$ (12)

Interestingly, FMD can be preserved if other axioms are maintained. This
requires the introduction of a new axiom.

######  Definition IV.13 (Generalized Dummy (GD)).

We say a BAM preserves dummy if $\forall f\in\mathcal{F}$, if
$f(\mathbf{x})=f(\mathbf{x}^{*})$, where $(\mathbf{x}_{*})_{j}=\mathbf{x}_{j}$
except for $i$ for all $\mathbf{x},\mathbf{x}^{*}$, then
$A_{i}(\overline{\mathbf{x}},\mathbf{x}^{\prime},f)=0$. Furthermore, let
$\mathbf{h}(\mathbf{x})=(x_{1},\dots,x_{i-1},x_{i+1},x_{n})$ and let $g$ be a
reduction of $f$ omitting dummy features. With loss of generality, let
$g(x_{1},\dots,x_{i-1},x_{i+1},\dots,x_{n})=f(x_{1},\dots,x_{i-1},0,x_{i+1},\dots,x_{n})$,
then we say a BAM preserves generalized dummy if
$A_{j}(\mathbf{h}(\overline{\mathbf{x}}),\mathbf{h}(\mathbf{x}^{\prime}),g)=A_{j}(\overline{\mathbf{x}},\mathbf{x}^{\prime},f)$
for $j\neq i$.

######  Theorem IV.14.

If a BAM preserves linearity, generalized dummy, and AIM, then it preserves
FMD.

######  Remark IV.15 (Hedge).

The FMD imposes implications for hedging. If there are two assets A and B, and
A is more sensitive to interest rates than B, then a BAM that preserves FMD
could reflect this. Furthermore, if we long A and short B, then the BAM would
be able to indicate that this strategy is still associated with a positive
risk of interest rate.

###  IV-D Axioms on the Domain

Last but not least, we present an axiom unrelated to mathematical derivatives.
In finance, certain characteristics, such as stock prices and volatility, have
a random nature. Because of this, even though their domains may be large, many
events may only occur with a low probability. ML models cannot accurately
predict such events if they have not been observed. As a rough guide, we can
divide the domain into training and out-of-training domains based on the
realization of the data. Furthermore, features are often correlated with each
other. As a result, features are not necessarily distributed equally among
themselves. When applying BAMs, we must ensure that functions are not
evaluated outside the training domain.

Consider Example III.2, for which we are interested in the option price based
on stock prices, volatility, interest rates, strike prices, and maturity
dates. Although the current market liquidity allows us to collect large
amounts of data with different strikes and maturity dates, stock prices and
volatility can only be observed over time. Thus, we may only have limited data
regarding stock prices and volatility, which are highly correlated. Therefore,
it is crucial to identify their training domain. We illustrate our point with
the example below.

We study S$\&$P 500 data during the 2008 financial crisis. More details can be
found in Appendix A-B and A-D. Imagine that we have a model $f$ for option
prices and we would like to determine the causes of the significant changes in
prices before and during the market crash. VIX is used as the approximation
for the volatility, with details in Appendix A-B. As an example, on August 1,
2008, we consider the baseline point $\mathbf{x}^{\prime}$ with the stock
price was about 1270 and the volatility was about 0.23, suggesting the market
was in a normal state; on November 21, 2008, we consider the explicand
$\overline{\mathbf{x}}$ with the volatility hit 0.81 and the stock price
plummeted to 756, suggesting the market is panicking. If we would like to
investigate the impact of stock prices and volatility, we will need to
identify the training domain. It is important to note that we did not train
the model in the entire rectangle determined by
$(S^{\prime},\sigma^{\prime}),(S^{\prime},\sigma),(S,\sigma^{\prime}),(S,\sigma)$,
as shown in Figure 1. Particularly, we do not have data on high stock prices
with high volatility or low stock prices with low volatility. It has been
consistently observed empirically that changes in an asset’s volatility are
negatively correlated with its return. In finance, this phenomenon is known as
the leverage effect [2, 5]. Therefore, BAMs should avoid using function values
in these out-of-training areas. Motivated by this, we propose the following
axiom regarding the geometry of a domain.

######  Definition IV.16 (Convex Geometry Axiom).

Suppose $\mathcal{X}$ is the training domain of $f(\mathbf{x})$. We say a BAM
preserves convex geometry (CG) if $\mathcal{X}$ is convex,
$\overline{\mathbf{x}},\mathbf{x}^{\prime}\in\mathcal{X}$, then the
calculation of $\mathbf{A}(\overline{\mathbf{x}},\mathbf{x}^{\prime},f)$ only
requires $f(\mathcal{X})$.

Figure 1: Stock price vs VIX

##  V Results for BShap and IG

The results of BShap and IG for axioms are summarized and compared in detail.
Proofs are left in Appendix A-C.

######  Theorem V.1.

BShap preserves average individual monotonicity (AIM), demand individual
monotonicity (DIM), (reverse) diminishing marginal effect ((R)DME), increasing
marginal effect (IME), and first-order monotonic dominance (FMD).

######  Theorem V.2.

IG preserves average individual monotonicity (AIM), (reverse) diminishing
marginal effect ((R)DME), increasing marginal effect (IME), first-order
monotonic dominance (FMD), and convex geometry (CG).

As a result of the comparison, IG fails to preserve the DIM, which may be one
of its greatest weaknesses. We provide a detailed example below.

######  Example V.3.

Consider Example III.1. Suppose we are interested in the explanation in terms
of $r$ and the baseline is $\mathbf{x}^{\prime}=\mathbf{0}$. By calculation,
we have
$\text{IG}_{\overline{r}}=\overline{c}\left(e^{-\overline{r}t}+\frac{e^{-\overline{r}t}}{\overline{r}t}-\frac{1}{\overline{r}t}\right).$
For $\overline{r}t\sim 0$, we have
$IG_{\overline{r}}\sim-\frac{\overline{c}\overline{r}t}{2}.$ As a result, DIM
is preserved by IG. However, for $\overline{r}t\rightarrow\infty$, we have
$\text{IG}_{\overline{r}}\sim-\frac{\overline{c}}{\overline{r}t},$ whereas the
DIM is violated. A similar event could occur for long-term bonds with a very
high interest rate. As an example, inflation in the U.S. reached an extremely
high level during the 1980s. The bond price decreases because of the rising
interest rate but might not be reflected by IG.

We examine when IG is capable of preserving DIM.

######  Theorem V.4.

If $f$ has the IME with respect to $x_{\alpha}$, then IG preserves DIM.

######  Example V.5.

Consider the problem of option pricing in Section III-B. As a reminder, Delta
and Gamma are the first derivative and second derivative of option prices with
respect to stock prices, respectively. According to the BSM, they are both
positive for the call options. Thus, DIM can be preserved for the stock price
by IG for call options.

In the case of BShap, its main weakness is the inability to preserve CG.

######  Example V.6.

BShap doesn’t preserve CG. Consider the example provided in Section IV-D. When
applying BShap to investigate the attributions of stock prices and volatility
before and during the market crash, we are required to provide
$f(S^{\prime},v^{\prime})$, $f(S^{\prime},v)$, $f(S,v^{\prime})$, and
$f(S,v)$. $f(S^{\prime},v)$ and $f(S,v^{\prime})$ are, however, outside the
training domain due to the leverage effect, as shown in Figure 1. In contrast,
the IG path appears to be a reasonable choice since there is some data closing
to it. Therefore, BShap may not be the best option in this situation.

##  VI Empirical Results

###  VI-A Option Pricing

We present an empirical study of option pricing in 2008 in the U.S., which is
considered the worst financial crisis of the 21st century. This period is used
to illustrate the results under extreme circumstances, but other periods can
also be analyzed similarly. We collect transaction data for European call and
put options. Models are based on five features, namely underlying stock
(index) prices $S$, risk-free interest rates $r$, time to expiration $\tau$,
strike prices $K$, and volatility $\sigma$ such that
$\mathbf{x}=(S,r,\tau,K,\sigma)$. For the sake of simplicity, we concentrate
on these features, since they are regarded as the most important factors when
pricing options. However, it may be possible to incorporate more features,
such as alternative data, in order to improve the model’s performance. These
five features are applied to neural networks. A detailed description of the
data and model setup is provided in Appendix A-D.

####  VI-A1 An Example of Explanations

We use the baseline point of

$\displaystyle\mathbf{x}^{\prime}=\left[\begin{matrix}1433.8&4.26&0.59&1396&0.23\end{matrix}\right],$
(13)

which is the average statistics on the first date for call options. It should
be noted that this is not a unique choice, we only use it as a demonstration.
We wish to explain the explicand

$\displaystyle\overline{\mathbf{x}}=\left[\begin{matrix}1344.8&3.09&0.2&1150&0.27\end{matrix}\right],$
(14)

which is a call option on February 5. As of this date, the market remains
relatively calm. IG and BShap results are shown in Figure 3 and 3. Overall,
our explanations are qualitatively similar. The major attributions are based
on the difference in stock prices and strike prices, as expected. As a result
of the shorter expiration date, stock prices are less likely to change, which
has a negative and significant impact. Interest rates have only a small
impact. Interest rates are usually not attributed to stock prices
significantly, as stock prices are much more volatile than interest rates. As
volatility has not changed too much, it is expected that attributions will be
small.

Figure 2: IG for Eq 13

Figure 3: BShap for Eq 13

####  VI-A2 Preservation/Violation of Risk Patterns

Risk patterns of option pricing could also be observed. In this example, we
vary the stock price from 1300 to 1500 and fix other parameters in (14) as
well as the baseline (13). Based on the finance theory, when it comes to Delta
$\frac{\partial V}{\partial S}$, it is positive for call options and negative
for put options. Accordingly, we observe consistent explanations (Definition
IV.10) in Figure 6. Gamma $\frac{\partial^{2}V}{\partial S^{2}}$ are positive
for both call and put options. Thus, we observe both convexity (Definition
IV.7) for IG and BShap for a put option in Figure 6. The results validated the
Theorem V.1, V.2, and demonstrated the potential of both IG and BShap in
preserving risk patterns. It is also possible to observe other risk patterns.

Figure 4: Preservation of IME

Figure 5: Preservation of RDME

Figure 6: Violation of DIM by IG

However, as discussed in Section V, not all risk patterns are preserved by IG.
As an example, we look at a put option with negative Delta $\frac{\partial
V}{\partial S}$ and positive Gamma $\frac{\partial^{2}V}{\partial S^{2}}$. Our
analysis indicates that DIM (Definition IV.4) is preserved for BShap, but not
necessarily for IG. As an example, we consider the baseline point of

$\displaystyle\mathbf{x}^{\prime}=\left[\begin{matrix}1250&4.2&0.3&1240&0.25\end{matrix}\right],$
(15)

an explicand

$\displaystyle\overline{\mathbf{x}}=\left[\begin{matrix}1300&4.0&0.3&1300&0.4\end{matrix}\right],$
(16)

and we vary stock prices between 1220 and 1320. The risk pattern is plotted in
Figure 6. DIM for IG is violated here since the option exhibits large Gamma at
the money ($S$ close to $K$). BShap is preferred over IG in such cases.

##  VII Discussion and Future Work

Our analysis indicates that BShap and IG are able to reflect risks in most
situations when it comes to attribution problems. However, there are certain
situations in which we should be more cautious. First, IG does not preserve
DIM in general, which is why BShap is preferred when such a risk pattern is
crucial to an application. Second, BShap does not preserve CG, which is why it
should not be used when out-of-training samples are required for calculations,
such as during periods of financial crises.

It would be interesting, in light of the limitations, to explore other
attribution methods that could preserve desired axioms in the future. It might
be possible, for example, to further incorporate the time series property to
avoid out-of-training samples when dealing with time series data. In a
different direction, it would be interesting to examine risk attribution with
applications such as portfolio analysis [26].

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##  Appendix A Option Pricing

###  A-A Greeks

Here, we provide information regarding some commonly used Greeks. We denote
$V$ as the price of a general option.

  * •

Delta: $\frac{\partial V}{\partial S}$. Positive for a call and negative for a
put.

  * •

Vega: $\frac{\partial V}{\partial\sigma}$. Vega is always positive for both
call and put options as larger volatility implies more possibilities of stock
prices.

  * •

Rho: $\frac{\partial V}{\partial r}$. Rho is positive for a call option and
negative for a put option. Rho is less sensitive than others since stock
prices are much more volatile than interest rates.

  * •

Gamma: $\frac{\partial^{2}V}{\partial S^{2}}$. Positive for both call and put
options.

  * •

Vomma: $\frac{\partial^{2}V}{\partial\sigma^{2}}$. Positive for both call and
put options.

###  A-B VIX

VIX is the ticker symbol for the Chicago Board Options Exchange’s CBOE
Volatility Index. It is used as a model-free measure of the stock market’s
expectation of volatility based on S$\&$P 500 index options. The VIX we use is
the 30-day expected volatility of the S$\&$P 500 index, more precisely the
square root of a 30-day expected realized variance of the index. It is
calculated as a weighted average of out-of-the-money calls and put options on
the S$\&$P 500,

$\displaystyle\text{VIX}=\sqrt{\frac{2e^{r\tau}}{\tau}\left(\int_{0}^{F}\frac{P(K)}{K^{2}}\
dK+\int_{F}^{\infty}\frac{C(K)}{K^{2}}\ dK\right)},$ (17)

where $F$ is the 30-day forward price on the S$\&$P 500. More details about
the calculation can be found in [32].

###  A-C Proofs

###### Proof of Theorem IV.14.

Consider $\mathbf{z}$ that generalize $\mathbf{x}$ and $\mathbf{y}$ and new
functions $\widetilde{f}(\mathbf{z})$ and $\widetilde{g}(\mathbf{z})$ whereas
features that are not used as dummy features. Due to the generalized dummy
axiom, we know feature attributions are the same for new functions. Let
$h(\mathbf{z})=\widetilde{f}(\mathbf{z})-\widetilde{g}(\mathbf{z})$, due to
the linearity and dummy axioms, we have

$\displaystyle A_{\alpha}(\overline{\mathbf{z}},\mathbf{z}^{\prime},h)$
$\displaystyle=A_{\alpha}(\overline{\mathbf{z}},\mathbf{z}^{\prime},\widetilde{f})-A_{\alpha}(\overline{\mathbf{z}},\mathbf{z}^{\prime},\widetilde{g})$
$\displaystyle=A_{\alpha}(\overline{\mathbf{x}},\mathbf{x}^{\prime},f)-A_{\alpha}(\overline{\mathbf{y}},\mathbf{y}^{\prime},g).$

Due to AIM, we have
$A_{\alpha}(\overline{\mathbf{z}},\mathbf{z}^{\prime},h)\geq 0.$ Thus, we
conclude. ∎

######  Lemma A.1.

BShap preserves AIM and DIM.

###### Proof.

Proof in [7]. ∎

######  Lemma A.2.

BShap preserves DME, RDME, and IME.

###### Proof.

We prove the results for DME, the reverse case and IME are similar. Suppose
$f$ has the DME with respect to $x_{\alpha}$, then $\frac{\partial}{\partial
x_{\alpha}}f(x_{\alpha},\mathbf{x}_{\neg\alpha})$ is monotonically decreasing
with respect to $x_{\alpha}$. Suppose we have explicands
$\overline{\mathbf{x}}=(\overline{x}_{\alpha},\overline{\mathbf{x}}_{\neg})$
and $\mathbf{x}^{*}=(x_{\alpha}^{*},\overline{\mathbf{x}}_{\neg})$. WLOG,
suppose $\mathbf{x}^{\prime}=\mathbf{0}$ (BShap preserves affine
transformation [29]) and $f(\mathbf{x}^{\prime})=0$. By the mean value
theorem, we have

$\displaystyle\frac{\partial}{\partial
x_{\alpha}}f(c,\overline{\mathbf{x}}_{\neg\alpha})$
$\displaystyle=\frac{f(\overline{x}_{\alpha},\overline{\mathbf{x}}_{\neg\alpha})-f(x^{*}_{\alpha},\overline{\mathbf{x}}_{\neg\alpha})}{\overline{x}_{\alpha}-x_{\alpha}^{*}},$
$\displaystyle\frac{\partial}{\partial
x_{\alpha}}f(d,\overline{\mathbf{x}}_{\neg\alpha})$
$\displaystyle=\frac{f(x_{\alpha}^{*},\overline{\mathbf{x}}_{\neg\alpha})-0}{x_{\alpha}^{*}-0},$

where $\overline{x}_{\alpha}>c>x_{\alpha}^{*}>d>0$. Since
$\frac{\partial}{\alpha x_{\alpha}}f(x_{\alpha},\mathbf{x}_{\neg})$ is
monotonically decreasing, $\frac{\partial}{\partial
x_{\alpha}}f(c,\mathbf{x}_{\neg\alpha})\leq\frac{\partial}{\partial
x_{\alpha}}f(d,\mathbf{x}_{\neg\alpha})$. Therefore,

$\displaystyle\frac{f(\overline{x}_{\alpha},\overline{\mathbf{x}}_{\neg\alpha})-f(x^{*}_{\alpha},\overline{\mathbf{x}}_{\neg\alpha})}{\overline{x}_{\alpha}-x_{\alpha}^{*}}\leq\frac{f(x_{\alpha}^{*},\overline{\mathbf{x}}_{\neg\alpha})}{x_{\alpha}^{*}}.$

Now in the calculation of Shapley value, we have

$\displaystyle\frac{s_{\alpha}-s_{\alpha}^{*}}{\overline{x}_{\alpha}-x_{\alpha}^{*}}$
$\displaystyle=\sum_{S\subseteq
N\backslash\alpha}\frac{|S|!(|N|-|S|-1)!}{N!}\frac{v(S\cup\alpha)-v^{*}(S\cup\alpha)}{\overline{x}_{\alpha}-x_{\alpha}^{*}}$
$\displaystyle\leq\sum_{S\subseteq
N\backslash\alpha}\frac{|S|!(|N|-|S|-1)!}{N!}\frac{f(x^{*}_{\alpha},\overline{x}_{S};\mathbf{x}^{\prime}_{N\backslash(S\cup\alpha))})}{x_{\alpha}^{*}}$
$\displaystyle=\sum_{S\subseteq
N\backslash\alpha}\frac{|S|!(|N|-|S|-1)!}{N!}\frac{v^{*}(S\cup\alpha)}{x_{\alpha}^{*}}=\frac{s_{\alpha}^{*}}{x_{\alpha}^{*}}.$

This implies that

$\displaystyle\frac{s_{\alpha}}{\overline{x}_{\alpha}}\leq\frac{s_{\alpha}^{*}}{x_{\alpha}^{*}}.$

Thus, we conclude. ∎

######  Lemma A.3.

BShap preserves FMD.

###### Proof.

The proof is followed by the preservation of linearity and the generalized
dummy of Shapley Values and Theorem IV.14. For the generalized dummy, Shapley
value collects the marginal contribution of all orders of players. Suppose now
there is an additional dummy feature, the presence of dummy features can be
removed without affecting the calculation. Therefore, the calculation of non-
dummy features is the same as the game omitting the dummy feature. ∎

###### Proof of Theorem V.1.

By Lemmas A.1, A.2, A.3. ∎

######  Lemma A.4.

IG preserves AIM, DME, RDME, and IME.

###### Proof.

Proof for the AIM is in [7]. We prove the result for DME, the reverse case and
IME case are similar. WLOG, suppose $\mathbf{x}^{\prime}=\mathbf{0}$ (IG
preserves affine transformation [19]). We calculate that

$\displaystyle\frac{1}{\overline{x}_{\alpha}}\text{IG}_{\alpha}(\overline{\mathbf{x}})-\frac{1}{x_{\alpha}^{*}}\text{IG}_{\alpha}(\mathbf{x}^{*})$
$\displaystyle=\int_{0}^{1}\frac{\partial}{\partial
x_{\alpha}}f(t\overline{x}_{\alpha},t\overline{\mathbf{x}}_{\neg\alpha})-\frac{\partial}{\partial
x_{\alpha}}f(tx_{\alpha}^{*},t\overline{\mathbf{x}}_{\neg\alpha})\ dt.$

Since

$\displaystyle\frac{\partial}{\partial
x_{\alpha}}f(t\overline{x}_{\alpha},t\overline{\mathbf{x}}_{\neg\alpha})-\frac{\partial}{\partial
x_{\alpha}}f(tx_{\alpha}^{*},t\overline{\mathbf{x}}_{\neg\alpha})\leq
0,\forall t\in[0,1]$

by mean-value theorem. We conclude. ∎

######  Lemma A.5.

IG preserves FMD.

###### Proof.

The proof is followed by the preservation of linearity and the generalized
dummy of IG and Theorem IV.14. For generalized dummy, since
$g(\mathbf{h}(\mathbf{x}))=f(\mathbf{x})$, for $j\neq i$, by chain rule, we
have

$\displaystyle\frac{\partial g}{\partial h_{j}}\frac{\partial h_{j}}{\partial
x_{j}}=\frac{\partial f}{\partial x_{j}}\Rightarrow\frac{\partial g}{\partial
h_{j}}=\frac{\partial f}{\partial x_{j}}.$

∎

######  Lemma A.6.

IG preserves CG.

###### Proof.

By Definition. ∎

###### Proof of Theorem V.2.

By Lemma A.4, A.5, A.6. ∎

###### Proof of Theorem V.4.

Since $x_{\alpha}^{*}\geq x_{\alpha}$ and $\frac{\partial f}{\partial
x_{\alpha}}\geq 0$, we have

$\displaystyle\text{IG}_{\alpha}(\mathbf{x}^{*},\mathbf{x}^{\prime},f)-\text{IG}_{\alpha}(\overline{\mathbf{x}},\mathbf{x}^{\prime},f)\geq$
$\displaystyle(\overline{x}_{\alpha}-x_{\alpha}^{\prime})\int_{0}^{1}\frac{\partial
f}{\partial
x_{\alpha}}(\mathbf{x}^{\prime}+t(\mathbf{x}^{*}-\mathbf{x}^{\prime}))-\frac{\partial
f}{\partial
x_{\alpha}}(\mathbf{x}^{\prime}+t(\overline{\mathbf{x}}-\mathbf{x}^{\prime}))\
dt.$

By mean-value theorem and $\frac{\partial^{2}}{\partial x_{\alpha}^{2}}f\geq
0$, we have

$\displaystyle\frac{\partial f}{\partial
x_{\alpha}}(\mathbf{x}^{\prime}+t(\mathbf{x}^{*}-\mathbf{x}^{\prime}))\geq\frac{\partial
f}{\partial
x_{\alpha}}(\mathbf{x}^{\prime}+t(\overline{\mathbf{x}}-\mathbf{x}^{\prime})),\
\ \forall t\in[0,1].$

Thus, we conclude.

∎

###  A-D Data and Neural Networks Setup for Option Pricing

We collected call and option data from Wharton Research Data Services in 2008.
There are 253 trading days with 123969 records of option data in total. The
London Interbank Offered Rate (LIBOR) is used to represent risk-free interest
rates. The LIBOR served as the benchmark interest rate at which major global
banks lent to one another in the international interbank market for short-term
loans. This key benchmark interest rate served as an indication of borrowing
costs between banks throughout the world. Daily volatility is measured by the
VIX index. At each date $i$, we collect option data as
$V(S_{i,j},r_{i,j},T_{i,j},K_{i,j},\sigma_{i})$. Here are more explanations.
On each date, options data with different stock prices, strike prices, and
maturity dates are collected. Interest rates fluctuate over time. In addition,
interest yields vary for different maturity times as a result of risk
premiums. During a given day, volatility is assumed to be constant since the
calculation of VIX requires access to a large number of option prices and is
not possible to measure instantly.

For neural networks, we use the architecture of $[32,16]$ with Relu
activations and $l2$ regularization with $\lambda=10^{-3}$. We solve the
optimization problem using the conjugate gradient, and we stop iterating after
1000 steps. We randomly split data into $75\%$ training data and $25\%$ test
data. The error is measured by the squared root of the mean squared error. Two
different neural networks are used to train call and put options. Call and put
options result in errors of 3.73 and 3.85, respectively. This is somewhat
larger than other periods given the volatility of the market in 2008. The
neural network used here is only intended for demonstration purposes. More
advanced models may provide better results, for example in [10].