# Generalized measure Black-Scholes equation: Towards option self-similar
pricing

Nizar Riane${{}^{\dagger}}{{}^{\ddagger}}$, Claire David‡

###### Abstract

In this work, we give a generalized formulation of the Black-Scholes model.
The novelty resides in considering the Black-Scholes model to be valid on
’average’, but such that the pointwise option price dynamics depends on a
measure representing the investors’ ’uncertainty’.

We make use of the theory of non-symmetric Dirichlet forms and the abstract
theory of partial differential equations to establish well posedness of the
problem. A detailed numerical analysis is given in the case of self-similar
measures.

† Université Mohammed V de Rabat, FSJESR-Agdal<EMAIL_ADDRESS>

‡ Sorbonne Université

CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, 4, place Jussieu 75005,
Paris, France222Corresponding author<EMAIL_ADDRESS>

Keywords: Generalized measure Black-Scholes equation; self-similar measure;
non-symmetric Dirichlet form; fractal differential equations; finite
difference method.

AMS Classification: 28A80 - 91G20 - 65L12 - 65L20.

##  1 Introduction

The Black-Scholes model arises as one of the most important application of
mathematics to economics and finance of the 20th century, allowing the
emergence of the theory of financial partial differential equations and the
associated numerical analysis.

Yet, despite its phenomenal success in the financial market, the model suffers
from deficiencies, for instance, when it comes to the modeling of real market
options with erratic behaviors [For96]. One can argue that the model does not
capture all the factors influencing investor decisions.

From a practical point of view, investors employ technical analysis to detect
patterns in price evolution graphs to beat the market; since those patterns
seem to appear in small and large scales, this could justify a self-similar
modelling of the market. Investigation of fractals in finance is not new, we
can refer, for instance, to Benoît Mandelbrot’s work in [Man97], the aspects
of which are not taken into account by the classical model.

In this paper, we give a definition of a general version of the Black-Scholes
model, based on the theory of non-symmetric Dirichlet forms, and on the
abstract theory of partial differential equations. The key point is to
consider the Black-Scholes equation describing an average evolution, while the
exact dynamics depends on uncertainty captured by a mathematical measure. The
analysis goes so far as proving the existence of a _Generalized Black-Scholes
operator_.

Special treatment is given to the self-similar case by writing an explicit
formula, which enables computation of the solution. The CFL333Courant-
Friedrichs-Lewy. convergence condition for the associated finite difference
scheme is determined and written explicitly. For the proper calibration, our
model can avoid overpricing options at the money, and underpricing options at
the ends, either deep in the money444When the option has what is also called
an _intrinsic value_ , i.e. the real value of the option, that is to say the
profit that could be made in the event of immediate exercise. It means that
the value is at a favorable strike price relative to the prevailing market
price of the underlying asset. Yet, this does not mean that the trader will be
making profit, since the expense of buying, and the commission prices have
also to be considered., or deep out of the money555When the option has what is
also called an _extrinsic value_ , i.e. a value at a strike price higher than
the market price of the underlying asset. In such a case, the Delta, i.e. the
Greek which quantifies the risk, is less than 50.. This work opens the way to
an empirical investigation and an inverse problem of the probability measure
$\mu$.

##  2 Generalized Measure Black-Scholes Model

######  Definition 2.1 (Generalized Measure Black-Scholes Equation).

Let’s introduce the generalized measure Black-Scholes equation, for European
options, in the sense of distributions:

$\left\\{\begin{array}[]{ccccc}\displaystyle\frac{\partial u}{\partial
t}(t,x)\,d\mu&=&\displaystyle\left(-r(t)\,x\,\frac{\partial u}{\partial
x}(t,x)-\frac{\sigma^{2}(t)}{2}\,x^{2}\,\Delta
u(t,x)+r(t)\,u(t,x)\right)\,dx&\quad\forall\,(t,x)\,\in\,\left[0,T\right]\times\left]L,M\right[\\\
u(T,x)&=&h(x)&\quad\forall\,x\,\in\,\left[L,M\right]\end{array}\right.$

where the variable $0<L<x<M$ is the price of the underlying financial
instrument, $\sigma$ denotes the volatility, $r$ the risk-free interest rate,
$T$ the maturity of the option, $\mu$ a non atomic finite measure with total
mass $\mathbb{M}=M-L$ and $u$ represents the option price. The real valued
function $h$ takes the values $h(x)=(x-K)^{+}$ for a call666The _call_ is an
option on a financial instrument, which consists in a right to buy.
Concretely, it consists in a contract which allows the subscriber to get the
targeted financial product, at a price fixed in advance - the _strike_ price -
at a given date - the expiry one, or _maturity_ of the call., and
$h(x)=(K-x)^{+}$ for a _put_ 777As for the _put_ , it is this time a right to
sell - or not - at the maturity date., given a constant $K$: “the strike".

In order to prove that the problem has a solution, technical results
associated to the classical Black-Scholes model are required. We refer to
[AP05] for the proof of the following assertions.

###  2.1 The Black-Scholes Model

######  Notation (Space of Test Functions).

Given a continuous subset $E$ of ${\mathbb{R}}$, we will denote by ${\cal
D}(E)$ the space of test functions on $E$, i.e. the space of smooth functions
with compact support in $E$.

######  Notation (Black-Scholes Bilinear Form).

For any pair
$(u,v)\,\in\,\mathcal{D}({\mathbb{R}}_{+})\times\mathcal{D}({\mathbb{R}}_{+})$,
set:

$\displaystyle B(u,v)$
$\displaystyle=\displaystyle\int_{{\mathbb{R}}_{+}}\frac{\sigma^{2}x^{2}}{2}\frac{\partial
u}{\partial x}\frac{\partial v}{\partial
x}dx+\int_{{\mathbb{R}}_{+}}\left(\sigma^{2}-r\right)x\frac{\partial
u}{\partial x}\,v\,dx+\int_{{\mathbb{R}}_{+}}r\,u\,v\,dx\,\cdot$

and note $B(u,u)=B(u)$.

######  Property 2.1.

The bilinear form $B(\cdot,\cdot)$ is non-symmetric.

Define its symmetric part, for any pair
$(u,v)\,\in\,\mathcal{D}({\mathbb{R}}_{+})\times\mathcal{D}({\mathbb{R}}_{+})$,
through:

$\displaystyle\tilde{B}(u,v)$
$\displaystyle=\displaystyle\frac{1}{2}\,\left(B(u,v)+B(v,u)\right)$
$\displaystyle=\displaystyle\int_{{\mathbb{R}}_{+}}\frac{\sigma^{2}x^{2}}{2}\frac{\partial
u}{\partial x}\,\frac{\partial v}{\partial
x}dx+\frac{1}{2}\,\int_{{\mathbb{R}}_{+}}\left(\sigma^{2}-r\right)\,x\,\left(\frac{\partial
u}{\partial x}\,v+u\,\frac{\partial v}{\partial
x}\right)\,dx+\int_{{\mathbb{R}}_{+}}r\,u\,v\,dx$
$\displaystyle=\displaystyle\frac{\sigma^{2}}{2}\int_{{\mathbb{R}}_{+}}x^{2}\,\frac{\partial
u}{\partial x}\frac{\partial v}{\partial
x}\,dx+\frac{3r-\sigma^{2}}{2}\int_{{\mathbb{R}}_{+}}u\,v\,dx$

######  Notations.

Set

$V=\left\\{v\,\in\,L^{2}({\mathbb{R}}_{+})\quad,\quad
x\,\displaystyle\frac{\partial v}{\partial
x}\,\in\,L^{2}({\mathbb{R}}_{+})\right\\}\quad,\quad
W=\left\\{v\,\in\,L^{2}({\mathbb{R}}_{+})\quad,\quad
x^{2}\,\displaystyle\frac{\partial^{2}v}{\partial
x^{2}}\,\in\,L^{2}({\mathbb{R}}_{+})\right\\}\,\cdot$

######  Property 2.2.

The space

$V=\left\\{v\,\in\,L^{2}({\mathbb{R}}_{+})\quad,\quad
x\,\displaystyle\frac{\partial v}{\partial
x}\,\in\,L^{2}({\mathbb{R}}_{+})\right\\}$

endowed with the inner product

$(u,v)\mapsto\left\langle u,v\right\rangle_{V}=\left\langle
u,v\right\rangle_{L^{2}({\mathbb{R}}_{+})}+\left\langle
x\frac{du}{dx},x\,\displaystyle\frac{dv}{dx}\right\rangle_{L^{2}({\mathbb{R}}_{+})}\,\cdot$

is a Hilbert space.

######  Definition 2.2 (Black-Scholes Weak Formula).

The Black-Scholes variational formula reads: find $u\in
C([0,T];L^{2}(R_{+}))\cap L^{2}((0,T);V)$ such that $\partial_{t}u\in
L^{2}((0,T);V^{\star})$, satisfying:

$\displaystyle\begin{cases}B(u,v)&=\displaystyle\frac{d}{dt}\,\displaystyle\int_{{\mathbb{R}}_{+}}u(t,x)\,v(x)\,dx\,\quad,\quad\forall
v\,\in\,\mathcal{D}({\mathbb{R}}_{+})\\\ u(T,x)&=h(x)\end{cases}$

where $V^{\star}$ is the dual space of $V$.

######  Lemma 2.3 (Black-Scholes Operator).

There exists a unique bounded linear operator $\mathcal{BS}\,:\,V\rightarrow
V^{\prime}$, which will be called Black-Scholes operator, such that:

$\forall\,(u,v)\,\in\,V^{2}:\quad
B(u,v)=\left\langle\mathcal{BS}(u),v\right\rangle_{L^{2}({\mathbb{R}}_{+})}\,\cdot$

######  Definition 2.3.

Recall that the first derivative is always bounded in the Hilbert-Sobolev
space $H^{1}({\mathbb{R}}_{+})$, and that the second derivative is always
bounded in the Hilbert-Sobolev space $H^{2}({\mathbb{R}}_{+})$.

The Black-Scholes operator is given, for any $v$ in $W$, by

$\mathcal{BS}(v)=-\displaystyle\frac{x^{2}\sigma^{2}}{2}\frac{\partial^{2}v}{\partial
x^{2}}-r\,x\,\frac{\partial v}{\partial x}+r\,v\,\cdot$

######  Lemma 2.4.

The set

$W=\left\\{v\,\in V\quad,\quad
x^{2}\,\displaystyle\frac{\partial^{2}v}{\partial
x^{2}}\,\in\,L^{2}({\mathbb{R}}_{+})\right\\}$

is dense in $V$.

######  Lemma 2.5 (Black-Scholes Weak Solution).

For $h$ in $L^{2}({\mathbb{R}}_{+})$, the Black-Scholes problem has a unique
weak solution.

###  2.2 An Interpretation of the Generalized Measure Black-Scholes Model

The generalized measure Black-Scholes equation, for a probability measure
$\mu$, can be written, by choosing $v=u$, as

$\displaystyle\frac{\partial}{\partial t}\mathbb{E}\left(|u|^{2}\right)$
$\displaystyle=\frac{2}{\mathbb{M}}\,B\left(u\right)$

where $\mathbb{M}=M-L$, while $\mathbb{E}\left(.\right)$ is the expected value
and $B(\cdot,\cdot)$ denotes the involved Black-Scholes bilinear form. This
equation can be understood in the following way: the expected value
depreciation of $u^{2}$ (and, hence, of $u$, since it is positive) is
proportional to the Black-Scholes energy $B(u)$. This implies a law given by
the classical Black-Scholes theory which holds on _average_ and another
induced by investors’ perception of reality and reflected by $\mu$.

A direct implication of this vision is a local dependence of the option price
dynamics on the probability measure $\mu$, which can be interpreted as
a$\backslash$n confidence$\backslash$uncertainty measure.

##  3 Non-Symmetric Dirichlet Forms and Generalized Measure Black-Scholes
Operators

Given two strictly positive numbers $L<M$, if one replaces ${\mathbb{R}}_{+}$
with $\overline{\mathcal{M}}=\left[L,M\right]$ (where
$\mathcal{M}=\left]L,M\right[$) in the generalized measure Black-Scholes
model, it does not result in a dramatic effect on the mathematical or
economical foundations of the model from the following two points of view:

  1. 1.

Financial stability: One can suppose, in short run, boundedness of the
underlying financial instrument price.

  2. 2.

Numerical analysis: It is well known that infinite boundaries are replaced
with finite ones for numerical simulations (See [KN01] for error estimates).

In section 5.3, a tolerance level associated with the choice of the bounds $L$
and $M$ is calculated.

######  Notations.

Let’s introduce the following two spaces:

$\displaystyle V_{\mathcal{M}}$
$\displaystyle=\left\\{v\,\in\,L^{2}(\mathcal{M})\quad,\quad
x\,\displaystyle\frac{\partial v}{\partial
x}\,\in\,L^{2}(\mathcal{M})\right\\}\quad,$ $\displaystyle
L^{2}_{\mu}(\mathcal{M})$
$\displaystyle=\left\\{v\quad,\quad\int_{L}^{M}v^{2}\,d\mu<\infty\right\\}\quad\cdot$

The dual space of $V_{\mathcal{M}}$ will be denoted by
$V_{\mathcal{M}}^{\star}$ and the closure of $D(\mathcal{M})$ in
$V_{\mathcal{M}}$ by $V_{0,\mathcal{M}}$.

######  Proposition 3.1.

The space $\mathcal{D}(\mathcal{M})$ is dense in $L^{2}_{\mu}(\mathcal{M})$.

As in [HLN06], the fundamental conditions under which the solution exist are
obtained thanks to the following assumptions:

######  Assumptions 3.2.

For any $u$ in $\mathcal{D}(\mathcal{M})$, there exists a positive constant
$C_{0}$:

$\displaystyle\|u\|_{L^{2}_{\mu}(\mathcal{M})}$ $\displaystyle\leq
C_{0}\left\|x\,\displaystyle\frac{\partial u}{\partial
x}\right\|_{L^{2}(\mathcal{M})}\,,$ (Continuous injection condition) (1)
$\displaystyle\sigma^{2}$ $\displaystyle<4r\,\cdot$ (Coercivity condition) (2)

######  Proposition 3.3.

Under the first condition in assumption 3.2, there exists a unique
$L^{2}_{\mu}(\mathcal{M})$-representative $\tilde{u}$ of each equivalence
class of functions $u$ in $V_{\mathcal{M}}$ such that the above condition
holds. There also exists a $\mathcal{D}(\mathcal{M})$ sequence
$(u_{n})_{n\in{\mathbb{N}}}$ which converges towards $\tilde{u}$ both in
$V_{\mathcal{M}}$ and in $L^{2}_{\mu}(\mathcal{M})$, since
$\mathcal{D}(\mathcal{M})$ is dense in $V_{\mathcal{M}}$ and
$L^{2}_{\mu}(\mathcal{M})$ [Kil94].

Consider the bilinear form $B(\cdot,\cdot)$ with domain $\text{dom}\,(B)$ on
the Hilbert space $V_{\mathcal{M}}$. We introduce the bilinear form:

$B^{\star}(\cdot,\cdot)=B(\cdot,\cdot)+\left\langle\cdot,\cdot\right\rangle_{L^{2}_{\mu}(\mathcal{M})}$

and the symmetric one:

$\tilde{B}^{\star}(\cdot,\cdot)=\tilde{B}(\cdot,\cdot)+\left\langle\cdot,\cdot\right\rangle_{L^{2}_{\mu}(\mathcal{M})}\,\cdot$

We refer to [MR92] for further details on the theory of non-symmetric
Dirichlet forms.

######  Definition 3.1 (Symmetric Closed Form).

A pair $(B,\text{dom}\,(B))$ is a symmetric closed form (on $H$) if
$\text{dom}\,(B)$ is a dense linear subspace of $H$ and
$B\,:\,\text{dom}\,(B)\times\text{dom}\,(B)\,\rightarrow{\mathbb{R}}$ is a
positive definite bilinear which is symmetric and closed on $H$ (i.e.,
$\text{dom}\,(B)$ is complete with respect to the norm
${B^{\star}(\cdot,\cdot)}^{\frac{1}{2}}$).

######  Definition 3.2 (Sector Condition).

Let us denote by $B$ a bilinear form on the Hilbert space $H$, and by
$\text{dom}\,(B)$ its domain. The pair $(B,\text{dom}\,(B))$ is said to
satisfy:

  * _i_.

The weak sector condition if there exists $K>0$ such that:

$\displaystyle\forall\,(u,v)\,\in\,\text{dom}\,(B)\times\text{dom}\,(B):\quad|B^{\star}(u,v)|\leq
K\,\sqrt{B^{\star}(u,u)\,B^{\star}(v,v)}\cdot$ (3)

  * _ii_.

The strong sector condition if there exists $K>0$ such that

$\forall\,(u,v)\,\in\,\text{dom}\,(B)\times\text{dom}\,(B):|B(u,v)|\leq
K\,\sqrt{B(u,u)\,B(v,v)}\,\cdot$

######  Remark 3.1.

A coercive continuous bilinear form satisfies both conditions.

######  Definition 3.3 (Coercive Closed Form).

A pair $(B,\text{dom}\,(B))$ will be called a coercive closed form (on $H$) if
$\text{dom}\,(B)$ is a dense linear subspace of $H$ and
$B\,:\,\text{dom}\,(B)\times\text{dom}\,(B)\,\rightarrow{\mathbb{R}}$ is a
bilinear form such that the following two conditions hold:

  * _i_.

Its symmetric part $(\tilde{B},\text{dom}\,(B))$ is a symmetric closed form on
$H$.

  * _ii_.

$(B,\text{dom}\,(B))$ satisfies the weak sector condition inequality $i$.
given in (3).

######  Definition 3.4 (Symmetric Vs Non-Symmetric Dirichlet Form).

A coercive closed form $(B,\text{dom}\,(B))$ on $L^{2}_{\mu}(\mathcal{M})$,
for a given measure $\mu$, will be called a Dirichlet form if, for any $u$ in
$\text{dom}\,(B)$, one has:

$\tilde{u}\in\text{dom}\,(B)\quad\text{and}\quad\left\\{\begin{array}[]{ccc}B(u+\tilde{u},u-\tilde{u})&\geqslant&0\\\
B(u-\tilde{u},u+\tilde{u})&\geqslant&0\\\ \end{array}\right.$

where $\tilde{u}=\max(0,\min(u,1))$. If $(B,\text{dom}\,(B))$ is in addition
symmetric, this is equivalent to:

$\displaystyle B(\tilde{u},\tilde{u})\leq B(u,u)\,\cdot$

$B$ will be called a symmetric Dirichlet form.

######  Theorem 3.4.

[MR92]

Let us denote by $(B,\text{dom}\,(B))$ a coercive closed form on $H$, and $J$
a continuous linear functional on $\text{dom}\,(B)$. Then, there exists a
unique $u\,\in\,\text{dom}\,(B)$ such that

$\forall\,v\,\in\,\text{dom}\,(B):\quad B^{\star}(u,v)=J(v)\,\cdot$

######  Definition 3.5 (Non-Symmetric Dirichlet Forms).

A coercive closed form on $H$, $(B,\text{dom}\,(B))$, is said to be a non-
symmetric Dirichlet form (on $H$), if there exists a one-to-one correspondence
with a pair of bounded linear operators $(L,\tilde{L})$:

$\forall\,(u,v)\,\in\,\text{dom}\,(L)\times\text{dom}\,(L):\quad
B(u,v)=(-Lu,v)=(u,-\tilde{L}v)$

where $\text{dom}\,(L)$ is the domain of $L$. Also, $\text{dom}\,(L)$ is a
dense subset of $\text{dom}\,(B)$.
The operator $L$ (respectively $\tilde{L}$) is the generator of a strongly
continuous contraction semi-group $(T_{t})_{t>0}$ (respectively
$(\tilde{T})_{t>0}$).

The following result follows from [AP05].

######  Proposition 3.5 (Poincaré Inequality ).

The space $\mathcal{D}(\mathcal{M})$ is dense in $V_{\mathcal{M}}$, and, for
any $v\,\in\,\mathcal{D}(\mathcal{M})$, the following inequality is satisfied:

$\|v\|_{L^{2}(\mathcal{M})}\leq
2\,\left\|x\,\displaystyle\frac{dv}{dx}\right\|_{L^{2}(\mathcal{M})}$

This inequality induces a second norm on $V_{\mathcal{M}}$, given, for any $v$
in $V_{\mathcal{M}}$, by:

$|v|_{V_{\mathcal{M}}}=\left\|x\frac{dv}{dx}\right\|_{L^{2}(\mathcal{M})}\,\cdot$

######  Proposition 3.6.

(Continuity and Gårding Inequality)

The bilinear form $B(\cdot,\cdot)$ is continuous on $V_{\mathcal{M}}$, and
satisfies the Gårding inequality:

$\forall\,u\,\in\,V_{\mathcal{M}}:\quad
B(u)\geqslant\displaystyle\frac{\sigma^{2}}{2}|u|_{V_{\mathcal{M}}}^{2}-\lambda\,\parallel
u\parallel_{L^{2}(\mathcal{M})}^{2}$

where $\lambda=\displaystyle\frac{\left(\sigma^{2}-3r\right)}{2}$. Moreover,
$B(\cdot,\cdot)$ is coercive under the second assumption 3.2.

###### Proof.

For any pair
$(u,v)\,\in\,\mathcal{D}(\mathcal{M})\times\mathcal{D}(\mathcal{M})$, by using
the Poincaré inequality, we obtain that

$\displaystyle|B(u,v)|$
$\displaystyle=\left|\displaystyle\int_{L}^{M}\frac{\sigma^{2}x^{2}}{2}\frac{\partial
u}{\partial x}\frac{\partial v}{\partial
x}dx+\int_{L}^{M}\left(\sigma^{2}-r\right)x\frac{\partial u}{\partial
x}\,v\,dx+\int_{L}^{M}r\,u\,v\,dx\right|$
$\displaystyle\leq\frac{\sigma^{2}}{2}|u|_{V_{\mathcal{M}}}|v|_{V_{\mathcal{M}}}+\left(\sigma^{2}-r\right)|u|_{V_{\mathcal{M}}}\parallel
v\parallel_{L^{2}(\mathcal{M})}+\,r\,\parallel
u\parallel_{L^{2}(\mathcal{M})}\parallel v\parallel_{L^{2}(\mathcal{M})}$
$\displaystyle\leq C_{1}\,|u|_{V_{\mathcal{M}}}|v|_{V_{\mathcal{M}}}$

where $C_{1}=\displaystyle 2r+\frac{5\sigma^{2}}{2}$. For the coercivity, we
use again Poincaré inequality:

$\displaystyle B(u)$
$\displaystyle=\displaystyle\frac{\sigma^{2}}{2}|u|_{V_{\mathcal{M}}}^{2}+\int_{L}^{M}\left(\sigma^{2}-r\right)x\frac{\partial
u}{\partial x}\,u\,dx+r\parallel u\parallel_{L^{2}(\mathcal{M})}^{2}$
$\displaystyle=\frac{\sigma^{2}}{2}|u|_{V_{\mathcal{M}}}^{2}-\frac{\left(\sigma^{2}-3r\right)}{2}\parallel
u\parallel_{L^{2}(\mathcal{M})}^{2}$ $\displaystyle\geqslant
C_{2}\,|u|_{V_{\mathcal{M}}}^{2}$

where $C_{2}=\displaystyle 6r-\frac{3\sigma^{2}}{2}$.

∎

######  Definition 3.6.

We introduce the mapping $\iota\,:\,V_{\mathcal{M}}\rightarrow
L^{2}_{\mu}(\mathcal{M})$ by

$\iota(u)=\bar{u}$

where $\bar{u}$ is the unique $L^{2}_{\mu}(\mathcal{M})$-representative of
$u$, along with the closed set:

$\mathcal{N}=\left\\{v\in
V_{\mathcal{M}}\,:\,\|v\|_{L^{2}_{\mu}(\mathcal{M})}=0\right\\}\,\cdot$

######  Theorem 3.7.

The Black-Scholes Non-Symmetric Dirichlet Form

Under the assumption 3.2:

  1. i.

$\text{dom}(B)=V_{\mathcal{M}}$ is dense in $L^{2}_{\mu}(\mathcal{M})$.

  2. ii.

$\left(\tilde{B}^{\star},{V_{\mathcal{M}}}\right)$ is a Hilbert space.

  3. iiii.

$(B,\text{dom}(B))$ is a (non-symmetric) Dirichlet form.

###### Proof.

  * •

Let us consider a sequence $(u_{n})_{n\in{\mathbb{N}}}$ of
$\mathcal{D}(\mathcal{M})$, which converges towards $u$ in
$L^{2}_{\mu}(\mathcal{M})$.

We then consider two sequences,
$(a_{n})_{n\in{\mathbb{N}}}\,\in\,{\mathcal{N}}^{\mathbb{N}}$, and
$(b_{n})_{n\in{\mathbb{N}}}\,\in\,{V_{\mathcal{M}}}^{\mathbb{N}}$ such that,
for any natural integer $n$,

$u_{n}=a_{n}+b_{n}\,\cdot$

Then, the sequence $(b_{n})_{n\in{\mathbb{N}}}$ converges to $u$ in
$L^{2}_{\mu}(\mathcal{M})$.

  * •

Under the continuous injection condition (1) in assumption 3.2, the induced
norm ${\tilde{B}^{\star}(\cdot,\cdot)}^{\frac{1}{2}}$ is equivalent to the
norm $|\cdot|_{V_{\mathcal{M}}}$. Hence, $(B^{\star},\text{dom}(B))$ is
complete.

  * •

It follows from the coercivity of $B$ that:

$\displaystyle 0\leq C_{2}\,|u^{2}-\tilde{u}^{2}|^{2}_{V_{\mathcal{M}}}\leq
B(u\pm\tilde{u},u\mp\tilde{u})\,\cdot$

∎

######  Theorem 3.8 (Generalized Measure Black-Scholes Operator).

Under the assumption 3.2, there exists a bounded linear operators
$\mathcal{BS}_{\mu}$, that we will call generalized measure Black-Scholes
operator, such that, for any pair
$(u,v)\,\in\,\text{dom}(\mathcal{BS}_{\mu})\times\text{dom}(B)$:

$B(u,v)=\left\langle\mathcal{BS}_{\mu}(u),v\right\rangle_{L^{2}_{\mu}(\mathcal{M})}$

Moreover, we will say that $u\in\text{dom}(\mathcal{BS}_{\mu})$ and
$\mathcal{BS}_{\mu}(u)=f$ if and only if

$B(u,v)=\displaystyle\int_{\mathcal{M}}f\,v\,d\mu\,\quad,\quad\forall v\in
V_{0,\mathcal{M}}$

######  Remark 3.2.

The generalized measure Black-Scholes operator is bounded from
$V_{\mathcal{M}}$ to $V_{\mathcal{M}}^{\star}$ since, $\forall v\in
V_{0,\mathcal{M}}$

$\displaystyle\left|\left\langle\mathcal{BS}_{\mu}(u),v\right\rangle_{L^{2}_{\mu}(\mathcal{M})}\right|$
$\displaystyle=\left|B(u,v)\right|$ $\displaystyle\leq
C_{1}\,|u|_{V_{\mathcal{M}}}|v|_{V_{\mathcal{M}}}$

by continuity of $B(\cdot,\cdot)$.

######  Notations (Sobolev Spaces).

Given a continuous subset $E$ of ${\mathbb{R}}$, $k\,\in\,{\mathbb{N}}$, and
$p\geqslant 1$, we recall that the classical Sobolev spaces on $E$ are

$W^{k}_{p}\left(E\right)=\left\\{f\,\in\,L^{p}\left(E\right)\,,\,\forall\,j\leq
k\,:\,f^{(j)}\,\in\,L^{p}\left(E\right)\right\\}$

and

$H^{k}\left(E\right)=W^{k}_{2}\left(E\right)=\left\\{f\,\in\,L^{2}\left(E\right)\,,\,\forall\,j\leq
k\,:\,f^{(j)}\,\in\,L^{2}\left(E\right)\right\\}\,\cdot$

The subspace $H_{0}^{k}$ of functions which vanish on the boundary $\partial
E$ is

$H_{0}^{k}\left(E\right)=\left\\{f\,\in\,L^{2}\left(E\right)\,,\,f_{|\partial
E}=0\,\text{and }\forall\,j\leq
k\,:\,f^{(j)}\,\in\,L^{2}\left(E\right)\right\\}\,\cdot$

It directly comes form the abstract theory of partial differential equations
[Zei90], [Wlo87], [LM68] that:

######  Theorem 3.9 (Generalized Measure Black-Scholes Weak Solution).

Let us define the Gelfand triple (or equipped Hilbert space)
$V_{\mathcal{M}}\subset L^{2}_{\mu}(\mathcal{M})\subset
V_{\mathcal{M}}^{\star}$. For $h$ in $L^{2}_{\mu}(\mathcal{M})$, the
generalized measure Black-Scholes problem admits, under the assumptions 3.2, a
unique weak solution. Moreover, for $k\geqslant 1$, the solution map:

$\displaystyle L^{2}_{\mu}(\mathcal{M})$ $\displaystyle\longrightarrow
W^{k}_{2}\left([0,T];V_{\mathcal{M}}\right)$ $\displaystyle h(x)$
$\displaystyle\longrightarrow u(t,x)\qquad,\qquad
x\in\mathcal{M}\quad\text{and}\quad t\in[0,T]$

is continuous.

##  4 The Self-Similar Black-Scholes Operator - a Pointwise Formula

######  Definition 4.1 (Self-Similar Measure on a Real Interval, Associated to
a Set of Contractions [Str06]).

We hereafter consider the particular case where the real interval
$\mathcal{M}=\left]L,M\right[\subset{\mathbb{R}}_{+}$ is self-similar with
respect to the family of contractions $\\{f_{1},f_{2}\\}$, and where $f_{1}$
and $f_{2}$ are defined, for any real number $x$, by:

$f_{1}(x)=\displaystyle\frac{1}{2}\,(x+L)\quad,\quad
f_{2}(x)=\displaystyle\frac{1}{2}\,\left(x+M\right)\,\cdot$

A measure $\mu$ with full support on $\overline{\mathcal{M}}$ will be called
self-similar measure on $\overline{\mathcal{M}}$, relative to the set of
contractions $\left(f_{1},f_{2}\right)$ if, given a family of strictly
positive weights $\left(\mu_{1}\,,\mu_{2}\right)$ such that

$\mu_{1}+\mu_{2}=1\,,$

one has, then,

$\mu=\displaystyle\mu_{1}\,\mu\circ f_{1}^{-1}+\mu_{2}\,\mu\circ
f_{2}^{-1}\,\cdot$

Let us consider $u\,\in\,\text{dom}(\mathcal{BS}_{\mu})$ where $\mu$ is a
self-similar measure according to definition 4.1. $\mathcal{BS}_{\mu}$ is now
called the self-similar Black-Scholes operator and we set:
$\mathcal{BS}_{\mu}(u)=f$.

In order to compute the explicit formula of $\mathcal{BS}_{\mu}$, we set
$\mathbb{M}=M-L$ and we recall the self similar construction of $\mathcal{M}$:

$\overline{\mathcal{M}}=\displaystyle
f_{1}(\overline{\mathcal{M}})+f_{2}(\overline{\mathcal{M}})\,\cdot$

######  Definition 4.2 (Prefractal Graph Approximation).

We denote by $V_{0}$ the ordered set of the (boundary) points:

$\left\\{L,M\right\\}$

We build the graph ${{\cal M}_{0}}$ by connecting the two extremities of
$V_{0}$.

For any strictly positive integer $m$, we set:

$V_{m}=f_{1}\left(V_{m-1}\right)+f_{2}\left(V_{m-1}\right)\,\cdot$

The set of points $V_{m}$, where consecutive points are connected, will be
denoted by ${{\mathcal{M}}_{m}}$.

The set $V_{m}$ is called the set of vertices of the graph
${\mathcal{M}}_{m}$. By extension, we will write that

${\mathcal{M}}_{m}=f_{1}\left({\mathcal{M}}_{m-1}\right)+f_{2}\left({\mathcal{M}}_{m-1}\right)\,\cdot$

One can prove that the sequence $\\{V_{m}\\}_{m\in{\mathbb{N}}}$ is increasing
and its limit is dense in $\mathcal{M}$ ([Kig01]).

######  Proposition 4.1.

Given a natural integer $m$, we will denote by $\mathcal{N}_{m}$ the number of
vertices of the graph ${\mathcal{M}}_{m}$. One has:

$\mathcal{N}_{0}=2$

and, for any strictly positive integer $m$:

$\mathcal{N}_{m}=2^{m}+1\,\cdot$

We recall the following definitions:

######  Definition 4.3 (Word).

Given a strictly positive integer $m$, we will call number-letter any integer
${\mathcal{W}}_{i}$ of $\left\\{1,2\right\\}$, and word of length
$|{\mathcal{W}}|=m$, on the graph ${\mathcal{M}}_{m}$, any set of number-
letters of the form:

${\mathcal{W}}=\left({\mathcal{W}}_{1},\ldots,{\mathcal{W}}_{m}\right)\,\cdot$

We will write:

$f_{\mathcal{W}}=f_{{\mathcal{W}}_{1}}\circ\ldots\circ
f_{{\mathcal{W}}_{m}}\,\cdot$

######  Definition 4.4 (Addresses).

Given a natural integer $m$, and a vertex $X$ of ${\cal M}_{m}$, we will call
address of the vertex $X$ an expression of the form

$X=f_{\cal W}\left(L\right)\quad\text{or}\quad X=f_{\cal
W^{{}^{\prime}}}\left(M\right)$

where $\cal W$ and ${\cal W}^{\prime}$ denote words of length $m$. The vertex
$X$ has thus a double address.

######  Property 4.2 (Space of Harmonic Splines).

Given a strictly positive integer $m$, we introduce the space of harmonic
splines of order $m$, denoted by $\mathcal{H}^{m}(\left[L,M\right])$, as the
space of functions $\psi^{m}_{X}$, $X\,\in\,[L,M]$, such that:

$\forall\,Y\,\in\,\mathcal{M}_{m}\quad\psi^{m}_{X}(Y)=\delta_{XY}\,\cdot$

For $k\,\in\left\\{1,\ldots,2^{m}-1\right\\}$, and $Y\,\in\,[L,M]$:

$\displaystyle\psi^{m}_{L+\frac{k\,\mathbb{M}}{2^{m}}}(Y)$
$\displaystyle=\begin{cases}\displaystyle\frac{2^{m}}{\mathbb{M}}(Y-L)-(k-1)\qquad
L+\frac{(k-1)\mathbb{M}}{2^{m}}\leq Y\leq L+\frac{k\,\mathbb{M}}{2^{m}}\\\
-\displaystyle\frac{2^{m}}{\mathbb{M}}(Y-L)+(k+1)\qquad\displaystyle
L+\frac{k\,\mathbb{M}}{2^{m}}\leq Y\leq L+\frac{(k+1)\,\mathbb{M}}{2^{m}}\\\
0\qquad\text{otherwise}\end{cases}$

and

$\displaystyle\displaystyle\psi^{m}_{L}(Y)$
$\displaystyle=\begin{cases}-\displaystyle\frac{2^{m}}{\mathbb{M}}(Y-L)+1\qquad
L\leq Y\leq L+\frac{\mathbb{M}}{2^{m}}\\\
0\qquad\text{otherwise}\end{cases}\quad,$ $\displaystyle\psi^{m}_{M}(Y)$
$\displaystyle=\begin{cases}\displaystyle\frac{2^{m}}{\mathbb{M}}(Y-M)+1\qquad
M-\frac{\mathbb{M}}{2^{m}}\leq Y\leq M\\\ 0\qquad\text{otherwise}\end{cases}$

######  Proposition 4.3 (Integration of Harmonic Splines).

Let us consider a strictly positive integer $m$. For
$k\,\in\left\\{1,\ldots,2^{m}-1\right\\}$, we denote by
${\mathcal{V_{\mathcal{M}}}}_{k}$ and ${\mathcal{W}}_{k}$ the unique indices
such that

$f_{{\mathcal{V_{\mathcal{M}}}}_{k}}\left(\left[L,M\right]\right)=\displaystyle\left[L+\frac{(k-1)\mathbb{M}}{2^{m}},L+\frac{k\,\mathbb{M}}{2^{m}}\right]\quad\text{and}\quad
f_{{\mathcal{W}}_{k}}\left(\left[L,M\right]\right)=\displaystyle\left[L+\frac{k\,\mathbb{M}}{2^{m}},L+\frac{(k+1)\,\mathbb{M}}{2^{m}}\right]\,\cdot$

Then,

$\displaystyle\int_{L}^{M}\psi^{m}_{L+\frac{k\,\mathbb{M}}{2^{m}}}\,d\mu={\mu_{1}}^{s_{{\mathcal{V_{\mathcal{M}}}}_{k}}}{\mu_{2}}^{m+1-s_{{\mathcal{V_{\mathcal{M}}}}_{k}}}+\mu_{1}^{s_{{\mathcal{W}}_{k}}+1}\,\mu_{2}^{m-s_{{\mathcal{W}}_{k}}}$

and

$\displaystyle\int_{L}^{M}\psi^{m}_{L}\,d\mu={\mu_{1}}^{m+1}\quad\quad\int_{L}^{M}\psi^{m}_{M}\,d\mu={\mu_{2}}^{m+1}\,\cdot$

In addition, if ${\mu_{1}}<\displaystyle\frac{1}{2}$:

$\displaystyle\displaystyle\int_{L}^{M}\psi^{m}_{L}d\mu<\int_{L}^{M}\psi^{m}_{L+\frac{k\,\mathbb{M}}{2^{m}}}\,d\mu<\int_{L}^{M}\psi^{m}_{M}\,d\mu\,\cdot$

######  Property 4.4.

Given a strictly positive integer $m$, we set, for any integer
$k\,\in\,\left\\{0,\ldots,2^{m}\right\\}$:

$x_{k}=\displaystyle L+\left(k\,\frac{\mathbb{M}}{2^{m}}\right)$

and, for any $u\in\text{dom}(\mathcal{BS}_{\mu})$:

$\displaystyle B(u,\psi^{m}_{x_{k}})$
$\displaystyle=\displaystyle\int_{L}^{M}\left(-\frac{\sigma^{2}x^{2}}{2}\frac{\partial^{2}u}{\partial
x^{2}}-rx\frac{\partial u}{\partial x}+r\,u\right)\,\psi^{m}_{x_{k}}\,dx$
$\displaystyle=\lim_{m\rightarrow+\infty}\sum_{j=0}^{2^{m}}\,\mathcal{BS}_{m}\,(u(t,x_{j}))\,\psi^{m}_{x_{k}}\,\left(\frac{\mathbb{M}}{2^{m}}\right)$
$\displaystyle=\lim_{m\rightarrow+\infty}\,\mathcal{BS}_{m}\,(u(t,x_{k}))\,\left(\frac{\mathbb{M}}{2^{m}}\right)$

where $\mathcal{BS}_{m}$ is the discrete Black-Scholes operator defined, for
any $t$ in $[0,T]$, by:

$\displaystyle\mathcal{BS}_{m}\,u(t,x_{k})$
$\displaystyle=-\displaystyle\frac{\sigma^{2}}{2}\,x_{k}^{2}\left(\displaystyle\frac{u(t,x_{k+1})-2\,u(t,x_{k})+u(t,x_{k-1})}{\left(\frac{\mathbb{M}}{2^{m}}\right)^{2}}\right)$
$\displaystyle-r\,x_{k}\left(\frac{u(t,x_{k+1})-u(t,x_{k-1})}{2\left(\frac{\mathbb{M}}{2^{m}}\right)}\right)+r\,C(t,x_{k})$
$\displaystyle=-\frac{\sigma^{2}}{2}\,k^{2}\left(u(t,x_{k+1})-2\,u(t,x_{k})+u(t,x_{k-1})\right)$
$\displaystyle-r\,{k}\left(\frac{u(t,x_{k+1})-u(t,x_{k-1})}{2}\right)+r\,C(t,x_{k})$

The mean value formula yields asymptotically

$\displaystyle\displaystyle\int_{L}^{M}\mathcal{BS}_{\mu}u(x)\,\psi^{m}_{x_{k}}\,d\mu$
$\displaystyle\approx\mathcal{BS}_{\mu}u(x_{k})\,\int_{L}^{M}\psi^{m}_{x_{k}}\,d\mu\,\cdot$

######  Theorem 4.5 (Self-Similar Black-Scholes Operator Pointwise Formula).

Let us consider $u\,\in\,\text{dom}(\mathcal{BS}_{\mu})$. Then, for any
$x\,\in\,\overline{\mathcal{M}}$, and any sequence
$(x_{m})_{m\in{\mathbb{N}}}$ of $V_{m}\setminus V_{0}$ which uniformly
converges towards $x$:

$\displaystyle\mathcal{BS}_{\mu}(u)(x)=\displaystyle\lim_{m\rightarrow+\infty}2^{-m}\,\left(\displaystyle\int_{L}^{M}\psi^{m}_{x_{m}}\,d\mu\right)^{-1}\,\mathcal{BS}_{m}(u)(x_{m})\,\cdot$

###### Proof.

The uniform convergence directly comes from the fact that

$\displaystyle\left(\frac{\mathbb{M}}{2^{m}}\right)\left(\displaystyle\int_{L}^{M}\psi^{m}_{x_{m}}d\mu\right)^{-1}\mathcal{BS}_{m}(u)(x_{m})=C\,\frac{\displaystyle\int_{L}^{M}\mathcal{BS}_{\mu}\,\psi^{m}_{x_{m}}\,d\mu}{\displaystyle\int_{L}^{M}\psi^{m}_{x_{m}}d\mu}\\\
=C\,\mathcal{BS}_{\mu}(u)(x)\,\cdot$

For $\mu_{1}=\mu_{2}=\displaystyle\frac{1}{2}$, one recovers the classical
Black-Scholes operator, which implies that: $C=\mathbb{M}$. ∎

##  5 Proof of the Assumptions

We hereafter consider the general case of non atomic finite measure $\mu$ with
total mass $\mathbb{M}=M-L$ (In the case of self-similar measures, it suffice
to multiply the measure by $\mathbb{M}$).

###  5.1 First assumption 3.2

######  Notation (Space of Weighted Continuous Functions).

We will denote by $C(\overline{\mathcal{M}})$ the space of weighted continuous
functions on $\overline{\mathcal{M}}$ endowed with the norm

$\|u\|_{\eta,\infty}=\underset{x\,\in\,\overline{\mathcal{M}}}{\max}\,|x\,u(x)|\,\cdot$

###### Proof.

of the first assumption

Let us consider $u\in V_{0,\mathcal{M}}$. On the one hand, we have that

$\displaystyle\left|x\,u(x)\right|$
$\displaystyle=\left|\displaystyle\int_{L}^{x}\,(su(s))^{\prime}\,ds\right|$
$\displaystyle=\left|\int_{L}^{x}u(s)\,ds+\int_{L}^{x}su^{\prime}(s)\,ds\right|$
$\displaystyle\leq\sqrt{\mathbb{M}}\left(\|u\|_{L^{2}}+|u|_{V_{\mathcal{M}}}\right)$
$\displaystyle\leq\sqrt{\mathbb{M}}\|u\|_{V_{\mathcal{M}}}$

We deduce the continuity of the injection
$\iota\,:\,(V_{\mathcal{M}},\|\cdot\|_{V_{\mathcal{M}}})\rightarrow(C_{\eta}(\overline{\mathcal{M}}),\|\cdot\|_{\eta,\infty})$.
On the other hand, for $u\in C_{\eta}(\overline{\mathcal{M}})$, one also has
that

$\|u\|_{L^{2}_{\mu}(\mathcal{M})}=\left(\int_{L}^{M}\frac{1}{x^{2}}\left(x\,u\right)^{2}\,d\mu\right)^{\frac{1}{2}}\leq\|u\|_{\eta,\infty}\,\left(\int_{L}^{M}\frac{1}{x^{2}}\,d\mu\right)^{\frac{1}{2}}\leq\|u\|_{\eta,\infty}\,\frac{\mu(\overline{\mathcal{M}})^{\frac{1}{2}}}{L}$

the injection
$\iota\,:\,(C_{\eta}(\overline{\mathcal{M}}),\|\cdot\|_{\eta,\infty})\rightarrow(L^{2}_{\mu}(\mathcal{M}),\|\cdot\|_{L^{2}_{\mu}(\mathcal{M})})$
is continuous, so we obtain that

$\|u\|_{L^{2}_{\mu}(\mathcal{M})}\leq C_{0}\,\|u\|_{V_{\mathcal{M}}}\,$

for
$C_{0}=\displaystyle\frac{\sqrt{\mathbb{M}\,\mu(\overline{\mathcal{M}})}}{L}=\frac{\mathbb{M}}{L}$.

∎

###  5.2 Commentary on the second assumption 3.2

The assumption $4r>\sigma^{2}$ is not that restrictive as it may seem in the
first sight, for example, if we give a look to sample from Vance L. Martin
data [LMM05]. The sample consist of $\mathbf{N}=269$ observations on the
European call options written on the S$\&$P$500$ stock index on the
$4^{\textrm{th}}$ of April, $1995$. One can calculate (see [RD21])

  * •

The interest rate $r=0.0591$.

  * •

The volatility $\sigma=0.076675$.

which means that

$\displaystyle 4r=0.2364\gg 0.00587906=\sigma^{2}\,\cdot$

The second assumption could also be replaced by the following condition: for
any $u$ in $\mathcal{D}(\mathcal{M})$,

$\displaystyle\|u\|_{L^{2}(\mathcal{M})}\leq
C_{3}\|u\|_{L^{2}_{\mu}(\mathcal{M})}\,\cdot$

Given $u$ in $\text{dom}\,(B)$, set, in the spirit of [Wlo87]:

$\tilde{\lambda}=\lambda\,C_{3}\quad,\quad
w(t,x)=e^{-\tilde{\lambda}\,t}u(T-t,x)$

where $\lambda$ is Gårding’s inequality constant in proposition 3.6. Then
$\forall\,v\,\in\,\text{dom}\,(B)$, with $w(0,x)=h(x)$, one has:

$\displaystyle\int_{L}^{M}\frac{d}{dt}w(t,x)\,v(x)\,d\mu=-B(w,v)-\tilde{\lambda}\,\int_{L}^{M}w(t,x)\,v(x)\,d\mu=-\widehat{B}(w,v)\quad\,\cdot$

Without affecting the solution space, one obtains the continuity and the
coercivity of the form $\widehat{B}$:

$\displaystyle|\widehat{B}(u,v)|$ $\displaystyle\leq
C\,|u|_{V_{\mathcal{M}}}\,|v|_{V_{\mathcal{M}}}+\tilde{\lambda}\,\parallel
u\parallel_{L^{2}_{\mu}(\mathcal{M})}\,\parallel
v\parallel_{L^{2}_{\mu}(\mathcal{M})}$
$\displaystyle\leq(C+\tilde{\lambda}\,C_{0})\,|u|_{V_{\mathcal{M}}}\,|v|_{V_{\mathcal{M}}}$

and:

$\displaystyle\widehat{B}(u,u)$
$\displaystyle\geqslant\displaystyle\frac{\underline{\sigma}^{2}}{4}\,|u|_{V_{\mathcal{M}}}^{2}-\lambda\parallel
u\parallel_{L^{2}(\mathcal{M})}^{2}+\tilde{\lambda}\,\parallel
u\parallel_{L^{2}_{\mu}(\mathcal{M})}^{2}$
$\displaystyle\geqslant\frac{\underline{\sigma}^{2}}{4}|\,u|_{V_{\mathcal{M}}}^{2}$

###  5.3 Underlying asset price bounds

An important question arises: how to choose $M$ and $L$ ? one answer is to use
the law of $S_{t}$, the underlying asset price, then choose $M(\alpha)$
(respectively $L(\alpha)$) using the rule:

$1-\mathbb{P}\left(L(\alpha)\leq S_{t}\leq
M(\alpha)\right)\leq\alpha\quad,\quad 0\leq t\leq T$

for some tolerance level $\alpha$.

One can deduce a similar rule in the case of boundary conditions. For example,
in the case of a call option:

$\mathbb{P}\left(S_{T}>K\,|\,S_{0}=L(\alpha)\right)\leq\alpha$

and

$\mathbb{P}\left(S_{T}\leq K\,|\,S_{0}=M(\alpha)\right)\leq\alpha$

##  6 Numerical Simulation of Self-Similar European Options

Let us consider as in section 4, the self-similar case, for an European
options call, defined by the system:

$\displaystyle\displaystyle\frac{\partial C}{\partial t}(t,S)$
$\displaystyle=\mathcal{BS}_{\mu}(C)(t,S)$
$\displaystyle\forall\,t\,\in\,\left[0,T\right],\,\forall\,S\,\in\,\mathcal{M}$
$\displaystyle C(T,S)$ $\displaystyle=h(S)$
$\displaystyle\forall\,S\,\in\,\mathcal{M}$ $\displaystyle C(t,L)$
$\displaystyle=0$ $\displaystyle\forall\,t\,\in\,\left[0,T\right]$
$\displaystyle C(t,M)$ $\displaystyle=g(t)$
$\displaystyle\forall\,t\in\left[0,T\right]$

for a self-similar measure $\mu$ on $\overline{\mathcal{M}}$, under the
condition

$4\,r>\sigma^{2}\,\cdot$

where $h(S)=(S-K)^{+}$ and $g(t)=M-K\,\exp(-r(T-t))$, and where the constant
$\sigma$ is the volatility, $r$ the risk-free interest rate, $T$ the maturity
of the option and $C$ represents the call price.

We will use the following change of variables: $\tau=T-t$, which leads to the
following equation (with the same notations):

$\displaystyle-\displaystyle\frac{\partial C}{\partial t}(t,S)$
$\displaystyle=\mathcal{BS}_{\mu}(C)(t,S)$
$\displaystyle\forall\,t\,\in\,\left[0,T\right],\,\forall\,S\,\in\,\mathcal{M}$
$\displaystyle C(0,S)$ $\displaystyle=h(S)$
$\displaystyle\forall\,S\,\in\,\mathcal{M}$ $\displaystyle C(t,L)$
$\displaystyle=0$ $\displaystyle\forall\,t\,\in\,\left[0,T\right]$
$\displaystyle C(t,M)$ $\displaystyle=g(t)$
$\displaystyle\forall\,t\,\in\,\left[0,T\right]$

for $g(t)=M-K\,\exp(-r\,t)$.

######  Remark 6.1.

The results of section 3 still hold, if we write $\tilde{C}=C-\tilde{g}$,
where
$\tilde{g}(t,x)=\displaystyle(x-L)\left(\frac{M-K\,\exp(-r\,t)}{M-L}\right)$,
and replace the space $V_{\mathcal{M}}$ by $V_{0,\mathcal{M}}$, then we solve
the non homogeneous problem

$\int_{L}^{M}\,\frac{\partial\tilde{C}}{\partial
t}\,v\,d\mu+B(\tilde{C},v)=\int_{L}^{M}\frac{d\tilde{g}}{dt}\,v\,d\mu$

applying abstract theory of partial differential equations [Wlo87].

###  6.1 The Finite Difference Method

In the spirit of our previous work [RD19], we fix a strictly positive integer
$N$, and set:

$\displaystyle{h=\displaystyle\frac{T}{N}}$

We will write, for a function $f$, $n\,\in\,\left\\{0,\ldots,N\right\\}$ and
$k\,\in\,\left\\{0,\ldots,2^{m}\right\\}$:

$f(h,k)=f(n\,h,L+k\,\displaystyle\frac{\mathbb{M}}{2^{m}})\,\cdot$

We use the Euler implicit scheme, for any integer $n$ belonging to
$\left\\{0,\ldots,N-1\right\\}$:

$\forall\,k\,\in\,\\{0,\ldots,2^{m}\\}\,:\quad\displaystyle\frac{\partial
C}{\partial
t}(n,k)\approx\displaystyle\frac{1}{h}\,\left(C(n+1,k)-C(n,k)\right)$

The self-similar Black-Scholes operator for $k=\\{0,\ldots,2^{m}\\}$ is
approximated through

$\displaystyle\mathcal{BS}_{\mu}\,C(n,k)$
$\displaystyle\approx\displaystyle\left(\frac{2^{-m}}{{\mu_{1}}^{s_{\mathcal{V_{\mathcal{M}}}}}{\mu_{2}}^{m-s_{\mathcal{V_{\mathcal{M}}}}+1}+{\mu_{1}}^{s_{\mathcal{W}}+1}{\mu_{2}}^{m-s_{\mathcal{W}}}}\right)\mathcal{BS}_{m}\,C(n,k)$
$\displaystyle\approx\displaystyle\delta_{m}\,\mathcal{BS}_{m}\,C(n,k)$

where $\mathcal{BS}_{m}$ is the discretized Black-Scholes operator given by

$\displaystyle\mathcal{BS}_{m}\,C(n,k)$
$\displaystyle=-\frac{\sigma^{2}}{2}\left(\frac{k\,\mathbb{M}}{2^{m}}\right)^{2}\left(\frac{C(n,k+1)-2\,C(n,k)+C(n,k-1)}{\left(\frac{\mathbb{M}}{2^{m}}\right)^{2}}\right)$
$\displaystyle-r\,\left(\frac{k\,\mathbb{M}}{2^{m}}\right)\left(\frac{C(n,k+1)-C(n,k-1)}{2\left(\frac{\mathbb{M}}{2^{m}}\right)}\right)+r\,C(n,k)\,\cdot$

For $0\leq n\leq N-1$, and $1\leq k\leq 2^{m}-1$, we define the following
scheme:

$\left({\mathcal{S}}_{\mathcal{B}S}\right)\quad\left\\{\begin{array}[]{cccc}\displaystyle-\frac{\mathcal{C}_{h,m}(n+1,k)-\mathcal{C}_{h,m}(n,k)}{h}&=&\displaystyle\delta_{m}\,\mathcal{BS}_{m}\,\mathcal{C}_{h,m}(n,k)&\\\
\mathcal{C}_{h,m}(n,0)&=&0&\\\ \mathcal{C}_{h,m}(n,2^{m})&=&g(n)\\\
\mathcal{C}_{h,m}(0,k)&=&h(k)\,\cdot&\end{array}\right.$

For $n\,\in\,\left\\{0,\ldots,N-1\right\\}$, we set:

$\mathcal{C}(n)=\left(\begin{matrix}\mathcal{C}_{h,m}(n,1)\\\ \vdots\\\
\mathcal{C}_{h,m}(n,2^{m}-1)\\\ \end{matrix}\right)\\\
=\left(\begin{matrix}\mathcal{C}_{h,m}(n,f_{\mathcal{W}^{1}}(M))\\\ \vdots\\\
\mathcal{C}_{h,m}(n,f_{\mathcal{W}^{2^{m}-1}}(M))\\\
\end{matrix}\right)\quad,\quad\left\\{\mathcal{W}^{1},\ldots,\mathcal{W}^{2^{m}-1}\right\\}\,\in\,\left\\{1,2\right\\}^{m}\,\cdot$

We have the following recurrence relation:

$\mathcal{C}(n+1)=A\,\mathcal{C}(n)+B(n)$

where the ${(2^{m}-1)}\times{(2^{m}-1)}$ matrix $A$ is given by:

$\displaystyle A$ $\displaystyle=I_{2^{m}-1}-h\,2^{-m}\,\Psi_{m}^{-1}\,BS_{m}$

and where $I_{2^{m}-1}$ denotes the ${(2^{m}-1)}\times{(2^{m}-1)}$ identity
matrix, ${\Psi}_{m}$ and $BS_{m}$ the ${(2^{m}-1)}\times{(2^{m}-1)}$ matrices:

$\displaystyle{\Psi}_{m}$ $\displaystyle=\left(\begin{matrix}\ddots&&&0\\\
&{\mu_{1}}^{s_{\mathcal{V_{\mathcal{M}}}}}{\mu_{2}}^{m+1-s_{\mathcal{V_{\mathcal{M}}}}}+{\mu_{1}}^{s_{\mathcal{W}}+1}{\mu_{2}}^{m-s_{\mathcal{W}}}&\\\
0&&&\ddots\\\ \end{matrix}\right)$

$\displaystyle{BS}_{m}$
$\displaystyle=\left(\begin{matrix}-\sigma^{2}-r&\frac{\sigma^{2}}{2}+\frac{r}{2}&0&\ldots&0&0\\\
2\,\sigma^{2}-r&-4\sigma^{2}-r&2\,\sigma^{2}+r&\ldots&0&0\\\
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\
0&0&0&\ldots&-(2^{m}-2)^{2}\sigma^{2}-r&(2^{m}-2)^{2}\frac{\sigma^{2}}{2}+(2^{m}-2)\frac{r}{2}\\\
0&0&0&\ldots&(2^{m}-1)^{2}\frac{\sigma^{2}}{2}-(2^{m}-1)\frac{r}{2}&-(2^{m}-1)^{2}\sigma^{2}-r\\\
\end{matrix}\right)$

and

$\displaystyle B(n)$
$\displaystyle=h\,2^{-m}\,{\mu_{2}}^{-(m+1)}\left((2^{m}-1)^{2}\,\frac{\sigma^{2}}{2}+(2^{m}-1)\,\frac{r}{2}\right)\left(\begin{matrix}0\\\
\vdots\\\ 0\\\ g(n)\\\ \end{matrix}\right)\,\cdot$

###  6.2 Numerical Analysis

#####  6.2.0.1 The Scheme Error and Consistency

Let us consider a function $V_{\mathcal{M}}$ defined on $\mathcal{M}$. For any
integer $n$ in $\left\\{0,\ldots,N-1\right\\}$, and any $X$ in
${\mathcal{M}}$:

$\displaystyle\frac{\partial v}{\partial
t}(nh,X)=\displaystyle\frac{1}{h}\,\left(v((n+1)h,X)-v(nh,X)\right)+{\mathcal{O}}(h)$

As in [RD19], for any strictly positive integer $m$, and any $X$ in
$V_{m}\setminus V_{0}$, we can prove that:

$2^{-m}\,\displaystyle\left(\int_{L}^{M}\psi^{m}_{X}d\mu\right)^{-1}\mathcal{BS}_{m}\,v(X)=\displaystyle\frac{\int_{L}^{M}\mathcal{BS}_{\mu}\,\psi^{m}_{X}\,d\mu}{\displaystyle\int_{L}^{M}\psi^{m}_{X}\,d\mu}$

and that there exists a vertex $Z$ in the $m$-cell
$f_{\mathcal{W}}\left(\left[L,M\right]\right)$ containing $X$ such that

$\displaystyle\left|\mathcal{BS}_{\mu}\,v(X)-2^{-m}\,\left(\int_{L}^{M}\psi_{X}^{m}d\mu\right)^{-1}\mathcal{BS}_{m}\,v(X)\right|$
$\displaystyle=\left|\mathcal{BS}_{\mu}\,v(X)-\mathcal{BS}_{\mu}\,v(Z)\right|$
$\displaystyle\lesssim\left|v(X)-v(Z)\right|$ $\displaystyle\lesssim|X-Z|$
$\displaystyle\lesssim\left(\frac{1}{2}\right)^{m}\,\cdot$

using remark 3.2 and the uniform continuity of $v\in V_{\mathcal{M}}$. Thus,

$\mathcal{BS}_{\mu}\,v(X)=2^{-m}\,\left(\int_{L}^{M}\psi_{X}^{m}d\mu\right)^{-1}\mathcal{BS}_{m}\,v(X)+\mathcal{O}(2^{-m})\,\cdot$

The consistency error of our scheme is given by :

$\varepsilon^{h,m}_{n,k}=\mathcal{O}(h)+\mathcal{O}(2^{-m})\quad 0\leq n\leq
N,\,0\leq k\leq 2^{m}\,\cdot$

We can check that the scheme is consistent:

$\lim_{h\rightarrow 0,m\rightarrow\infty}\varepsilon^{h,m}_{n,k}=0\,\cdot$

#####  6.2.0.2 Stability

We hereafter prove that the scheme is conditionally stable for the
$\|\cdot\|_{\infty}$ norm.

Let us recall that, for $0\leq n\leq N$, and $0\leq k\leq 2^{m}$:

$\displaystyle\displaystyle\mathcal{C}_{h,m}(n+1,k)$
$\displaystyle=\mathcal{C}_{h,m}(n,k)\left(1-h\,\delta_{m}\,\sigma^{2}\,k^{2}\right)$
$\displaystyle+\mathcal{C}_{h,m}(n,k+1)\left(h\,\delta_{m}\,\frac{\sigma^{2}}{2}\,k^{2}+h\,\delta_{m}\,\frac{r}{2}\,k\right)$
$\displaystyle+\mathcal{C}_{h,m}(n,k-1)\left(h\,\delta_{m}\,\frac{\sigma^{2}}{2}\,k^{2}-h\,\delta_{m}\,\frac{r}{2}\,k\right)$
$\displaystyle-\mathcal{C}_{h,m}(n,k)\,h\,\delta_{m}\,r$
$\displaystyle=\mathcal{C}_{h,m}(n,k)\left(1-\alpha_{k}\right)+\mathcal{C}_{h,m}(n,k+1)\left(\frac{\alpha_{k}}{2}+\beta_{k}\right)+\mathcal{C}_{h,m}(n,k-1)\left(\frac{\alpha_{k}}{2}-\beta_{k}\right)$
$\displaystyle-\mathcal{C}_{h,m}(n,k)\,\gamma$

If we consider $\gamma=0$, this is just an affine combination. Moreover, we
have:

$1-\alpha_{k}\geqslant
1-\sigma^{2}\,\displaystyle\frac{h\,2^{2m}}{2^{m}\left(\displaystyle\int_{L}^{M}\psi_{X}^{m}d\mu\right)}\geqslant
0\quad,\quad\displaystyle\frac{\alpha_{k}}{2}-\beta_{k}\geqslant 0\quad,\quad
1-\gamma\geqslant 0\,\cdot\\\ $

May we suppose that $\sigma^{2}\geqslant r$ and that the following CFL
condition

$\displaystyle\frac{h\,2^{m}}{\left(\displaystyle\int_{L}^{M}\psi_{X}^{m}d\mu\right)}\leq\displaystyle\frac{1}{\sigma^{2}}$

is satisfied, the combination is then convex, and the scheme is stable for the
norm $\|\cdot\|_{\infty}$.

For $\gamma\neq 0$, one has:

$\displaystyle(1-\gamma)\min_{0\leq j\leq
2^{m}}\,\mathcal{C}_{h,m}(n,j)\leq\mathcal{C}_{h,m}(n+1,k)$
$\displaystyle\leq\max_{0\leq j\leq
2^{m}}\,\mathcal{C}_{h,m}(n,j)-\gamma\,\min_{0\leq j\leq
2^{m}}\,\mathcal{C}_{h,m}(n,j)$ $\displaystyle\leq\max_{0\leq j\leq
2^{m}}\,\mathcal{C}_{h,m}(n,j)-\gamma\,(1-\gamma)^{n}\,\min_{0\leq j\leq
2^{m}}\mathcal{C}_{h,m}(0,j)$ $\displaystyle\leq\max_{0\leq j\leq
2^{m}}\,\mathcal{C}_{h,m}(n,j)$

and the scheme is $\|\cdot\|_{\infty}$-stable under the same conditions.

#####  6.2.0.3 Convergence

######  Theorem 6.1.

If the above CFL condition holds, the scheme is convergent for the norm
$\|\cdot\|_{2,\infty}$ given by:

$\|\left(\mathcal{C}_{h,m}(n,k)\right)_{0\leq n\leq N,\,0\leq k\leq
2^{m}}\|_{2,\infty}=\max_{0\leq n\leq N}\left(\sum_{0\leq k\leq
2^{m}}\mu(f_{\mathcal{W}^{k}}(\mathcal{M}))\left(\mathcal{C}_{h,m}(n,k)\right)^{2})\right)^{\frac{1}{2}}$

where $\mu\left(f_{\mathcal{W}^{k}}(\mathcal{M})\right)$ is the measure of the
$f_{\mathcal{W}^{k}}(\mathcal{M})$.

###### Proof.

For $0\leq n\leq N$ and $0\leq k\leq 2^{m}$, we set:

$w^{n}_{k}=C(n,k)-\mathcal{C}_{h,m}(n,k)$

and

$\displaystyle\mathcal{BS}_{m}\,C(n,k)$
$\displaystyle=-\displaystyle\frac{\sigma^{2}}{2}\left(\frac{k\,\mathbb{M}}{2^{m}}\right)^{2}\left(\frac{C(n,k+1)-2\,C(n,k)+C(n,k-1)}{\left(\frac{\mathbb{M}}{2^{m}}\right)^{2}}\right)$
$\displaystyle-r\,\left(\frac{k\,\mathbb{M}}{2^{m}}\right)\left(\frac{C(n,k+1)-C(n,k)}{\left(\frac{\mathbb{M}}{2^{m}}\right)}\right)+r\,C(n,k)\,\cdot$

We can check that

$\displaystyle\frac{w^{n+1}_{k}-w^{n}_{k}}{h}-\frac{\sigma^{2}}{2}\,k^{2}\,\delta_{m}\left(w^{n}_{k+1}-2w^{n}_{k}+w^{n}_{k-1}\right)-r\,k\,\delta_{m}\left(w^{n}_{k+1}-w^{n}_{k}\right)+r\,\delta_{m}\,w^{n}_{k}$
$\displaystyle=\varepsilon^{h,m}_{n,k}\quad$ $\displaystyle 0\leq n\leq
N-1,\,1\leq k\leq 2^{m}-1$ $\displaystyle w^{n}_{0}=w^{n}_{2^{m}}$
$\displaystyle=0\qquad$ $\displaystyle 0\leq n\leq N$ $\displaystyle
w^{0}_{k}$ $\displaystyle=0\qquad$ $\displaystyle 1\leq k\leq 2^{m}-1\,\cdot$

Let us set, for any integer $n$ in $\left\\{0,\ldots,N\right\\}$:

$W^{n}=\left(\begin{matrix}w^{n}_{1}\\\ \vdots\\\
w^{n}_{2^{m}-1}\end{matrix}\right)\quad,\quad
E^{n}=\left(\begin{matrix}\varepsilon^{h,m}_{n,1}\\\ \vdots\\\
\varepsilon^{h,m}_{n,2^{m}-1}\end{matrix}\right)\,\cdot$

One has:

$W^{0}=0\quad\text{and}\quad\forall\,n\,\in\,\left\\{0,\ldots,N-1\right\\}:\quad
W^{n+1}=A\,W^{n}+h\,E^{n}\,\cdot$

By induction, this yields, for any integer $n$ in
$\left\\{0,\ldots,N-1\right\\}$:

$W^{n+1}=A^{n}\,W^{0}+h\,\displaystyle\sum_{j=0}^{n}A^{j}\,E^{n-j}=h\,\displaystyle\sum_{j=0}^{n}A^{j}\,E^{n-j}$

Since $A$ is a symmetric matrix, the CFL stability condition yields, for any
integer $n$ in $\left\\{0,\ldots,N\right\\}$:

$\displaystyle|W^{n}|$ $\displaystyle\leq
h\,\displaystyle\left(\sum_{j=0}^{n-1}\|A\|^{j}\right)\left(\max_{0\leq j\leq
n-1}|E^{j}|\right)$ $\displaystyle\leq h\,n\,\displaystyle\left(\max_{0\leq
j\leq n-1}|E^{j}|\right)$ $\displaystyle\leq
h\,N\,\displaystyle\left(\max_{0\leq j\leq n-1}|E^{j}|\right)$
$\displaystyle\leq T\,\displaystyle\left(\max_{0\leq j\leq
n-1}\left(\sum_{k=1}^{2^{m}-1}|\varepsilon^{h,m}_{j,k}|^{2}\right)^{\frac{1}{2}}\right)\,\cdot$

By assuming $\displaystyle{\mu_{1}}\geqslant\displaystyle\frac{1}{2}$ (the
same result holds for $\displaystyle{\mu_{1}}\leq\displaystyle\frac{1}{2}$ by
changing ${\mu_{1}}$ into ${\mu_{2}}$), we deduce then that:

$\displaystyle\max_{0\leq n\leq
N}\left(\sum_{k=1}^{2^{m}-1}\mu\left(f_{\mathcal{W}^{k}}(\mathcal{M})\right)|w^{n}_{k}|^{2}\right)^{\frac{1}{2}}$
$\displaystyle\leq\max_{1\leq k\leq
2^{m}-1}\left(\mu\left(f_{\mathcal{W}^{k}}(\mathcal{M})\right)\right)^{\frac{1}{2}}\max_{1\leq
n\leq N}|W^{n}|$ $\displaystyle\leq\max_{1\leq k\leq
2^{m}-1}\left(\mu\left(f_{\mathcal{W}^{k}}(\mathcal{M})\right)\right)^{\frac{1}{2}}T\,\left(\max_{0\leq
n\leq
N-1}\left(\sum_{k=1}^{2^{m}-1}|\varepsilon^{h,m}_{n,k}|^{2}\right)^{1/2}\right)$
$\displaystyle\leq\displaystyle\max_{1\leq k\leq
2^{m}-1}\left(\mu\left(f_{\mathcal{W}^{k}}(\mathcal{M})\right)\right)^{\frac{1}{2}}T\,\left((2^{m}-1)^{\frac{1}{2}}\max_{0\leq
n\leq N-1,\,1\leq k\leq 2^{m}-1}|\varepsilon^{h,m}_{n,k}|\right)$
$\displaystyle\leq\sqrt{\left({\mu_{1}}\times
2\right)^{m}}T\,\left(\max_{0\leq n\leq N-1,\,1\leq k\leq
2^{m}-1}|\varepsilon^{h,m}_{n,k}|\right)$
$\displaystyle=\sqrt{\left({\mu_{1}}\times
2\right)^{m}}\left(\mathcal{O}(h)+\mathcal{O}(2^{-m})\right)$
$\displaystyle=\mathcal{O}\left(\left(\sqrt{\frac{{\mu_{1}}}{2}}\right)^{m}\right)$

The scheme is then convergent. ∎

###  6.3 Self-Similar Pricing

In the sequel, we give a numerical simulation of the self-similar pricing, in
the case of a call:

$\displaystyle\frac{\partial C}{\partial t}(t,S)$
$\displaystyle=\mathcal{BS}_{\mu}(C)(t,S)$
$\displaystyle\forall\,t\,\in\,\left[0,T\right],\,\forall\,S\,\in\,\left[L,M\right[$
$\displaystyle C(T,S)$ $\displaystyle=(S-K)^{+}$
$\displaystyle\forall\,x\,\in\,\left[L,M\right[$ $\displaystyle C(t,L)$
$\displaystyle=0$ $\displaystyle\forall\,t\,\in\,v\left[0,T\right]$
$\displaystyle C(t,M)$ $\displaystyle=M-K\exp(-r(T-t))$
$\displaystyle\forall\,t\,\in\,\left[0,T\right]$

for a self-similar measure $\mu$ on $\left[L,M\right]$. The solutions are
generated using the finite difference method, for

$T=1\quad,\quad K=150\quad,\quad\sigma=0.3\quad,\quad r=0.1\,\cdot$

Figure 1: Black-Scholes solution $u(0,x)$ for different values of the weights.

The self-similar Black-Scholes model generates an exotic pricing for different
values of $\mu_{1}$, including the classical one for
$\mu_{1}=\displaystyle\frac{1}{2}$.

$\mu_{1}=\frac{1}{2}$.

$\mu_{1}=0.2$.

$\mu_{1}=0.8$.

Figure 2: The call value for different values of the weights.

###  6.4 The Greeks

As in [Dav17], we recall that, in finance, the sensitivity of a portfolio to
changes in parameters values can be measured through what commonly call “the
Greeks", i.e.,

  1. _i_.

The Delta, $\Delta=\displaystyle\frac{\partial C}{\partial S}\,\in\,[0,1]$,
which enables one to quantify the risk, and is thus the most important Greek.
It can also be interpreted as a probability that the option will expire in the
money.

  2. _ii_.

The Gamma, $\Gamma=\displaystyle\frac{\partial^{2}C}{\partial S^{2}}\geqslant
0$, which measures the rate of the acceleration of the option price, with
respect to changes in the underlying price.

  3. _iii_.

The Vega (the name of which comes from the form of the Greek letter $\nu$),
$\nu=\displaystyle\frac{\partial C}{\partial\sigma}$, which measures the
sensitivity to volatility.

  4. _iv_.

The Theta $\Theta=\displaystyle\frac{\partial C}{\partial t}$, which is the
time cost of holding an option.

  5. _v_.

The rho, $\rho=\displaystyle\frac{\partial C}{\partial r}$, which measures the
sensitivity to the risk-free interest rate.

The good strategy, for traders, is to have delta-neutral positions at least
once a day, and, whenever the opportunity arises, to improve the Gamma and the
Vega.

####  6.4.1 The Delta

$\mu_{1}=\frac{1}{2}$.

$\mu_{1}=\frac{1}{3}$.

$\mu_{1}=\frac{2}{3}$.

Figure 3: The $\Delta$, for different values of the weights.

####  6.4.2 The Gamma

$\mu_{1}=\frac{1}{2}$.

$\mu_{1}=\frac{1}{3}$.

$\mu_{1}=\frac{2}{3}$.

Figure 4: The $\Gamma$, for different values of the weights.

####  6.4.3 The Theta

$\mu_{1}=\frac{1}{2}$.

$\mu_{1}=\frac{1}{3}$.

$\mu_{1}=\frac{2}{3}$.

Figure 5: The $\Theta$, for different values of the weights.

####  6.4.4 The Vega

$\mu_{1}=\frac{1}{2}$.

$\mu_{1}=\frac{1}{3}$.

$\mu_{1}=\frac{2}{3}$.

Figure 6: The $\nu$, for different values of the weights.

####  6.4.5 The rho

$\mu_{1}=\frac{1}{2}$.

$\mu_{1}=\frac{1}{3}$.

$\mu_{1}=\frac{2}{3}$.

Figure 7: The $\rho$, for different values of the weights.

###  6.5 Discussion

Let us make a few remarks about the behavior of the solution with respect to
the weight $\mu$ at $t=0$:

  1. _i_.

For $\displaystyle\mu_{1}=\frac{1}{3}$: the premium is greater than that of
the classical model under the strike and smaller above. The Greeks value shows
a drastic increase in the strike neighbor, and self-similarity clearly affects
_in the money_ region. The Theta shows a slower premium expected decrease.

  2. _ii_.

For $\displaystyle\mu_{1}=\frac{2}{3}$: the premium is everywhere greater than
that of the classical model. The Greeks value increases progressively in the
strike neighbor, and self-similarity affects _in the money_ region. The Theta
indicates a slower premium expected decrease _in the money_ and a greater
decrease _deep in the money_.

The dynamic generated by the self-similar Black-Scholes model is exotic and
enables the emergence of non-standard behaviors, the parameter
$\displaystyle\mu_{1}$ can capture the behavior of non confident investors
under uncertainty and other factors influencing their perception of future.

The self-similar Black-Scholes equation can be understood as a diffusion
equation with a time change through self-similar probability $\mu$, where the
cumulative distribution function satisfies for $x\in\left]0,1\right[$,
$\left[0,x\right]=f_{\mathcal{W}}(\left[0,1\right])$ for some word
$\mathcal{W}$,

$\displaystyle\mu\left[0,x\right]$
$\displaystyle=\Pi_{i\in\mathcal{W}}\,\mu_{i}$

depending on the address (path) of $x$. According to this remark, we can
create more exotic behavior by using a self-similar measure $\mu$ with many
weights or enable the weights to change over time.

## References

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# Closing of the Mott gap near step edges in NiS2

Yuuki Yasui<EMAIL_ADDRESS>RIKEN Center for Emergent Matter Science,
Wako, Saitama 351-0198, Japan  Department of Advanced Materials Science, The
University of Tokyo, Kashiwa, Chiba 277-8561, Japan  Kota Iwata  Department of
Advanced Materials Science, The University of Tokyo, Kashiwa, Chiba 277-8561,
Japan  Shota Okazaki  Laboratory for Materials and Structures, Tokyo Institute
of Technology, Yokohama, Kanagawa, 226-8501, Japan  Shigeki Miyasaka
Department of Physics, Osaka University, Toyonaka, Osaka, 560-0043, Japan
Yoshiaki Sugimoto  Department of Advanced Materials Science, The University of
Tokyo, Kashiwa, Chiba 277-8561, Japan  Tetsuo Hanaguri<EMAIL_ADDRESS>RIKEN Center for Emergent Matter Science, Wako, Saitama 351-0198, Japan
Hidenori Takagi  Department of Physics, Graduate School of Science, The
University of Tokyo, Bunkyo, Tokyo 113-0033, Japan  Institute for Functional
Matter and Quantum Technologies, University of Stuttgart, 70569 Stuttgart,
Germany  Max Planck Institute for Solid State Research, Heisenbergstraße 1,
70569 Stuttgart, Germany  Takao Sasagawa  Laboratory for Materials and
Structures, Tokyo Institute of Technology, Yokohama, Kanagawa, 226-8501, Japan
Research Center for Autonomous Systems Materialogy, Tokyo Institute of
Technology, Yokohama, Kanagawa 226-8501, Japan

###### Abstract

A prototypical charge-transfer type Mott insulator NiS2 pyrite exhibits a
metal-insulator transition with bandwidth control. Recent discoveries on
surface-specific electronic states on other $3d$ transition-metal disulfide
pyrites motivate us to further investigate the surface of NiS2, where metallic
surface conduction is discussed. Here, the spectroscopic-imaging scanning-
tunneling-microscopy observations revealed that the surface is not metallic,
contrary to the expectation. Instead, the Mott gap is closed near step edges,
suggesting possible electrical conduction from one-dimensional channels. The
edge anomaly was observed irrespective of its magnetic order and is limited to
the insulator phases.

Transition-metal (TM) disulfides in the pyrite structure, in which TM2+ and
S2^2- crystalize in the NaCl structure with space group $Pa\bar{3}$ [Figs.
1(a) and (b)], offer a variety of electric and magnetic properties depending
on its $d$-band filling [1, 2, 3]. FeS2 $(t_{2g}^{6}e_{g}^{0})$ is a
diamagnetic semiconductor, CoS2 $(t_{2g}^{6}e_{g}^{1})$ is a ferromagnetic
metal, NiS2 $(t_{2g}^{6}e_{g}^{2})$ is an antiferromagnetic Mott insulator
with half-filled $e_{g}$ bands, CuS2 $(t_{2g}^{6}e_{g}^{3})$ is a
superconductor, and ZnS2 $(t_{2g}^{6}e_{g}^{4})$ is a semiconductor.

NiS2 undergoes a metal-to-insulator transition with bandwidth control, and it
is considered to be a typical model as it does not change the crystal
structure [4]. The bandwidth control is achieved by Se substitution with S [5,
6, 7, 8, 9] and an application of hydrostatic pressure [10, 11]. The pristine
NiS2 at an ambient pressure exhibits a noncolinear antiferromagnetic order
below $39\text{\,}\mathrm{K}$, and a weak ferromagnetic order below
$30\text{\,}\mathrm{K}$ [Fig. 1(c)].

Figure 1:  (a) The crystal structure of NiS2 [12, 13]. Ni atoms are
represented in green, and S atoms are in yellow. (b) A model for a cleaved
surface seen from the c-axis. The zigzag structure of the topmost S atoms is
highlighted with dashed lines. (c) A schematic phase diagram of NiS_2-xSe_x
reproduced from [5]. PM, paramagnetic metal; AFI, antiferromagnetic insulator.
The conditions investigated in the present manuscript are marked by black
dots. (d) A constant-current topographic image of NiS2. (e) A spatially
averaged $\derivative*{I}{V}$ spectrum at a 2D surface. The set-point bias
voltage $V_{\mathrm{s}}=$+1\text{\,}\mathrm{V}$$, set-point current
$I_{\mathrm{s}}=$100\text{\,}\mathrm{pA}$$, and measurement temperature
$T=$77\text{\,}\mathrm{K}$$.

The resistivity of NiS2 tends to saturate at low temperatures even though it
is an insulator. It has been argued that the surface may be metallic and
dominate the electric conduction at low temperatures [14]. The multi-channel
conductance is supported by changing the crystalline grain size [15] and with
heat treatments [16]. The neutron scattering measurements with susceptibility
measurements suggest the surface contributes also to its magnetic state [17].
Additionally, a finite density of states (DOS) at the Fermi level by the
photoemission spectroscopy [18, 7] further implies its metallic surface
states. More recent researches focusing on the surface anomalies convince the
additional metallic channel [19] and imply magnetic anomalies at step edges
[20, 21].

As surface conduction is implied in NiS2, it is interesting to investigate the
surface directly by using the spectroscopic-imaging scanning tunneling
microscopy (STM). An STM observation was reported for the metallic compounds
NiS_2-xSe_x $(x\geq 0.45)$ [22] whereas its insulating state has not been
investigated. In this manuscript, we report STM results for the Mott
insulating side ($x=0$ and $0.45$). In the three-dimensional (3D) crystal
structure, we found that its two-dimensional (2D) surface is indeed
insulating, contrary to the expectation from previous reports. Interestingly,
the Mott gap becomes smaller as approaches to one-dimensional (1D) step edges
in the weak ferromagnetic insulator (WFI) and the paramagnetic insulator (PI)
phases. Thus, we infer that the suggested conductance is happening at the 1D
step edges rather than 2D surfaces. Such edge effect was not observed in Se
substituted metallic crystals NiS_1.55Se_0.45.

Figure 2:  Edge states in NiS2 measured at $77\text{\,}\mathrm{K}$ (PI phase).
(a) A topographic image of a pseudo-island structure. (b) A
$\derivative*{I}{V}$ map at $+200\text{\,}\mathrm{mV}$ within the surface Mott
gap taken in the same field of view as (a). Positions where the tip changed
are marked with black arrows. $V_{\mathrm{s}}=$+0.9\text{\,}\mathrm{V}$$ and
$I_{\mathrm{s}}=$100\text{\,}\mathrm{pA}$$.  Figure 3:  The edge anomaly in
NiS2 measured at $77\text{\,}\mathrm{K}$ (PI phase). (a) A topographic image
near a step edge. (b) A $\derivative*{I}{V}$ map at $+200\text{\,}\mathrm{mV}$
within the surface Mott gap taken in the same field of view as (a). Points at
$z<$-50\text{\,}\mathrm{pm}$$ are not shown. (c) $\derivative*{I}{V}$ spectra
near (red circle) and away (blue square) from the step edge. (d) Tunneling
conductance spectra along the green line in (b). (e) The variation of the
insulating gap defined at $1\text{\,}\mathrm{pS}$. (f) The line profile of the
simultaneously taken topographic image.
$V_{\mathrm{s}}=$+1.0\text{\,}\mathrm{V}$$ and
$I_{\mathrm{s}}=$100\text{\,}\mathrm{pA}$$.  Figure 4:  The edge anomaly in
NiS2 measured at $5\text{\,}\mathrm{K}$ (WFI phase). (a) A topographic image
near a step edge. The image is Fourier-filtered to suppress high-frequency
noise. (b) A $\derivative*{I}{V}$ map at $+200\text{\,}\mathrm{mV}$ within the
surface Mott gap in the same field of view as (a). Points at
$z<$-50\text{\,}\mathrm{pm}$$ are not shown. (c) $\derivative*{I}{V}$ spectra
near (red circle) and away from the step (blue square). (d) Tunneling
conductance spectra along the green line in (b). (e) The variation of the
insulating gap defined at $2.5\text{\,}\mathrm{pS}$. (f) The line profile of
the simultaneously taken topographic image.
$V_{\mathrm{s}}=$+1.5\text{\,}\mathrm{V}$$ and
$I_{\mathrm{s}}=$200\text{\,}\mathrm{pA}$$.  Figure 5:  Absence of a
noticeable edge effect at the boundary of the PI and the AFM phases in
NiS_1.55Se_0.45 measured at $77\text{\,}\mathrm{K}$. (a) A topographic image
near a step edge. Se atoms appear as bright protrusions while S atoms appear
darker in the same zigzag structure. (b) A $\derivative*{I}{V}$ map at
$+200\text{\,}\mathrm{mV}$ in the same field of view as (a). Points at $z<-50$
pm are not shown. (c) $\derivative*{I}{V}$ spectra near (red circle) and away
from the step (blue square). (d) Tunneling conductance spectra along the green
line in (b). (e) The variation of the insulating gap defined at
$2.5\text{\,}\mathrm{pS}$. (f) The line profile of the simultaneously taken
topographic image. $V_{\mathrm{s}}=$+1.0\text{\,}\mathrm{V}$$ and
$I_{\mathrm{s}}=$100\text{\,}\mathrm{pA}$$.

NiS_2-xSe_x $(x=0\,\mathrm{and}\,0.45)$ single crystals were grown with the
chemical vapor transport method [6]. For STM measurements at
$77\text{\,}\mathrm{K}$, the samples were cleaved in an ultra-high vacuum
chamber ($\sim{10}^{-8}\text{\,}\mathrm{Pa}$) at around
$77\text{\,}\mathrm{K}$ to expose clean and flat (0 0 1) surfaces and then
transferred to a microscope without breaking vacuum. As scanning tips,
tungsten wires were used after electro-chemical etching in KOH aqueous
solution, followed by tuning using field ion microscopy and controlled
indentation at clean Cu(1 1 1) surfaces. Measurements at
$5\text{\,}\mathrm{K}$ were performed with the Omicron low-temperature
scanning tunneling microscopy (LT-STM) system. The crystals were cleaved in
the ultra-high vacuum chamber at room temperature. A tungsten tip attached to
a qPlus sensor was used after controlled indentation at clean Au(1 1 1)
surfaces. Tunneling-conductance $\derivative*{I}{V}$ curves were obtained with
numerical differentials of tunneling current $I$ versus bias voltage $V$
curves.

An STM topographic image of the cleaved surface is shown in Fig. 1(d). One
type of surface is available in this material because the S-S dimer bonds may
not be broken. The zigzag structure does not contradict the model structure as
in Fig. 1(b). The observed lattice constant is around 560 pm, which agrees
with the literature value of NiS2 [23].

Tunneling conductance curves were measured on such cleaved surfaces at
$77\text{\,}\mathrm{K}$, in the PI phase [Fig. 1(e)]. The observed
$\derivative*{I}{V}$ spectra are approximately proportional to the local
density of states (LDOS), and hence we can directory refer to the insulating
gap. The measured insulating gap width is about $0.3\text{\,}\mathrm{eV}$ at
$77\text{\,}\mathrm{K}$, which is comparable to the reported gap width of the
bulk [24]. Contrary to the speculations of the previous reports [14, 15, 16,
17, 18, 7, 19], the 2D surfaces are found to be insulating.

As the surface of NiS2 is found to be insulating, metallic states may
originate from some defects or from some specific structures. We investigated
the electronic states near step edges because the lack of a good cleavage
plane in NiS2 results in a large number of step-terrace structures at
cleavages. The electronic state just below step edges cannot be investigated
with STM since a finite radius of a scanning tip causes tunneling current
through the side of the tip. Such artifacts of the tip shape are avoided on
terraces above steps. Thus, we focus on the electronic states above the steps
in the following, and the data are considered only for the topographic height
$\quantity|z|<$50\text{\,}\mathrm{pm}$$.

A step-terrace structure with 2-unit-cell hight [Fig. 2(a)] was investigated
with the spectroscopic-imaging technique, where the $\derivative*{I}{V}$
conductance curves are measured at each pixel of the image. Figure 2(b) shows
a $\derivative*{I}{V}$ map at $+200\text{\,}\mathrm{mV}$, representing the
local DOS distribution within the insulating gap. The in-gap states appear all
around the step edges. Thus, the behavior is not constrained to a specific
orientation of the step with respect to the crystalline axis. Although the tip
condition changed several times during the measurements as marked with black
arrows, the in-gap state is observed irrespective of the tip condition.

A magnified view shows a more detailed structure of the in-gap states. The
step was obtained from the same crystal at a different cleavage, and the step
height corresponds to 1 unit cell. The topographic image is shown in Fig.
3(a), and a DOS map at $+200\text{\,}\mathrm{mV}$ representing the in-gap edge
states is shown in Fig. 3(b). These images are deformed due to the thermal
drift but the atomic registry is maintained. Figure 3(c) compares the
tunneling spectra near and away from the step edge. The insulating gap is
almost closed near the step edge, resulting in a quasimetallic state in the 1D
step edge. The spatial distribution of the tunneling spectra near the step
edge is shown in Figs. 3(d) – (f). The in-gap states near the step edge are
not restricted to the step edge, but they spread over several nanometers into
the terrace.

Some bright spots in $\derivative*{I}{V}$ are observed on the terraces [Fig.
2(b)]. These spots probably originated from impurities embedded under the
surface. Although the $\derivative*{I}{V}$ spectra on these impurities are
very similar to those near the step edges, it is unlikely that the observed
gap collapse is associated with the impurities. This is because the in-gap
states homogeneously distribute around step edges regardless of the position
of impurities. Therefore, the metallic state is intrinsic to step edges of
NiS2.

The location of the gap with respect to the chemical potential depends on
samples (compare Fig. 1(e) and Fig. 3(c)). They probably originated from some
doping effects from impurities. We confirmed that the gap closes toward the
Fermi energy near step edges wherever the surface gap is.

To investigate the effect of magnetic order on the in-gap state near the step
edge, we performed spectroscopic-imaging measurements near step edges also in
the WFI phase at $5\text{\,}\mathrm{K}$. A NiS2 crystal from a different batch
was used. Atomically sharp step edges are resolved in the topographic image in
Fig. 4(a), where the tip probably feels the symmetry of Ni atoms as well. The
step height is $4\text{\,}\mathrm{nm}$, and the zigzag structure of S atoms is
perpendicular to the step. As shown in Figs. 4(b) – (f), the insulating gap
gradually collapses over a few nanometers as approaching the step edge,
similar to the paramagnetic phase.

A natural question is whether the edge-induced effect is observed when the
insulating gap is suppressed by the bandwidth control. To this end, we
investigated NiS_2-xSe_x (nominal $x=0.45$), which is located around the
boundary of the PI and the antiferromagnetic metal (AFM) phases [Fig. 1(c)].
As shown in Fig. 5, the topographic image exhibits two atomic sites with
different heights. Se atoms are considered to appear higher in constant-
current topographic images [22]. From the ratio between the numbers of higher
and lower atoms on the surface, $x$ is estimated to be 0.43, which agrees with
the nominal amount of Se. A $\derivative*{I}{V}$ map is measured at a step
edge with 0.5-unit-cell hight [Figs. 5(a) and (b)]. There is no gap in the
tunneling spectra as shown in Fig. 5(c). A line profile clarifies that any
edge effect was not evident at any bias voltage within $\pm
0.5\text{\,}\mathrm{V}$ [Figs. 5(d) – (f)].

The surface conduction in NiS2 has been proposed in the previous reports [14,
17, 15, 16, 19]. The present observations clarify that the 2D surfaces are as
insulating as their bulk, and in fact, their 1D edges are more metallic. Since
the in-gap states near step edges can provide an additional channel to the
conductance, whose amount is directly related to the surface area, we consider
that the previous observations are consistent for 1D-edge conductance. The
1D-edge channel can also infer why the previous angle-resolved photoemission
spectroscopy measurements did not report dispersive features within the Mott
gap even though a finite DOS at the Fermi level is observed [25, 18, 7, 9].

The question is what causes the observed metallic edge state. One possible
scenario concerns a bandwidth-controlled Mott insulator-to-metal transition,
which occurs when the onsite Coulomb repulsion $U$ becomes smaller with
respect to the bandwidth $W$. In bulk NiS2, physical or chemical pressures
induce the bandwidth-controlled Mott insulator-to-metal transition. However,
$U$ and $W$ are expected to become larger and smaller, respectively, at the
step edges due to weaker screening and hopping, which contradict the observed
metallic edge state. Particular edge termination may effectively cause local
doping, resulting in a metallic edge, as in the case of the metallic surface
of V_2-xCr_xO_3 with excess vanadyl cations [26]. In the present case,
however, the metallic edge states appear irrespective of the edge direction,
ruling out the possibility of a termination-dependent metallic state. Another
possible cause of the metallic state is topological edge states [27, 28, 29,
30, 31]. Although the topological aspect of the band structure has
theoretically been argued in NiS2 for $U=0$ [29], we are not aware of such an
expectation for a more realistic situation with $U\neq 0$. More experimental
and theoretical work is indispensable to clarify the underlying mechanism of
the 1D metallic edge states in pyrite NiS2.

In summary, the spectroscopic-imaging STM measurements on the charge-transfer
type Mott insulator NiS2 clarified that the 2D surface is as gapped as the 3D
bulk. The Mott gap collapses near the step edges in the WFI and the PI phases,
exhibiting a quasimetallic DOS. This behavior was not observed at the phase
boundary between the PI and the AFM phases. The spatially resolved results
obtained in this work indicate that the surface conduction so far inferred
from macroscopic measurements [14, 15, 16, 19] are not related to the surface
but associated with the step edges. Different types of “surface” effects have
been found in various $3d$ transition-metal disulfides [32, 33, 34, 35, 36,
37, 38]. We anticipate that their systematic spatially-resolved investigations
may help to understand the role of step edges in the pyrite structure.

## Acknowledgement

The authors acknowledge T. Yoshida, Y. Kohsaka, T. Machida, and C. J. Butler
for discussion. This work was supported by Grant-in-Aid JSPS KAKENHI Grant
Nos. JP19H01855, JP21H04652, JP21H05236, JP21K18181, JP22H04496, and
JP22H05448.

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principal $K\rtimes H$-bundle.

Using this correspondence, we can write down a classifying map for $P.$ That
is, a map $f\colon P\to EK\times EH$ such that $f(p(k,h))=f(p)(k,h).$ Firstly,
note that if $P\xrightarrow{\pi}M$ is a ${\mathcal{G}}$-bundle then we can
write the classifying map as follows: Let $\\{U_{\alpha}\\}$ be an open cover
of $M$ over which $P$ is trivial. Then $\pi^{-1}(U_{\alpha})$ is isomorphic to
$U_{\alpha}\times{\mathcal{G}}.$ Now choose local sections $s_{\alpha}\colon
U_{\alpha}\to\pi^{-1}(U_{\alpha})$ and define the functions
$g_{\alpha}\colon\pi^{-1}(U_{\alpha})\to{\mathcal{G}}$ by
$s_{\alpha}(m)=(m,g_{\alpha}(s_{\alpha}(m))),$ where we have used the
isomorphism to identify $\pi^{-1}(U_{\alpha})$ with
$U_{\alpha}\times{\mathcal{G}}.$ Now, let $\\{\psi_{\alpha}\\}$ be a partition
of unity subordinate to $\\{U_{\alpha}\\}.$ Then define the map
$f_{{\mathcal{G}}}\colon P\to E{\mathcal{G}}$ by

$f_{{\mathcal{G}}}(p)=\sum\psi_{\alpha}(\pi(p))g_{\alpha}(p).$

This is clearly ${\mathcal{G}}$-equivariant and so defines the classifying map
for $P.$

Now consider again the case where ${\mathcal{G}}=K\rtimes H.$ Write the
classifying map $f$ as a pair of functions $(f_{K},f_{H}).$ Then we require
that

$(f_{K}(p(k,h)),f_{H}(p(k,h)))=(\tilde{\varphi}_{h^{-1}}(f_{K}(p)k),f_{H}(p)h).$

Using the correspondence above, we can construct a pair of bundles
$P\xrightarrow{\pi_{K}}P/K\xrightarrow{\pi_{H}}M.$ Define $f_{H}$ to be the
classifying map of the $H$-bundle $P/K.$ To define $f_{K},$ consider an open
cover $\\{U_{\alpha}\\}$ of $M$ as above. Consider the cover
$\\{V_{\alpha}\\}$ of $P/K$ where $V_{\alpha}=\pi_{H}^{-1}(U_{\alpha}).$ $P$
is trivial over $V_{\alpha}$ since we can construct a local section as
follows. Identify $V_{\alpha}$ with $U_{\alpha}\times H.$ Then over the subset
$U_{\alpha}\times\\{1\\},$ $P$ has a section, say $\sigma_{\alpha}.$ We can
define a section of $P$ over $U_{\alpha}\times H$ by forcing $H$-equivariance.
That is, by defining
$\chi_{\alpha}(s_{\alpha}(m)h)\colon=\sigma_{\alpha}(m)h,$ where $s_{\alpha}$
is a local section of $P/K.$ So $\pi_{K}^{-1}(V_{\alpha})\simeq
U_{\alpha}\times H\times K.$ Then we can define the functions $k_{\alpha}$ as
above and we see that $k_{\alpha}(ph)=\varphi_{h^{-1}}(k_{\alpha}(p))$ (which
follows from the fact that $(pk)h=(ph)\varphi_{h^{-1}}(k)$). Therefore, if we
choose partitions of unity $\\{\psi_{a}\\}$ subordinate to $\\{U_{\alpha}\\}$
and $\\{\chi_{\alpha}\\}$ subordinate to $\\{V_{\alpha}\\},$ we can define

$f(p)=\left(\sum\chi_{\alpha}(\pi(p))k_{\alpha}(p),\sum\psi_{\alpha}(\pi(p))h_{\alpha}(\pi_{K}(p))\right),$

which is $K\rtimes H$-equivariant because

$\displaystyle f(p(k,h))$
$\displaystyle=\left(\sum\chi_{\alpha}(\pi(p))k_{\alpha}((pk)h),\sum\psi_{\alpha}(\pi(p))h_{\alpha}(\pi_{K}(ph))\right)$
$\displaystyle=\left(\sum\chi_{\alpha}(\pi(p))\varphi_{h^{-1}}(k_{\alpha}(p)k),\sum\psi_{\alpha}(\pi(p))h_{\alpha}(\pi_{K}(p))h\right)$
$\displaystyle=f(p)(k,h).$

Thus $f$ is a classifying map for $P.$

###  B.2 $LG\rtimes S^{1}$-bundles

We have shown in the previous section that principal $K\rtimes H$-bundles are
equivalent to $K$-bundles with a twisted equivariant $H$ action over
$H$-bundles. Consider now the case where $K=LG$ and $H=S^{1},$ as in chapter
4. We have already seen (see section 4.2) that there is a bijective
correspondence between isomorphism classes of principal $LG\rtimes
S^{1}$-bundles and isomorphism classes of principal $G$-bundles over
$S^{1}$-bundles. The result from section B.1, however, implies that we could
construct a principal $LG$-bundle over a circle bundle. Namely, the bundle
$P\to P/LG=(P\times S^{1})/LG\rtimes S^{1}$ is a principal $LG$ bundle. We
would like to understand the relationship between the $LG$-bundle we have
constructed and the $G$-bundle we have constructed in section 4.2. Consider
the map

$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\widetilde{P}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P/LG\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{P}/G}$

given by $f(p)=[p,1,1]$ (and where the induced map $LG\to G$ is the
homomorphism $\gamma\mapsto\gamma(1)$). This is a bundle map since

$\displaystyle f(p\gamma)$ $\displaystyle=[p(\gamma,1),1,1]$
$\displaystyle=[p,\gamma(1),1]$ $\displaystyle=[p,1,1]\gamma(1)$
$\displaystyle=f(p)\gamma(1).$

Therefore, we see that $\widetilde{P}\simeq P\times_{LG}G$ (via the
isomorphism $[p,g]\mapsto[p,g,1]$). So $\widetilde{P}$ is given by extending
the structure group of $P$ from $LG$ to $G.$

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