# Deep learning for quadratic hedging in incomplete jump market

Nacira Agram1, Bernt Øksendal2 & Jan Rems3

###### Abstract

We propose a deep learning approach to study the minimal variance pricing and
hedging problem in an incomplete jump diffusion market. It is based upon a
rigorous stochastic calculus derivation of the optimal hedging portfolio,
optimal option price, and the corresponding equivalent martingale measure
through the means of the Stackelberg game approach. A deep learning algorithm
based on the combination of the feedforward and LSTM neural networks is tested
on three different market models, two of which are incomplete. In contrast,
the complete market Black-Scholes model serves as a benchmark for the
algorithm’s performance. The results that indicate the algorithm’s good
performance are presented and discussed.

In particular, we apply our results to the special incomplete market model
studied by Merton and give a detailed comparison between our results based on
the minimal variance principle and the results obtained by Merton based on a
different pricing principle. Using deep learning, we find that the minimal
variance principle leads to typically higher option prices than those deduced
from the Merton principle. On the other hand, the minimal variance principle
leads to lower losses than the Merton principle.

11footnotetext: Department of Mathematics, KTH Royal Institute of Technology
100 44, Stockholm, Sweden.
Email<EMAIL_ADDRESS>Work supported by the Swedish Research Council grant
(2020-04697).22footnotetext:  Department of Mathematics, University of Oslo,
Oslo, Norway.
Email<EMAIL_ADDRESS>Department of Mathematics,
University of Ljubljana, Ljubljana, Slovenia.
Email<EMAIL_ADDRESS>

Keywords : Option pricing; Incomplete market; Equivalent martingale measure;
Merton model; Deep learning; LSTM.

##  1 Introduction

Many jump diffusion models have been extensively studied in the past. Of
particular interest to us is the Merton model proposed by Merton (1976), where
the jump distribution is log-normal. Merton derived a closed-form solution to
the hedging problem for the European option, assuming diversifiabilty of the
jump component. There are also other popular jump diffusion models. Kou (2002)
proposed an alternative approach where double exponential jump distribution is
assumed. The variance gamma model proposed in (Madan and Seneta, 1990) also
supports the inclusion of jumps.

Regarding the use of machine learning in stochastic control theory, there have
been a lot of new developments in the past few years. With new techniques, it
has now become possible to solve problems that do not obtain closed-form
solutions numerically. In (Han et al., 2016), a deep learning approach for
solving stochastic control problems was proposed. The idea is that one
discretizes time and, at each timestep, approximates the control with a feed-
forward neural network. In the papers (Buehler, Gonon, Teichmann, and Wood,
2019; Carbonneau and Godin, 2021), the authors extend these results to
reinforcement learning algorithms for hedging in high dimensions. Methods
proposed in (Han et al., 2016) also provide a strong tool for finding the
numerical solutions to high dimensional partial differential equations. In a
seminal paper (Han, Jentzen, and E, 2018) and later on in (Beck, E, and
Jentzen, 2019; Chan-Wai-Nam, Mikael, and Warin, 2019), the authors transform
PDEs to backward stochastic differential equations and use deep learning to
solve the associated stochastic control problem.

Since control problems related to hedging rely heavily on time series
dynamics, it is convenient to consider recurrent neural networks (RNN) instead
of feed-forward ones. These types of networks were proposed in (Rumelhart,
Hinton, and Williams, 1986). The recurring nature of the network can contain
cycles, which makes them suitable for problems that have sequential inputs. A
particular case of RNN called long short-term memory (LSTM) was first
introduced in (Hochreiter and Schmidhuber, 1997) and became very prominent in
the last few years for time series modelling. The main advantage over RNN is
LSTM’s ability to provide a memory structure that can store information over a
large number of timesteps. It has been heavily used in financial time series
prediction; for instance, see (Cao, Li, and Li, 2019; Selvin, Vinayakumar,
Gopalakrishnan, Menon, and Soman, 2017; Siami-Namini, Tavakoli, and Namin,
2018). One of the main advantages of LSTM networks is that they can also be
used for non-Markovian models. In particular, Han et al. used them in (2021)
to apply the methods proposed in (Han et al., 2016) to stochastic control
problems with delay. Deep learning for mean-field stochastic control has been
studied by Agram et al. (2020) where the initial state was a part of the
control process and then extended to mean-field systems with delay in (Agram,
Grid, Kebiri, and Øksendal, 2022). Lai et al. (2022) also used LSTM networks
to develop a data-driven approach for options market making.

Since our main concern is hedging in an incomplete market, we also refer to
(2020), where Fecamp et al. present several deep learning algorithms for
discrete-time hedging, where the sources of incompleteness are illiquidity,
non-tradable risk factors, and transaction costs. They propose a modified
global LSTM approach where a single LSTM net is used that takes the whole time
series as an input. The above approach was adopted in (Gao, Wu, and Duan,
2021), where they study the double exponential jump distribution model
proposed in (Kou, 2002).

Furthermore, we do not consider the calibration of SDEs to fit market data in
this paper. For promising approaches, one can check (Boursin, Remlinger, and
Mikael, 2022; Remlinger, Mikael, and Elie, 2022) where generative adversarial
networks are used. Another approach could feature neural SDEs. A good survey
is (Kidger, Foster, Li, and Lyons, 2021).

In what follows, let us describe our purpose in the present paper.
We consider a financial market where the underlying price of a risky asset
follows a jump diffusion process

$\displaystyle dS(t)$
$\displaystyle=S(t^{-})\left[\alpha(t)dt+\sigma(t)dB(t)+\int_{\mathbb{R^{*}}}\gamma(t,\zeta)\widetilde{N}(dt,d\zeta)\right],\quad
S(0)>0,$ (1.1)

with $\mathbb{R}^{*}=\mathbb{R}\setminus\\{0\\}$, and $S_{0}(t)=1$ for all $t$
denotes the value of a risk-free money market account.

Suppose now that the interest rate in the risk free asset equals 0 and the
time to maturity is finite and denoted by $T$.
For simplicity of notation, we will write $S(t)$ instead of $S(t^{-})$ in the
following.
Here
$\alpha(t)\in\mathbb{R},\sigma(t)=(\sigma_{1}(t),...,\sigma_{m}(t))\in\mathbb{R}^{m},$
$\gamma(t,\zeta)=(\gamma_{1}(t,\zeta),...,\gamma_{k}(t,\zeta))\in\mathbb{R}^{k}$,
and $B(t)=(B_{1}(t),...,B_{m}(t))^{{}^{\prime}}\in\mathbb{R}^{m}$ and
$\widetilde{N}(dt,d\zeta)=(\widetilde{N}_{1}(dt,d\zeta),...,\widetilde{N}_{k}(dt,d\zeta))^{{}^{\prime}}\in\mathbb{R}^{k}$
are independent Brownian motions and compensated Poisson random measures,
respectively, defined on a complete filtered probability space
$(\Omega,\mathcal{F},\mathbb{F}=\\{\mathcal{F}_{t}\\}_{t\geq 0},P)$, where
$\mathbb{F}=\\{\mathcal{F}_{t}\\}_{t\geq 0}$ is the filtration generated by
$\\{B(s)\\}_{s\leq t}$ and $\\{N(s,\zeta)\\}_{s\leq t}$. We are using the
matrix notation, i.e.

$\displaystyle\sigma(t)dB(t):$
$\displaystyle=\sum_{i=1}^{m}\sigma_{i}(t)dB_{i}(t),\quad\gamma(t,\zeta)\widetilde{N}(dt,d\zeta)=\sum_{j=1}^{k}\gamma_{j}(t,\zeta)\widetilde{N}_{j}(dt,d\zeta),$

and we assume that

$\gamma(t,\zeta)>-1\quad\text{ and
}\quad\sigma^{2}(t)+\int_{\mathbb{R^{*}}}\gamma^{2}(t,\zeta)\nu(d\zeta)>0,\;\text{
for a.a.}\;t,\text{ a.s.}$ (1.2)

The coefficients $\alpha,\sigma$ and $\gamma$ are supposed to be deterministic
and bounded.

The results in this paper can easily be extended to an arbitrary number of
risky assets, but since the features of incomplete markets we are dealing
with, can be fully illustrated by one jump diffusion risky asset only, we will
for simplicity concentrate on this case in the following. We emphasise
however, that we deal with an arbitrary number $m$ of independent Brownian
motions and an arbitrary number $k$ of independent Poisson random measures in
the representation (1.1).

Let $z\in\mathbb{R}$ be an initial endowment and let $\pi(t)\in\mathbb{R}$ be
a self-financing portfolio, representing the fraction of the total wealth
$X(t)=X_{z,\pi}(t)$ invested in the risky asset at time $t$. We say that $\pi$
is admissible if in addition $\pi$ is a predictable process in $L^{2}(dt\times
dP)$. The set of admissible portfolios is denoted by $\mathcal{A}$. We
associate to an admissible portfolio $\pi$ the wealth dynamics

$\displaystyle
dX(t)=X(t)\pi(t)\left[\alpha(t)dt+\sigma(t)dB(t)+\int_{\mathbb{R^{*}}}\gamma(t,\zeta)\widetilde{N}(dt,d\zeta)\right],\quad
X(0)=z.$ (1.3)

Here and in the following, we write for notational simplicity $X(t)$ instead
of $X(t^{-})$, in agreement with our convention for $S(t)$.

Fix $T$ be the time to maturity and let $F$ be a given $T$-claim, i.e. $F\in
L^{2}(P)$ is an $\mathcal{F}_{T}$-measurable random variable, representing the
payoff at that time. Then for each initial wealth $z\in\mathbb{R}$ and each
portfolio $\pi\in\mathcal{A}$, we want to minimize the expected squared
hedging error

$\displaystyle
J(z,\pi)=E\left[\frac{1}{2}\left(X_{z,\pi}(T)-F\right)^{2}\right].$

Then the problem we consider is the following.

###### Problem 1.1

Find the optimal initial endowment $\widehat{z}\in\mathbb{R}$ and the optimal
portfolio $\widehat{\pi}\in\mathcal{A}$, such that

$\underset{z,\pi}{\inf}J(z,\pi)=J(\widehat{z},\widehat{\pi}).$

Heuristically, this means that we define the price of the option with payoff
$F$ to be the initial endowment $\widehat{z}$ needed to get the terminal
wealth $X(T)$ as close as possible to $F$ in quadratic mean by an admissible
portfolio $\widehat{\pi}$.

We may regard the minimal variance problem (Problem 1.1) as a Stackelberg
game, in which the first player chooses the initial endowment $z$, followed by
the second player choosing the optimal portfolio $\pi$ based on this initial
endowment. Knowing this response $\pi=\pi_{z}$ from the follower, the first
player chooses the initial endowment $\widehat{z}$ which leads to a response
$\pi=\pi_{\widehat{z}}$ which is optimal, in the sense that
$J(\widehat{z},\pi_{\widehat{z}})\leq J(z,\pi)$ over all admissible pairs
$(z,\pi)$.

For a general (discontinuous) semimartingale market, this is already a known
result; we refer, for example, to (Černý and Kallsen, 2007) where the authors
have proved that the optimal quadratic hedging strategy $\widehat{\pi}$ can
first be characterized in a linear feedback form which is independent of the
initial endowment $z$, and that $\widehat{z}$ can then be identified in a
second step. Therefore, the results of Section 2 of the current paper might be
derived from already existing results in the literature (Černý and Kallsen,
2007) but we found it convenient for the reader to find explicit expressions
of the optimal quadratic hedging problem for a jump diffusion market (Lévy
process).
Schweizer (1996) proves (under some conditions, including a non-arbitrage
condition) that there exists a signed measure $\tilde{P}$, called the
variance-optimal measure, such that

$\displaystyle\widehat{z}=E_{\tilde{P}}[F].$ (1.4)

We consider in the present paper a jump diffusion market and we show that
$\tilde{P}$ is a positive measure and we find it explicitly.

In the next section, we prove the existence of the optimal hedging strategy
and the optimal initial endowment for jump diffusion market.
We show that the minimal variance price can be represented as the expected
value of the option’s (discounted) payoff under some equivalent martingale
measures (which is given explicitly). The analysis concludes with looking at
specific examples of jump diffusion market models.

Section 3 is devoted to the deep learning approach. Since the option price and
the optimal hedging portfolio can not always be computed explicitly, or it is
not feasible, we need to use some other numerical methods. We propose a deep
learning algorithm that approximates option price and hedging portfolio. We
propose a joint feed-forward and multilayered LSTM network that adopts the
”online” approximation approach proposed in (Han et al., 2016), where a neural
network is used at each timestep to predict the control process.

First, we test this algorithm in the case of a complete market, namely in the
case of the Black Scholes (BS) model. This is done in order to estimate its
performance since both option price and hedging portfolio can be obtained
explicitly in the BS case.

Then we apply it to models that assume an incomplete market. First, a
continuous model where the underlying asset depends on multiple independent
Brownian motions is used in order to study the algorithm’s scalability to
multi-dimensional inputs. The same is then also done for the jump diffusion
Merton model.

We show the algorithm’s success in all three models and discuss its
performance. Since Merton’s reasoning behind his hedging strategy is not
consistent with the behaviour of financial markets, we also compare our
results to those obtained by Merton and discuss how the approaches differ. In
particular, we show how the minimal variance approach provides a safer hedging
strategy compared to the one proposed by Merton. Finally, the algorithm’s
performance is also tested for another jump diffusion model, namely the Kou
model.

##  2 Optimal quadratic hedging portfolio/optimal initial endowment

In this section, we find the optimal strategy pair to Problem 1.1.

###  2.1 Equivalent martingale measures (EMMs)

Since EMMs play a crucial role in our discussion, we start this section by
recalling that an important group of measures $Q\in\mathbb{M}$ can be
described as follows (we refer to Chapter 1 in (Øksendal and Sulem, 2007) for
more details):
Let $\theta_{0}(t)$ and $\theta_{1}(t,\zeta)>-1$ be $\mathbb{F}$-predictable
processes such that

$\displaystyle\alpha(t)+\theta_{0}(t)\sigma(t)+\int_{\mathbb{R^{*}}}\theta_{1}(t,\zeta)\gamma(t,\zeta)\nu(d\zeta)=0\text{,
}\quad t\geq 0.$ (2.1)

Define the local martingale $Z(t)=Z^{\theta_{0},\theta_{1}}(t)$, by

$\displaystyle
dZ(t)=Z(t)\left[\theta_{0}(t)dB(t)+\int_{\mathbb{R^{*}}}\theta_{1}(t,\zeta)\widetilde{N}(dt,d\zeta)\right],\quad
Z(0)=1,$

i.e.,

$\displaystyle Z(t)$
$\displaystyle=\exp\left(\int_{0}^{t}\theta_{0}(s)dB(s)-\frac{1}{2}\int_{0}^{t}\theta_{0}^{2}(s)ds\right.+\int_{0}^{t}\int_{\mathbb{R^{*}}}\left\\{\ln(1+\theta_{1}(s,\zeta))-\theta_{1}(s,\zeta)\right\\}\nu(d\zeta)ds$
$\displaystyle\left.+\int_{0}^{t}\int_{\mathbb{R^{*}}}\ln(1+\theta_{1}(s,\zeta))\widetilde{N}(ds,d\zeta)\right).$

A sufficient condition for $Z(\cdot)$ being a true martingale is

$\displaystyle
E\Big{[}\exp\Big{(}\frac{1}{2}\int_{0}^{T}\theta_{0}^{2}(s)ds+\int_{0}^{T}\int_{\mathbb{R^{*}}}\theta_{1}^{2}(s,\zeta)N(ds,d\zeta)\Big{)}\Big{]}<\infty.$
(2.2)

For proof see (Kallsen and Shiryaev, 2002). Then the measure
$Q^{\theta_{0},\theta_{1}}$ defined by

$\displaystyle dQ^{\theta_{0},\theta_{1}}(\omega)=Z(T)dP(\omega)\text{ on
}\mathcal{F}_{T}$

is in $\mathbb{M}$.

###  2.2 The optimal portfolio

Assume as before that the wealth process $X(t)=X_{z,\pi}(t)$, corresponding to
an initial wealth $z$ and a self-financing portfolio $\pi$, is given by

$\displaystyle
dX(t)=X(t)\pi(t)\Big{[}\alpha(t)dt+\sigma(t)dB(t)+\int_{\mathbb{R^{*}}}\gamma(t,\zeta)\widetilde{N}(dt,d\zeta)\Big{]},\quad
X(0)=z.$

Let $F$ be a given $T$-claim representing the terminal payoff of the option.
By the Itô/martingale representation theorem for jump diffusions (see (Løkka,
2004)) we can write $F=F(T)$, where the martingale
$F(t):=E[F|\mathcal{F}_{t}],$ $t\in[0,T]$ has the Itô-Lévy representation

$\displaystyle
dF(t)=\beta(t)dB(t)+\int_{\mathbb{R^{*}}}\kappa(t,\zeta)\widetilde{N}(dt,d\zeta),\quad
E[F]=F(0),$

for unique $\mathbb{F}$-predictable processes $\beta(t)\in
L^{2}(\lambda_{0}\times P),\kappa(t,\zeta)\in L^{2}(\lambda_{0}\times\nu\times
P)$, where $\lambda_{0}$ denotes Lebesgue measure.
Note that $\beta$ and $\kappa$ depend linearly on $F$ (they can be given in
terms of Malliavin derivatives of $F$).

Then by the Itô formula for jump diffusions (see e.g. Theorem 1.14 in
(Øksendal and Sulem, 2007)) we get

$\displaystyle d(X(t)F(t))$ $\displaystyle=X(t)dF(t)+F(t)dX(t)+d[X,F]_{t}$
$\displaystyle=X(t)\Big{[}\beta(t)dB(t)+\int_{\mathbb{R^{*}}}\kappa(t,\zeta)\widetilde{N}(dt,d\zeta)\Big{]}$
$\displaystyle+X(t)F(t)\Big{[}\pi(t)\alpha(t)dt+\pi(t)\sigma(t)dB(t)+\int_{\mathbb{R^{*}}}\pi(t)\gamma(t,\zeta)\widetilde{N}(dt,d\zeta)\Big{]}$
$\displaystyle+X(t)\Big{[}\pi(t)\sigma(t)\beta(t)dt+\int_{\mathbb{R^{*}}}\pi(t)\gamma(t,\zeta)\kappa(t,\zeta)\widetilde{N}(dt,d\zeta)$
$\displaystyle+\int_{\mathbb{R^{*}}}\pi(t)\gamma(t,\zeta)\kappa(t,\zeta)\nu(d\zeta)dt\Big{]}.$

Hence

$\displaystyle E[X(T)F(T)]$
$\displaystyle=zF(0)+\int_{0}^{T}E\Big{[}X(t)\Big{\\{}F(t)\pi(t)\alpha(t)+\pi(t)\sigma(t)\beta(t)$
$\displaystyle+\int_{\mathbb{R^{*}}}\pi(t)\gamma(t,\zeta)\kappa(t,\zeta)\nu(d\zeta)\Big{\\}}\Big{]}dt.$

Similarly,

$\displaystyle
E[X^{2}(T)]=z^{2}+\int_{0}^{T}E\Big{[}X^{2}(t)\Big{\\{}2\pi(t)\alpha(t)+\pi^{2}(t)\sigma^{2}(t)+\int_{\mathbb{R^{*}}}\pi^{2}(t)\gamma^{2}(t,\zeta)\nu(d\zeta)\Big{\\}}\Big{]}dt,$

and

$\displaystyle
E[F^{2}(T)]=F(0)^{2}+\int_{0}^{T}E\Big{[}\beta^{2}(t)+\int_{\mathbb{R^{*}}}\kappa^{2}(t,\zeta)\nu(d\zeta)\Big{]}dt.$

This gives

$\displaystyle J(\pi)$
$\displaystyle=E\Big{[}\frac{1}{2}(X(T)-F)^{2}\Big{]}=\frac{1}{2}\Big{(}E[X^{2}(T)]-2E[X(T)F(T)]+E[F^{2}(T)]\Big{)}$
$\displaystyle=\frac{1}{2}(z-F(0))^{2}+\frac{1}{2}E\Big{[}\int_{0}^{T}\Big{\\{}2X^{2}(t)\pi(t)\alpha(t)+X^{2}(t)\pi^{2}(t)\Big{(}\sigma^{2}(t)+\int_{\mathbb{R^{*}}}\gamma^{2}(t,\zeta)\nu(d\zeta)\Big{)}$
$\displaystyle-2X(t)\pi(t)\Big{(}F(t)\alpha(t)+\sigma(t)\beta(t)+\int_{\mathbb{R^{*}}}\gamma(t,\zeta)\kappa(t,\zeta)\nu(d\zeta)\Big{)}$
$\displaystyle+\beta^{2}(t)+\int_{\mathbb{R^{*}}}\kappa^{2}(t,\zeta)\nu(d\zeta)\Big{\\}}dt\Big{]}.$

We can minimize $J(\pi)$ by minimizing the $dt$-integrand pointwise for each
$t$. This gives the following result:

###### Theorem 2.1

Recall that we have made the assumption (1.2).
a) For given initial value $X(0)=z>0$ the portfolio
$\widehat{\pi}=\widehat{\pi}_{z}$ which minimises

$\displaystyle\pi\mapsto E\Big{[}\frac{1}{2}(X_{z,\pi}(T)-F)^{2}\Big{]}$

is given in feedback form with respect to $X(t)=X_{z,\widehat{\pi}}$ by

$\displaystyle\widehat{\pi}(t)$
$\displaystyle=\widehat{\pi}(t,X(t))=\frac{F(t)\alpha(t)+\sigma(t)\beta(t)+\int_{\mathbb{R^{*}}}\gamma(t,\zeta)\kappa(t,\zeta)\nu(d\zeta)-X(t)\alpha(t)}{X(t)\Big{(}\sigma^{2}(t)+\int_{\mathbb{R^{*}}}\gamma^{2}(t,\zeta)\nu(d\zeta)\Big{)}},$
(2.3)

or, equivalently,

$\displaystyle\widehat{\pi}(t)X(t)=G(t)[X(t)-F(t)]+\frac{\sigma(t)\beta(t)+\int_{\mathbb{R^{*}}}\gamma(t,\zeta)\kappa(t,\zeta)\nu(d\zeta)}{\sigma^{2}(t)+\int_{\mathbb{R^{*}}}\gamma^{2}(t,\zeta)\nu(d\zeta)},$

where

$G(t)=-\alpha(t)\Big{(}\sigma^{2}(t)+\int_{\mathbb{R^{*}}}\gamma^{2}(t,\zeta)\nu(d\zeta)\Big{)}^{-1}.$
(2.4)

b) Given an initial value $z>0$, the corresponding optimal wealth
$X_{\widehat{\pi}}(t)=\widehat{X}(t)$ solves the SDE

$\displaystyle d\widehat{X}(t)$
$\displaystyle=\widehat{X}(t)\widehat{\pi}(t,\widehat{X}(t))\Big{[}\alpha(t)dt+\sigma(t)dB(t)+\int_{\mathbb{R^{*}}}\gamma(t,\zeta)\widetilde{N}(dt,d\zeta)\Big{]}$
$\displaystyle=\frac{F(t)\alpha(t)+\sigma(t)\beta(t)+\int_{\mathbb{R^{*}}}\gamma(t,\zeta)\kappa(t,\zeta)\nu(d\zeta)-X(t)\alpha(t)}{\sigma^{2}(t)+\int_{\mathbb{R^{*}}}\gamma^{2}(t,\zeta)\nu(d\zeta)}\times$
$\displaystyle\times\Big{[}\alpha(t)dt+\sigma(t)dB(t)+\int_{\mathbb{R^{*}}}\gamma(t,\zeta)\widetilde{N}(dt,d\zeta)\Big{]}.$
(2.5)

###  2.3 The optimal initial endowment

Completing the Stackelberg game, we now proceed to find the initial endowment
$\widehat{z}$ which leads to a response $\widehat{\pi}=\pi_{\widehat{z}}$ that
is optimal for Problem 1.1, in a sense that $J(\widehat{z},\widehat{\pi})\leq
J(z,\pi)$ over all pairs $(z,\pi)$. We shall first find the explicit solution
for $X$.

Writing $X=\widehat{X}$ for notational simplicity, equation (2.1) is of the
form

$dX\left(t\right)=C\left(t\right)d\Lambda_{t}+X\left(t\right)d\Gamma_{t};\quad
X\left(0\right)=z,$ (2.6)

where

$C\left(t\right)=\frac{F\left(t\right)\alpha\left(t\right)+\sigma\left(t\right)\beta\left(t\right)+\int_{\mathbb{R^{*}}}\gamma\left(t,\zeta\right)\kappa\left(t,\zeta\right)\nu\left(d\zeta\right)}{\sigma^{2}\left(t\right)+\int_{\mathbb{R^{*}}}\gamma^{2}\left(t,\zeta\right)\nu\left(d\zeta\right)},$
(2.7)
$d\Lambda_{t}=\alpha\left(t\right)dt+\sigma\left(t\right)dB\left(t\right)+\int_{\mathbb{R^{*}}}\gamma\left(t,\zeta\right)\widetilde{N}\left(dt,d\zeta\right),$
(2.8)
$d\Gamma_{t}=\alpha_{1}\left(t\right)dt+\sigma_{1}\left(t\right)dB\left(t\right)+\int_{\mathbb{R^{*}}}\gamma_{1}\left(t,\zeta\right)\widetilde{N}\left(dt,d\zeta\right),$

with

$\alpha_{1}\left(t\right)=G\left(t\right)\alpha\left(t\right);\quad\sigma_{1}\left(t\right)=G\left(t\right)\sigma\left(t\right);\quad\gamma_{1}\left(t,\zeta\right)=G\left(t\right)\gamma\left(t,\zeta\right).$

We rewrite (2.6) as

$dX\left(t\right)-X\left(t\right)d\Gamma_{t}=C\left(t\right)d\Lambda_{t},$
(2.9)

and multiply this equation by a process of the form

$Y_{t}=Y_{t}^{\left(\rho,\lambda,\theta\right)}=\exp\left(A_{t}^{\left(\rho,\lambda,\theta\right)}\right),$

with

$A_{t}^{\left(\rho,\lambda,\theta\right)}=\int_{0}^{t}\rho\left(s\right)ds+\int_{0}^{t}\lambda\left(s\right)dB\left(s\right)+\int_{0}^{t}\int_{\mathbb{R^{*}}}\theta\left(s,\zeta\right)\widetilde{N}\left(ds,d\zeta\right),$

where $\rho,\lambda$ and $\theta$ are processes to be determined.
Then (2.9) gets the form

$Y_{t}dX\left(t\right)-Y_{t}X\left(t\right)d\Gamma_{t}=Y_{t}C\left(t\right)d\Lambda_{t}.$
(2.10)

We want to choose $\rho,\lambda$ and $\theta$ such that $Y_{t}$ becomes an
integrating factor, in the sense that

$d\left(Y_{t}X\left(t\right)\right)=Y_{t}dX\left(t\right)-Y_{t}X\left(t\right)d\Gamma_{t}+\text{terms
not depending on }X.$

To this end, note that by the Itô formula for Lévy processes, we have

$\displaystyle dY_{t}$ $\displaystyle=$ $\displaystyle
Y_{t}\left[\rho\left(t\right)dt+\lambda\left(t\right)dB\left(t\right)\right]+\frac{1}{2}Y_{t}\lambda^{2}\left(t\right)dt$
$\displaystyle+\int_{\mathbb{R^{*}}}\left\\{\exp\left(A_{t}+\theta\left(t,\zeta\right)\right)-\exp\left(A_{t}\right)-\exp\left(A_{t}\right)\theta\left(t,\zeta\right)\right\\}\nu\left(d\zeta\right)dt$
$\displaystyle+\int_{\mathbb{R^{*}}}\left\\{\exp\left(A_{t}+\theta\left(t,\zeta\right)\right)-\exp\left(A_{t}\right)\right\\}\widetilde{N}\left(dt,d\zeta\right)$
$\displaystyle=$ $\displaystyle
Y_{t}\left[\left\\{\rho\left(t\right)+\frac{1}{2}\lambda^{2}\left(t\right)+\int_{\mathbb{R^{*}}}\left(e^{\theta\left(t,\zeta\right)}-1-\theta\left(t,\zeta\right)\right)\nu\left(d\zeta\right)\right\\}dt\right.$
$\displaystyle\left.+\lambda\left(t\right)dB\left(t\right)+\int_{\mathbb{R^{*}}}\left(e^{\theta\left(t,\zeta\right)}-1\right)\widetilde{N}\left(dt,d\zeta\right)\right].$

Therefore, again by the Itô formula, using (2.6),

$\displaystyle d\left(Y_{t}X\left(t\right)\right)$ $\displaystyle=$
$\displaystyle
Y_{t}dX\left(t\right)+X\left(t\right)dY_{t}+d\left[X,Y\right]_{t}$
$\displaystyle=$ $\displaystyle
Y_{t}dX\left(t\right)+Y_{t}X\left(t\right)\left[\left\\{\rho+\frac{1}{2}\lambda^{2}+\int_{\mathbb{R^{*}}}\left(e^{\theta\left(t,\zeta\right)}-1-\theta\left(t,\zeta\right)\right)\nu\left(d\zeta\right)\right\\}dt\right.$
$\displaystyle\left.+\lambda\left(t\right)dB\left(t\right)+\int_{\mathbb{R^{*}}}\left(e^{\theta\left(t,\zeta\right)}-1\right)\widetilde{N}\left(dt,d\zeta\right)\right]$
$\displaystyle+Y_{t}X\left(t\right)\left[\left\\{\lambda\left(t\right)\sigma_{1}\left(t\right)+\int_{\mathbb{R^{*}}}\left(e^{\theta\left(t,\zeta\right)}-1\right)\gamma_{1}\left(t,\zeta\right)\nu\left(d\zeta\right)\right\\}dt\right.$
$\displaystyle\left.+\int_{\mathbb{R^{*}}}\left(e^{\theta\left(t,\zeta\right)}-1\right)\gamma_{1}\left(t,\zeta\right)\widetilde{N}\left(dt,d\zeta\right)\right]+Y_{t}C\left(t\right)dK_{t},$

where

$dK_{t}=\left\\{\lambda\left(t\right)\sigma\left(t\right)+\int_{\mathbb{R^{*}}}\left(e^{\theta\left(t,\zeta\right)}-1\right)\gamma\left(t,\zeta\right)\nu\left(d\zeta\right)\right\\}dt+\int_{\mathbb{R^{*}}}\left(e^{\theta\left(t,\zeta\right)}-1\right)\gamma\left(t,\zeta\right)\widetilde{N}\left(dt,d\zeta\right).$
(2.11)

This gives

$\displaystyle
d\left(Y_{t}X\left(t\right)\right)-Y_{t}dX\left(t\right)+Y_{t}X\left(t\right)d\Gamma_{t}$
$\displaystyle=Y_{t}X\left(t\right)\Big{[}\Big{\\{}\rho+\alpha_{1}+\frac{1}{2}\lambda^{2}+\lambda\sigma_{1}+\int_{\mathbb{R^{*}}}\left(e^{\theta\left(t,\zeta\right)}-1-\theta\left(t,\zeta\right)\right)\nu\left(d\zeta\right)\Big{\\}}dt$
$\displaystyle+\left(\lambda\left(t\right)+\sigma_{1}\left(t\right)\right)dB\left(t\right)+\int_{\mathbb{R^{*}}}\left(e^{\theta\left(t,\zeta\right)}-1\right)\gamma_{1}\left(t,\zeta\right)\nu\left(d\zeta\right)dt$
$\displaystyle+\int_{\mathbb{R^{*}}}\Big{\\{}\left(e^{\theta\left(t,\zeta\right)}-1\right)\left(1+\gamma_{1}\left(t,\zeta\right)\right)+\gamma_{1}\left(t,\zeta\right)\Big{\\}}\widetilde{N}\left(dt,d\zeta\right)$
$\displaystyle+Y_{t}C\left(t\right)dK_{t}.$

Choose $\theta\left(t,\zeta\right)=\widehat{\theta}\left(t,\zeta\right)$, such
that

$(e^{\theta\left(t,\zeta\right)}-1)\left(1+\gamma_{1}\left(t,\zeta\right)\right)+\gamma_{1}\left(t,\zeta\right)=0,$

i.e.

$\widehat{\theta}\left(t,\zeta\right)=-\ln\left(1+\gamma_{1}\left(t,\zeta\right)\right).$

Next, choose $\lambda\left(t\right)=\widehat{\lambda}\left(t\right)$ such that

$\widehat{\lambda}\left(t\right)=-\sigma_{1}\left(t\right).$

Finally, choose $\rho\left(t\right)=\widehat{\rho}\left(t\right)$, such that

$\displaystyle\widehat{\rho}(t)$
$\displaystyle=-\Big{[}\alpha_{1}(t)+\frac{1}{2}\sigma_{1}^{2}(t)-\sigma_{1}^{2}(t)+\int_{\mathbb{R^{*}}}\Big{(}e^{\widehat{\theta}(t,\zeta)}-1-\widehat{\theta}(t,\zeta)+(e^{\widehat{\theta}(t,\zeta)}-1)\gamma_{1}(t,\zeta)\Big{)}\nu
d\zeta)\Big{]}$
$\displaystyle=-\left[\alpha_{1}\left(t\right)-\frac{1}{2}\sigma_{1}^{2}\left(t\right)+\int_{\mathbb{R^{*}}}\Big{(}\ln(1+\gamma_{1}(t,\zeta))-\gamma_{1}(t,\zeta)\Big{)}\nu(d\zeta)\right].$

Then

$\displaystyle\widehat{A}_{t}:$
$\displaystyle=A_{t}^{(\widehat{\rho},\widehat{\lambda},\widehat{\theta})}$
$\displaystyle=-\Big{[}\int_{0}^{t}\\{\alpha_{1}(s)-\frac{1}{2}\sigma_{1}^{2}(s)+\int_{\mathbb{R^{*}}}(\ln(1+\gamma_{1}(s,\zeta))-\gamma_{1}(s,\zeta))\nu(d\zeta)\\}ds$
$\displaystyle+\int_{0}^{t}\sigma_{1}(s)dB(s)+\int_{0}^{t}\int_{\mathbb{R^{*}}}\ln(1+\gamma_{1}(s,\zeta))\widetilde{N}(ds,d\zeta)\Big{]},$
(2.12)

with
$\widehat{Y}_{t}=Y_{t}^{\left(\widehat{\rho},\widehat{\lambda},\widehat{\theta}\right)}=\exp(\widehat{A}_{t})$,
we have, by (2.11)

$\displaystyle
d\left(\widehat{Y}_{t}X\left(t\right)\right)-\widehat{Y}_{t}dX\left(t\right)+\widehat{Y}_{t}X\left(t\right)d\Gamma_{t}=\widehat{Y}_{t}C\left(t\right)dK_{t}.$

Substituting this into (2.10), we get

$d\left(\widehat{Y}_{t}X\left(t\right)\right)-\widehat{Y}_{t}C\left(t\right)dK_{t}=\widehat{Y}_{t}C\left(t\right)d\Lambda_{t},$

which we integrate to, since $\widehat{Y}_{0}=1$,

$\widehat{Y}_{t}X\left(t\right)=z+\int_{0}^{t}\widehat{Y}_{s}C\left(s\right)d\left(K_{s}+\Lambda_{s}\right).$

Solving for $X\left(t\right)$, we obtain the following:

###### Theorem 2.2

Given initial value $z$ the corresponding optimal wealth process
$\widehat{X}_{z}(t)$ is given by

$\displaystyle\widehat{X}_{z}(t)$
$\displaystyle=z\widehat{Y}_{t}^{-1}+\widehat{Y}_{t}^{-1}\int_{0}^{t}\widehat{Y}_{s}C(s)d(K_{s}+\Lambda_{s})$
$\displaystyle=z\exp(-\widehat{A}_{t})+\exp(-\widehat{A}_{t})\int_{0}^{t}\exp(\widehat{A}_{s})C\left(s\right)d\left(K_{s}+\Lambda_{s}\right).$
(2.13)

In particular, note that

$\frac{d}{dz}\widehat{X}_{z}\left(t\right)=\exp(-\widehat{A}_{t}).$ (2.14)

Going back to our problem, choose $z\in\mathbb{R}$ and let $\widehat{\pi}_{z}$
be the corresponding optimal hedging portfolio given by (2.3) and let
$\widehat{X}_{z}$ be the corresponding optimal wealth process given by (2.1)
and (2.13) respectively. Then

$\displaystyle\inf_{z,\pi}J(z,\pi)=\inf_{z,\pi}E\Big{[}\frac{1}{2}(X_{z,\pi}(T)-F)^{2}\Big{]}=\inf_{z}E\Big{[}\frac{1}{2}(X_{z,\widehat{\pi}_{z}}(T)-F)^{2}\Big{]}=\inf_{z}E\Big{[}\frac{1}{2}(\widehat{X}_{z}(T)-F)^{2}\Big{]}.$

Note that, if we define

$\displaystyle R_{t}$
$\displaystyle:=\exp(-\widehat{A}_{t})=\exp\Big{[}\int_{0}^{t}\Big{\\{}\alpha_{1}(s)-\frac{1}{2}\sigma_{1}^{2}(s)+\int_{\mathbb{R^{*}}}\Big{(}\ln(1+\gamma_{1}(s,\zeta))-\gamma_{1}(s,\zeta)\Big{)}\nu(d\zeta)\Big{\\}}ds$
$\displaystyle+\int_{0}^{t}\sigma_{1}(s)dB(s)+\int_{0}^{t}\int_{\mathbb{R^{*}}}\ln(1+\gamma_{1}(s,\zeta))\widetilde{N}(ds,d\zeta)\Big{]},$

and

$\displaystyle Z^{*}(t)$
$\displaystyle:=\exp\Big{(}-\int_{0}^{t}\alpha_{1}(s)ds\Big{)}R_{t}$
$\displaystyle=\exp\Big{[}\int_{0}^{t}\Big{\\{}-\frac{1}{2}\sigma_{1}^{2}(s)+\int_{\mathbb{R^{*}}}(\ln(1+\gamma_{1}(s,\zeta))-\gamma_{1}(s,\zeta))\nu(d\zeta)\Big{\\}}ds$
$\displaystyle+\int_{0}^{t}\sigma_{1}(s)dB(s)+\int_{0}^{t}\int_{\mathbb{R^{*}}}\ln(1+\gamma_{1}(s,\zeta))\widetilde{N}(ds,d\zeta)\Big{]},$

then we can verify by the Itô formula that

$\displaystyle dR_{t}$
$\displaystyle=R_{t}\Big{(}\alpha_{1}(t)dt+\sigma_{1}(t)dB(t)+\int_{\mathbb{R^{*}}}\gamma_{1}(t,\zeta)\widetilde{N}(dt,d\zeta)\Big{)}$
$\displaystyle=R_{t}G(t)\Big{(}\alpha(t)dt+\sigma(t)dB(t)+\int_{\mathbb{R^{*}}}\gamma(t,\zeta)\widetilde{N}(dt,d\zeta)\Big{)}$
$\displaystyle=R_{t}G(t)S^{-1}(t)dS(t),$

and

$\displaystyle dZ^{*}(t)$
$\displaystyle=Z^{*}(t)G(t)\Big{(}\sigma(t)dB(t)+\int_{\mathbb{R^{*}}}\gamma(t,\zeta)\widetilde{N}(dt,d\zeta)\Big{)}.$
(2.15)

###### Proposition 2.3

$Z^{*}(\cdot)$ is a $P$-martingale. (See (2.2).) In particular, it follows
that $Q^{*}$ defined by

$\displaystyle dQ^{*}(\omega)=Z^{*}(T)dP(\omega)\text{ on }\mathcal{F}_{T}.$
(2.16)

is an EMM for $S(\cdot)$.

Proof. To see this, we verify that the coefficients
$\theta_{0}(t):=G(t)\sigma(t)$ and $\theta_{1}(t,\zeta):=G(t)\gamma(t,\zeta)$
satisfy condition (2.1):

$\displaystyle\alpha(t)+\theta_{0}(t)\sigma(t)+\int_{\mathbb{R^{*}}}\theta_{1}(t,\zeta)\gamma(t,\zeta)\nu(d\zeta)$
$\displaystyle=\alpha(t)-\frac{\alpha(t)}{\sigma^{2}(t)+\int_{\mathbb{R^{*}}}\gamma^{2}(t,\zeta)\nu(d\zeta)}\sigma^{2}(t)-\frac{\alpha(t)}{\sigma^{2}(t)+\int_{\mathbb{R^{*}}}\gamma^{2}(t,\zeta)\nu(d\zeta)}\int_{\mathbb{R^{*}}}\gamma^{2}(t,\zeta)\nu(d\zeta)$
$\displaystyle=\alpha(t)-\alpha(t)=0\text{, }\quad t\geq 0.$

$\square$

Using this, we obtain the following, which is the main result in this section:

###### Theorem 2.4

(i) The unique minimal variance price $\widehat{z}$ of a European option with
terminal payoff $F$ at time $T$ is given by

$\displaystyle\widehat{z}=\frac{E\Big{[}exp(-\widehat{A}_{T})\Big{(}F-exp(-\widehat{A}_{T})\int_{0}^{T}exp(\widehat{A}_{s})C\left(s\right)d\left(K_{s}+\Lambda_{s}\right)\Big{)}\Big{]}}{E[exp(-2\widehat{A}_{T})]},$
(2.17)

where $\widehat{A}_{T}$ is given by (2.12), $C(s),\Lambda_{s}$ are given by
(2.7), (2.8) respectively, and $K$ is given by (2.11).
(ii) Assume that the coefficient $\alpha(t)$, $\sigma(t)$ and
$\gamma(t,\zeta)$ are bounded and deterministic functions. Then

$\displaystyle\widehat{z}=E_{Q^{*}}[F],$ (2.18)

where $Q^{*}$ is the EMM measure given by (2.16).

Proof. (i) To minimize
$J_{0}(z):=E\Big{[}\frac{1}{2}(\widehat{X}_{z}(T)-F)^{2}\Big{]}$ with respect
to $z$ we get by (2.13) and (2.14) that

$\displaystyle\frac{d}{dz}J_{0}(z)$
$\displaystyle=E\Big{[}(\widehat{X}_{z}(T)-F)\frac{d}{dz}\widehat{X}_{z}(T)\Big{]}$
$\displaystyle=E[(\widehat{X}_{z}(T)-F)exp(-\widehat{A}_{T})]$ (2.19)
$\displaystyle=E\Big{[}\Big{(}zexp(-\widehat{A}_{T})+exp(-\widehat{A}_{T})\int_{0}^{T}exp(\widehat{A}_{s})C\left(s\right)d\left(K_{s}+\Lambda_{s}\right)-F\Big{)}exp(-\widehat{A}_{T})\Big{]}.$

This is $0$ if and only if (2.17) holds. (ii) By (2.19), we get

$\displaystyle
E[\widehat{X}_{z}(T)\exp(-\widehat{A}_{T})]=E[F\exp(-\widehat{A}_{T})],$ i.e.
$\displaystyle
E\Big{[}\widehat{X}_{z}(T)\exp\Big{(}\int_{0}^{T}\alpha_{1}(s)ds\Big{)}Z^{*}(T)\Big{]}=E\Big{[}F\exp\Big{(}\int_{0}^{T}\alpha_{1}(s)ds\Big{)}Z^{*}(T)\Big{]}.$

If $\alpha_{1}$ is deterministic, we cancel out the factor
$\exp\Big{(}\int_{0}^{T}\alpha_{1}(s)ds\Big{)}$ and (2.18) follows. $\square$

###### Remark 2.5

In particular, if $F$ is a deterministic constant. Then $F(t)=F=E[F]$ for all
$t$, and $\beta=\kappa=0$. Hence the optimal portfolio is given in feedback
form by

$\widehat{\pi}(t)X(t)=G(t)[X(t)-F].$

Assume, for example, that $\alpha(t)>0$. Then $G(t)<0$, and we see that if
$X(t)<F$, then $\widehat{\pi}(t)X(t)>0$, and hence the optimal portfolio
pushes $X(t)$ upwards towards F. Similarly, if $X(t)>F$ then
$\widehat{\pi}(t)X(t)<0$ and the optimal push of $X(t)$ is downwards towards
$F$. This is to be expected since the portfolio tries to minimize the terminal
variance $E[(X(T)-F)^{2}]$. Moreover, if we start at $z=X(0)=F$, we can choose
$\pi=0$ and this gives
$J(z,\pi)=J(F,0)=E[\frac{1}{2}(X(T)-F)^{2}]=E[\frac{1}{2}(F-F)^{2}]=0$, which
is clearly optimal. By uniqueness of $(\widehat{z},\widehat{\pi})$ we conclude
that $(\widehat{z},\widehat{\pi})=(F,0)$ is the optimal pair in this case.

###### Remark 2.6

The price $\widehat{z}$ is an arbitrage-free price of $F$, follows by Theorem
2.4 (ii).

###### Remark 2.7

Note that, as remarked earlier, the coefficients $\beta$ and $\kappa$ depend
linearly on $F$. Therefore it follows from the formula (2.17) that the map
$\Phi:L^{2}(P,\mathcal{F}_{T})\mapsto\mathbb{R}$ defined by

$\Phi(F)=\widehat{z};\quad F\in L^{2}(P,\mathcal{F}_{T})$

is linear and bounded. By the Riesz representation theorem this map can be
represented by a random variable $Z\in L^{2}(P,\mathcal{F}_{T})$, in the sense
that

$\displaystyle\widehat{z}=E[ZF];\quad F\in L^{2}(P,\mathcal{F}_{T}).$

Therefore, if we define the (signed) measure $\tilde{Q}$ on $\mathcal{F}_{T}$
by

$\displaystyle d\tilde{Q}=ZdP$

then

$\displaystyle\widehat{z}=E_{\tilde{Q}}[F].$

Comparing with the Schweizer variance-optimal pricing measure $\tilde{P}$
(1.4) we conclude the following:

###### Corollary 2.8

  1. (i)

$\tilde{P}=\tilde{Q}$
In particular, $\tilde{P}$ always exists in this market, without any non-
arbitrage conditions.

  2. (ii)

Moreover, we have
$\tilde{P}=Q^{*}$, which is a positive EMM.

###  2.4 Example: European call option

We give some details about how to compute the minimal variance price
$\widehat{z}$ explicitly in the case of a European call option that will used
in Section 3.

  1. (i)

Note that the term $C(s)$ in Theorem 2.4 depends on the coefficients $\beta$
and $\kappa$ in the Itô representation of $F$. These coefficients can for
example be found by using the generalised Clark-Ocone formula for Lévy
processes, extended to $L^{2}(P)$. See Theorem 12.26 in (Nunno, Øksendal, and
Proske, 2008).
Let us find these coefficients in the case of a European call option, where

$F=(S(T)-K)^{+},$

where $K$ is a given exercise price. In this case $F(\omega)$ represents the
payoff at time $T$ (fixed) of a (European call) option which gives the owner
the right to buy the stock with value $S(T,\omega)$ at a fixed exercise price
$K$. Thus if $S(T,\omega)>K$ the owner of the option gets the profit
$S(T,\omega)-K$ and if $S(T,\omega)\leq K$ the owner does not exercise the
option and the profit is 0. Hence in this case

$F(\omega)=(S(T,\omega)-K)^{+}.$

Thus, we may write

$F(\omega)=f(S(T,\omega)),$

where

$f(x)=(x-K)^{+}.$

The function $f$ is not differentiable at $x=K$, so we cannot use the chain
rule directly to evaluate $D_{t}F$. However, we can approximate f by $C^{1}$
functions $f_{n}$ with the property that

$f_{n}(x)=f(x)\quad\text{ for }\quad|x-K|\geq\frac{1}{n},$

and

$0\leq f^{\prime}_{n}(x)\leq 1\text{ for all }x.$

Putting

$F_{n}(\omega)=f_{n}(S(T,\omega)),$

we see

$D_{t}F(\omega)=\lim_{n\to+\infty}D_{t}F_{n}(\omega).$

We get

$\displaystyle\beta(t)=E[D_{t}F|\mathcal{F}_{t}],\kappa(t,\zeta)=E[D_{t,\zeta}F|\mathcal{F}_{t}],$

where $D_{t}F$ and $D_{t,\zeta}F$ denote the generalised Malliavin derivatives
(also called the Hida-Malliavin derivative) of $F$ at $t$ and $(t,\zeta)$
respectively, with respect to $B(\cdot)$ and $N(\cdot,\cdot)$, respectively.
Combining this with the chain rule for the Hida-Malliavin derivative and the
Markov property of the process $S(\cdot)$, and assuming for simplicity that
$\sigma$ is constant and $\gamma(t,\zeta)=\gamma(\zeta)$ does not depend on
$t$, we obtain the following for $\beta$:

$\displaystyle\beta(t)$
$\displaystyle=E^{S_{0}}\Big{[}\mathbbm{1}_{[K,\infty)}(S(T))\sigma
S(T)\Big{|}\mathcal{F}_{t}\Big{]}$
$\displaystyle=E^{S(t)}\Big{[}\mathbbm{1}_{[K,\infty)}(S(T-t))\sigma
S(T-t)\Big{]}.$ (2.20)

To find the corresponding result for $\kappa$ we first use the chain rule for
$D_{t,\zeta}$ and get

$\displaystyle D_{t,\zeta}S(T)$
$\displaystyle=D_{t,\zeta}\Big{[}S_{0}\exp\Big{(}\alpha
T-\frac{1}{2}\sigma^{2}T+\sigma
B(T)+\int_{\mathbb{R^{*}}}(\log(1+\gamma(\zeta))-\gamma(\zeta))\nu(d\zeta)T$
$\displaystyle+\int_{0}^{t}\int_{\mathbb{R^{*}}}\ln(1+\gamma(\zeta))\widetilde{N}(ds,d\zeta)\Big{)}\Big{]}=S(T)\gamma(\zeta).$

Then we obtain

$\displaystyle\kappa(t,\zeta)$
$\displaystyle=E^{S_{0}}\Big{[}\mathbbm{1}_{[K,\infty)}(S(T)+D_{t,\zeta}S(T))-\mathbbm{1}_{[K,\infty)}(S(T))\Big{|}\mathcal{F}_{t}\Big{]}$
$\displaystyle=E^{S_{0}}\Big{[}\mathbbm{1}_{[K,\infty)}(S(T)+\gamma(\zeta)S(T))-\mathbbm{1}_{[K,\infty)}(S(T))\Big{|}\mathcal{F}_{t}\Big{]}$
$\displaystyle=E^{S(t)}\Big{[}\mathbbm{1}_{[K,\infty)}(S(T-t)+\gamma(\zeta)S(T-t))-\mathbbm{1}_{[K,\infty)}(S(T-t))\Big{]},$
(2.21)

where in general $E^{y}[h(S(u))]$ means $E[h(S^{y}(u))]$, i.e. expectation
when $S$ starts at $y$.

  2. (ii)

Since the coefficients $\alpha,\sigma$ and $\gamma$ of the process $S$ are
deterministic and bounded, we compute numerically the minimal variance price

$\widehat{z}=E\Big{[}(S(T)-K)^{+}Z_{T}^{*}\Big{]}$

of a European call option with payoff

$\displaystyle
F=(S(T)-K)^{+}=E[F]+\int_{0}^{T}\beta(t)dB(t)+\int_{0}^{T}\int_{\mathbb{R^{*}}}\kappa(t,\zeta)\widetilde{N}(dt,d\zeta),$

where $\beta,\kappa$ are given by (i),(i), respectively, we use the Itô
formula combined with (2.15) to obtain

$\displaystyle\widehat{z}=E[FZ_{T}^{*}]=E[F]+\int_{0}^{T}G(t)\Big{\\{}\sigma(t)E[Z_{t}^{*}\beta(t)]+\int_{\mathbb{R^{*}}}\gamma(t,\zeta)E[\kappa(t,\zeta)Z_{t}^{*}]\nu(d\zeta)\Big{\\}}dt,$

where $G(t)$ is given by (2.4).

####  2.4.1 Application to Black Scholes market with two independent Brownian
motions

In this example, we consider a market driven by two independent Brownian
motions, $B_{1}(t),B_{2}(t)$:

$\displaystyle
dS(t)=S(t)[\alpha_{0}dt+\beta_{1}dB_{1}(t)+\beta_{2})dB_{2}(t)],\quad S(0)>0,$

where the coefficients $\alpha_{0},\beta_{1},\beta_{2}$ are assumed to be
bounded constants. For a self-financing portfolio $\pi$ and an initial wealth
$z$, we have a wealth dynamic

$\displaystyle
dX(t)=X(t)\pi(t)[\alpha_{0}dt+\beta_{1}dB_{1}(t)+\beta_{2}dB_{2}(t)],\quad
X(0)=z.$

We want to find the pair $(\widehat{z},\widehat{\pi})$ which minimizes the
expected squared hedging error

$\displaystyle E\Big{[}\frac{1}{2}(X(T)-F)^{2}\Big{]}.$ (2.22)

###### Corollary 2.9

From Theorem 2.4 (ii), we conclude that

$\displaystyle E_{Q^{*}}[F]=E[FZ^{*}(T)],$ (2.23)

where

$\displaystyle Z^{*}(t)=$
$\displaystyle\exp\Big{(}-\frac{1}{2}G^{2}(\beta_{1}^{2}+\beta_{2}^{2})t+G\beta_{1}B_{1}(t)+G\beta_{2}B_{2}(t)\Big{)},$

and

$\displaystyle G=-\frac{\alpha_{0}}{\beta_{1}^{2}+\beta_{2}^{2}}.$ (2.24)

Looking at the above corollary, we can see that process $Z^{*}$ coincides with
the one in the Black-Scholes setting, where the volatility coefficient of the
single Brownian motion $B(t)$ is given by
$\beta=\sqrt{\beta_{1}^{2}+\beta_{2}^{2}}$. Since processes $\beta B(t)$ and
$\beta_{1}B_{1}(t)+\beta_{2}B_{2}(t)$ have the same distribution, so do the
option payoffs. This yields that the minimal-variance European call option
price in an incomplete market with two independent Brownian motions
$E_{Q^{*}}[F]$ coincides with the unique Black-Scholes price in a single
Brownian motion case for volatility
$\beta=\sqrt{\beta_{1}^{2}+\beta_{2}^{2}}$.

Furthermore, we know that the hedging portfolio in the BS model is only a
function of time and the underlying asset. Since for the choice of volatility
parameter as above, the stocks in both BS and two-Brownian motion model follow
the same distribution, we can deduce that for initial endowment $E_{Q^{*}}[F]$
from Equation (2.23) and for the BS hedging portfolio, we get

$\displaystyle E\Big{[}\frac{1}{2}(X(T)-F)^{2}\Big{]}=0.$

The above can, of course, be extended to the arbitrary number of Brownian
motions.

###  2.5 Application to Merton model

Here we consider the special case of the Merton model first proposed in
(Merton, 1976). It deals with European options in the market modeled by jump
diffusion. More precisely, the market consists of

(i)

a risk free asset, with unit price $S_{0}(t)=1$ for all $t$,

(ii)

a risky asset, with price $S(t)$ given by

$\displaystyle
dS(t)=S(t)\Big{[}\alpha_{0}dt+\sigma_{0}dB(t)+(y-1)\tilde{N}(dt,dy)\Big{]},\quad
S(0)=S_{0}>0.$ (2.25)

Here $\tilde{N}$ is a compensated Poisson random measure corresponding to the
compound Poisson process with intensity $\lambda$ and jump sizes $y=\exp(Y)$,
where $Y\sim\mathcal{N}(\mu,\delta^{2}),$ independent of $B$ and jump times.
Heuristically speaking, $y$ represents the absolute jump size, while
$\gamma_{0}(t,y)=y-1$ equals the relative jump size. The coefficients
$\alpha_{0}$ and $\sigma_{0}$ are assumed to be bounded and deterministic.
If we denote $k=E[y-1]=\exp(\mu+\frac{\delta^{2}}{2})-1$ and observe that
$y-1>-1$ a.s. we can use Itô formula and obtain the explicit solution

$\displaystyle S(t)$
$\displaystyle=S_{0}\exp\Big{(}\\{\alpha_{0}-\frac{\sigma_{0}^{2}}{2}-\lambda
k\\}t+\sigma_{0}B(t)+\sum^{N(t)}_{i=1}Y_{i}\Big{)}.$

Merton then proceeds to argue that the jumps in the asset price are not
systemic and, therefore, the risk related to the jumps is diversifiable. In
other words, this means that all the properties of the jump component of
$S(\cdot)$ under the risk neutral measure are the same as under the natural
measure. This argument yields the choice
$\theta_{0}=\frac{\alpha_{0}}{\sigma_{0}}$ and $\theta_{1}(y)=0$ which results
in EMM ${Q}^{M}$ corresponding to the Radon-Nikodým derivative

$\displaystyle
Z^{M}(t)=\exp\Big{(}-\frac{\alpha_{0}^{2}}{2\sigma_{0}^{2}}t-\frac{\alpha_{0}}{\sigma_{0}}B(t)\Big{)}$

and enables Merton to obtain the option price as

$\displaystyle E_{{Q}^{M}}[F]=\sum_{j\geq
0}\frac{e^{-\lambda(k+1)T}((\lambda(k+1)T)^{j}}{j!}BS\Big{(}0,S_{0},\sqrt{\sigma_{0}^{2}+\frac{j\delta^{2}}{T}},\frac{j\mu+j\delta^{2}/2}{T}-\lambda
k\Big{)},$ (2.26)

where $BS(t,S(t),\sigma_{0},r)$ denotes the price of the European call option
under the BS model at time $t$, with stock price $S(t)$, volatility $\sigma$,
and interest rate $r$.

This approach results in a non-symmetric loss. Merton argues that if no jump
occurs, then the owner of the option collects a small profit. However, in a
rare case when a jump does occur, the option holder suffers a significant
loss. The nonsymmetric loss distribution can, in some instances, be
nondesirable. Apart from that, the assumption of diversifiabilty of the jump
risk does not hold in practice; for example, market indexes do experience
occasional jumps in price.

Using the results from the previous section, we can obtain the optimal
portfolio, optimal wealth process, and the unique minimal variance price given
by (2.3), (2.13) and (2.17) respectively. To evaluate these expressions, one
has to compute coefficients $\beta(t)$ and $\kappa(t,y)$ given by equations
(i) and (i). If we denote
$L(t)=S(t)\exp\big{(}(\alpha-\frac{\sigma^{2}}{2}-\lambda k)(T-t)\big{)},$
$H(t)=\sigma B(t)+\sum_{i=1}^{N(t)}Y_{i}$ we get

$\displaystyle\beta(t)$
$\displaystyle=E^{S(t)}\Big{[}\mathbbm{1}_{[K,\infty)}(S(T-t))\sigma
S(T-t)\Big{]}$ $\displaystyle=\sigma
L(t)E[\mathbbm{1}_{[KL(t)^{-1},\infty)}(e^{H(T-t)})e^{H(T-t)}]$
$\displaystyle=\sigma
L(t)\sum_{j=0}^{\infty}E[\mathbbm{1}_{[KL(t)^{-1},\infty)}(e^{H(T-t)})e^{H(T-t)}\Big{|}N(T-t)=j]{P}(N(T-t)=j)$
$\displaystyle=\sigma
L(t)\sum_{j=0}^{\infty}\exp\Big{(}j\mu+\frac{j\delta^{2}+\sigma^{2}(T-t)}{2}\Big{)}\cdot$
$\displaystyle\cdot\Phi\Big{(}\frac{j\mu+j\delta^{2}+\sigma^{2}(T-t)-\log(K)+\log(L(t))}{\sqrt{j\delta^{2}+\sigma^{2}(T-t)}}\Big{)}\cdot\frac{(\lambda(T-t))^{j}\exp(-\lambda(T-t))}{j!},$

$\displaystyle\kappa(t,y)$
$\displaystyle=E^{S(t)}\Big{[}\mathbbm{1}_{[K,\infty)}(S(T-t)+\gamma(y)S(T-t))-\mathbbm{1}_{[K,\infty)}(S(T-t))\Big{]}$
$\displaystyle=E^{S(t)}\Big{[}\mathbbm{1}_{[K,\infty)}(S(T-t)+(y-1)S(T-t))-\mathbbm{1}_{[K,\infty)}(S(T-t))\Big{]}$
$\displaystyle=E^{S(t)}\Big{[}\mathbbm{1}_{[K,\infty)}(yS(T-t))-\mathbbm{1}_{[K,\infty)}(S(T-t))\Big{]}$
$\displaystyle=\mathbb{P}\Big{(}e^{H(T-t)}>\frac{K}{yL(t)}\Big{)}-\mathbb{P}\Big{(}e^{H(T-t)}>\frac{K}{L(t)}\Big{)}$
$\displaystyle=\sum_{j=0}^{\infty}\Bigg{(}\Phi\Big{(}\frac{\log(K)-\log(L(t))-j\mu}{\sqrt{j\delta^{2}+\sigma^{2}(T-t)}}\Big{)}$
$\displaystyle-\Phi\Big{(}\frac{\log(K)-\log(yL(t))-j\mu}{\sqrt{j\delta^{2}+\sigma^{2}(T-t)}}\Big{)}\Bigg{)}\cdot\frac{(\lambda(T-t))^{j}\exp(-\lambda(T-t))}{j!},$

where we use the fact that
$H(t)|\big{\\{}N(t)=j\big{\\}}\sim\mathcal{N}\big{(}j\mu,\sigma^{2}(T-t)+j\delta^{2}\big{)}.$
Unfortunately, this representation is not enough to obtain the desired results
explicitly. However, we may observe the following. The Lévy measure associated
with the compound process in equation (2.25) equals $\lambda\mu_{y}$, where
$\mu_{y}$ is the law of lognormally distributed random variable $y.$ Hence,
even though the coefficient $\gamma_{0}$ is stochastic, we have that
$m=\int_{\mathbb{R}^{*}}\gamma_{0}(y)^{2}\nu(dy)$ is deterministic. Since
$\alpha_{0}$ and $\sigma_{0}$ are also deterministic, so is

$\displaystyle G=-\frac{\alpha_{0}}{\sigma_{0}^{2}+\lambda m}.$ (2.27)

By part (ii) of the Theorem 2.4, we have an explicit representation of the EMM
$Q^{*}$ and the corresponding Radon-Nikodým derivative given by

$\displaystyle Z^{*}(t)=\exp\Big{(}\\{-\frac{1}{2}G^{2}\sigma_{0}^{2}-\lambda
Gk\\}t+G\sigma_{0}B(t)+\sum_{i=1}^{N(t)}\ln(1+G(y_{i}-1))\Big{)}.$

Moreover, the minimal variance price $\widehat{z}$ of an option with payoff
$F$ at time $T$ is

$\displaystyle\widehat{z}=E_{Q^{*}}(F)=E[FZ^{*}(T)].$ (2.28)

Figure 1: One realization of $Z^{M}$ and $Z^{*}$ for the choice of parameters
specified in Section 3.3.3

###### Remark 2.10

It must be stated that, in theory, we do not have a guarantee that
$G(y_{i}-1)>-1$ since lognormal distribution takes values on the whole
positive line. Consequently, the logarithm may not be defined, in which case
we use the alternative formulation

$\displaystyle
Z^{*}(t)=\Big{(}\prod_{i=1}^{N(t)}1+G(y_{i}-1)\Big{)}\exp\Big{(}\\{-\frac{1}{2}G^{2}\sigma_{0}^{2}-\lambda
Gk\\}t+G\sigma_{0}B(t)\Big{)}$

from (Lamberton and Lapeyre, 2011). This results in a signed measure $Q^{*}$.
Nonetheless, in practice, for a reasonable choice of parameters,
$G(y_{i}-1)>-1$ holds with probability practically equal to 1, and we do not
need to worry about this technical problem.

We can not derive the same formula as in (2.26) since the distribution of
$\ln(1+G(y-1))$ is unknown; however, we can use the Monte Carlo approach to
obtain the option price. This option price obviously differs from Merton’s, as
does the hedging portfolio that also considers the risk coming from the jumps
and results in a symmetric loss. Using the Euler-Maruyama discretization rule,
we simulate one realization of $Z^{M}$ and $Z^{*}$ and present it in Figure 1.
We can see that $Z^{*}$ has upward jumps, which are not present in $Z^{M}.$

In figures 2 and 3, we can see how the price of the European option changes
for both models depending on the parameters $\lambda,\,\mu,\,\sigma,\,\delta.$
The option price in both models grows together with the absolute value of the
parameters. Note that option prices coincide with the exclusion of jumps,
namely when $\lambda=0$ since the Merton model devolves into the BS model. It
must be noted how the changes in prices are higher in the case described by
Figure 2, where we consider the dependence on parameters $\lambda$ and $\mu$,
than in Figure 3, where $\sigma$ and $\delta$ are taken into consideration,
due to those parameters’ effect on stock and wealth dynamics. Large values of
parameters $\lambda$ and $\mu$ change dynamics directly, while $\sigma$ and
especially $\delta$ do it indirectly. In Figure 2, it is shown how model
prices are similar for small absolute values of parameters. When parameters
grow, minimal variance price grows faster; however, the difference stabilizes.
As a matter of fact, the difference appears to be most significant when
$\lambda$ is relatively small, while $|\mu|$ is large. This is because once
$\lambda$ is large, the compensated part of the jump component, which is taken
into account in both hedging strategies, takes over.

To continue with observations for the other two parameters, we can see that
even though absolute differences are smaller, relatively speaking, models
differ more when $\sigma$ and $\delta$ parameters are modified, as shown in
Figure 3. The difference between the prices seems to be biggest when jump
sizes are constant while volatility in the model is large.

(a) $E_{Q^{M}}[F]$

(b)  $E_{Q^{*}}[F]$

(c) $E_{Q^{*}}[F]-E_{Q^{M}}[F]$

Figure 2: Option price comparison depending on parameters $\lambda$ and $\mu$,
where $S_{0}=1$, $K=0.5,$ $\sigma=0.2$ and $\delta=0.05$

(a) $E_{Q^{M}}[F]$

(b)  $E_{Q^{*}}[F]$

(c) $E_{Q^{*}}[F]-E_{Q^{M}}[F]$

Figure 3: Option price comparison depending on parameters $\sigma$ and
$\delta$, where $S_{0}=1$, $K=0.5,$ $\lambda=10$ and $\mu=-0.2$

Numerical methods for obtaining the optimal portfolio are presented in the
next section.

##  3 Deep Learning Approach

Here we present the algorithm using deep learning methods that returns both
the optimal initial value and the optimal portfolio. In (Fecamp, Mikael, and
Warin, 2020), different types of neural networks (NN) are compared when
studying hedging in discrete-time incomplete markets. It turns out that Long
short-term memory (LSTM) networks, which are the particular case of Recurrent
neural networks (RNN), outperform standard feed-forward NN. Because the model
discussed in the aforementioned paper doesn’t incorporate jumps in the price
dynamics, we also conducted a comparison between the performance of LSTM and
feed-forward neural networks. However, we detected no significant difference
between the two approaches. A possible intuitive reason for the good
performance of the LSTM network in the jump diffusion models could be its
ability to learn the jump dynamics from the previous jump occurrences.
Furthermore, RNN and, in particular, LSTM networks seem to be the most natural
approach when working with time series data and, importantly, allow
generalization to non-Markovian models. Hence, we opted to focus exclusively
on the LSTM model in the subsequent work. In particular a possible extension
that would benefit from the use of LSTM networks is if we consider stochastic
volatility models. Historical information that LSTM keeps or forgets would
help us to store and use knowledge about the behaviour of the volatility
parameter.

The performance of our network is to be tested on four models. First, we
consider the BS model, where the market is complete. The reason for that is
that both option value and hedging portfolio are known and can be compared to
our results. Then, we consider three cases of incomplete markets. We start
with the BS model with multiple independent Brownian motions and continue with
two jump diffusion models, Merton and Kou double exponential models.

###  3.1 Data Generation

To generate the data, we discretize time interval $[0,T]$ into $R$ equidistant
points $t_{i}$ at a distance $\Delta_{t}$. We denote different realizations of
the process with $(j)$ in superscript, where $j=[M]$ and $M$ denotes the batch
size. When the superscript is omitted, we assume the vector notation.
Additionally, we here assume the most general case with $d_{B}$ dimensional
Brownian motion and $d_{N}$ dimensional Poisson random measure.

For initial wealth $x_{0}$ we construct under portfolio $\pi$ the wealth
process at time $t_{i}$ using the update rule

$x_{i+1}=x_{i}+x_{i}\pi_{i}\big{[}(\alpha_{0}-\lambda^{\top}k)\Delta_{t}+\sqrt{\Delta_{t}}B_{i}\sigma_{0}+\gamma_{0}J_{i}\big{]},\qquad
i\in[R-1].$ (3.1)

Here $\alpha_{0}\in\mathbb{R}$ and $\sigma_{0}\in\mathbb{R}^{d_{B}}$ are drift
and volatility parameters respectively. Parameter $\gamma_{0}\in\\{0,1\\}$
serves as a dummy variable that indicates whether a model is continuous or
not. Moreover $B_{i}$ are $M\times d_{B}$ dimensional independent
$\mathcal{N}(0,1)$ random variables, while
$J_{i}=\sum_{l=1}^{d_{N}}J_{i}^{l},$ where $J_{i}^{l}$ are independent and
given by

$J_{i}^{l}=\begin{cases}e^{Y_{i}^{l}}-1,&N^{l}(t)\text{ has a jump in
}[t_{i},t_{i+1}]\\\ 0,&\text{otherwise},\end{cases}$

where $Y_{i}^{l}$ is $M$ dimensional $\mathcal{N}(\mu_{l},\delta^{2}_{l})$ and
$N(t)$ is an $M\times d_{N}$ dimensional homogeneous Poisson process with
intensity $\lambda\in\mathbb{R}_{+}^{d_{N}}$ and independent components
$N^{l}(t).$ Parameter $k$ in Equation (3.1) has $d_{N}$ components
$k^{l}=\exp(\mu_{l}+\frac{\delta^{2}_{l}}{2})-1.$

This discretization coincides with continuous time SDEs in the Merton model
described in Section 2.5 and in the BS model when $\gamma_{0}=0$ and Brownian
motion is one dimensional.

To obtain a discrete stock process $s$, which we need to compute the loss
function later on, we choose the initial stock value $s_{0}$ and set $\pi=1$
in equation (3.1).

For option strike price $K$ we first compute option’s $t_{R}$ claim as
$F^{(j)}=(s^{(j)}_{R}-K)^{+}.$ Then we can define loss as

$loss(s_{R},x_{R})=\frac{1}{2}\frac{1}{M}\sum_{j=1}^{M}(x^{(j)}_{R}-F^{(j)})^{2}.$
(3.2)

Note that here we still, for brevity’s sake, assume that the interest rate
$r=0,$ since all the proceeding computations can be done with the same amount
of accuracy.

###  3.2 Network Architecture and Algorithm

As mentioned above, we make use of the LSTM networks. A feed-forward network
approach was also tested. However, it resulted in worse performance; hence we
decided to stick with LSTM to allow more generality. We divide our learning
task into two parts. First, we need to find the initial wealth value $x_{0}.$
This is a relatively simple task, so a simple one-layer linear feed-forward NN
with hidden dimension $d$ given by

$L(y)=Ay+b,\qquad y\in\mathbb{R}^{d},A\in\mathbb{R}^{M\times
d},b\in\mathbb{R}^{M}$ (3.3)

is used. Recall that we interpret initial wealth $x_{0}$ as an option price, a
deterministic quantity. Hence, $x_{0}^{j}=x_{0}$ for each $j\in[M]$ and we may
use a vector of initial stock prices $y=[s_{0},\ldots,s_{0}]^{\top}$ as an
input for NN $L$. Since the wealth process in Equation (1.3) is a geometric
jump diffusion where a positive initial value is assumed, we use the softplus
activation function $f(x)=\log(1+e^{x}).$

In the second part of the learning process, we strive to find the optimal
portfolio $\pi$. We use a stacked LSTM network with two layers. Each layer is
built from LSTM cells, one at each timestep. We do not give a detailed
description of LSTM cells here. An interested reader can find more information
in (Hochreiter and Schmidhuber, 1997). At each time step, the update rule
(3.1) is used to obtain the wealth state $x_{i}$, which we feed as an input
for the first LSTM layer, as shown in Figure 4. It is important to note that
learning the optimal portfolio $\pi$ is a challenging problem since the
portfolio at each timestep depends indirectly on the previous portfolio
values. Our network architecture captures this dependency through hidden
states $h_{i}^{j}$ and cell states $c_{i}^{j}$ for $i\in[R]$ and $j=1,2.$
Second LSTM layer is used mainly to take into account the non-linearity of our
problem and consequently the non-linearity of the solution $\pi$. After the
second LSTM layer returns its hidden state $h_{i}^{2}\in\mathbb{R}^{d}$, we
feed it to another linear network of the form (3.3), which at last yields
$\pi_{i}$. To summarize, our network takes as input the initial stock value
$[s_{0},\ldots,s_{0}]^{\top}\in\mathbb{R}^{d}$, Brownian motion increments
$B\in\mathbb{R}^{M\times d_{B}\times R}$ and jump increments
$J\in\mathbb{R}^{M\times d_{N}\times R}$ and returns an output
$(x_{0},\pi)\in\mathbb{R}^{M\times R+1},$ where we can interpret $x_{0}$ as
option price and $\pi$ as hedging portfolio.

$x_{0}$LSTM CellLSTM CellLinear$\pi_{0}$InputLinearSoftplus$x_{0}$$x_{1}$LSTM
CellLSTM
CellLinear$\pi_{1}$$\ldots$$\ldots$$\ldots$$\ldots$$\ldots$$x_{R}$LSTM
CellLSTM
CellLinear$\pi_{R}$$h_{1}^{1}$$h_{1}^{2}$$c_{1}^{1}$$h_{1}^{1}$$c_{1}^{2}$$h_{1}^{2}$$h_{2}^{1}$$h_{2}^{2}$$c_{2}^{1}$$h_{2}^{1}$$c_{2}^{2}$$h_{2}^{2}$$c_{R}^{1}$$h_{R}^{1}$$c_{R}^{2}$$h_{R}^{2}$$h_{R}^{1}$$h_{R}^{2}$
Figure 4: Neural network architecture with two LSTM layers. Dashed lines
represent the update rule in Equation (3.1)

Once done, we compute $loss(s_{R},x_{R})$ and use backpropagation to update
the weights. In particular, the Adam optimizer is used. It feels befit to note
here how the backpropagation approach to modifying the weights connects
naturally to the theoretical control problem we solve in Section 2. Recall, in
the Stackelberg game approach, we first determine the optimal portfolio
$\hat{\pi}_{z}$ given arbitrary initial state $z$, and only after that do we
also find the optimal initial state $\hat{z}$ where behavior under the above
portfolio is assumed. Similarly, we start with an arbitrary initial state and
portfolio when taking the deep learning approach. Then once the loss is
computed, we modify weights in a backward manner. Hence we first modify
weights responsible for the selection of portfolio $\pi_{x_{0}}$ given (at the
moment arbitrary) initial wealth $x_{0}$, and only after that do we also
modify the weights associated with the initial wealth.

###### Remark 3.1

Since the control process $\pi$ is $\mathbb{F}$-adopted, one may feel inclined
to use the Brownian motion sequence $(B_{i})_{i=1}^{R}$ as an input for the
LSTM network. Indeed, it holds $\mathcal{F}^{X}_{t}\subseteq\mathcal{F}_{t}$,
so it may happen that using the sequence $(x_{i})_{i=1}^{R}$ as an input may
result in sub-optimal controls. However, as it is shown in Equation (2.3), the
optimal control is of the feedback form, hence the choice of
$(x_{i})_{i=1}^{R}$ as the data input is reasonable. Apart from that, as
discussed in (Han and Hu, 2021), taking $(B_{i})_{i=1}^{R}$ as an input, does
not improve the performance.

To summarise, let us present our Algorithm 1 in a more compact way.

Algorithm 1 Price of a European option and a hedging portfolio in Merton model

Maturity time $T$, drift parameter $\alpha_{0}$, volatility parameter
$\beta_{0}$, jump parameter $\gamma_{0}$, initial stock price $s_{0}$, strike
price $K$, number of time steps $R$, batch size $M$ and number of epochs $P$

for $1\leq\text{epoch}\leq P$ do

Using discretization rules in Equation (3.1) and NN architecture in Figure 4
compute $M$ dimensional wealth sequence $(x_{i})_{i\in[R]},$ stock sequence
$(s_{i})_{i\in[R]}$ and control sequence $(\pi_{i})_{i\in[R]}$

Compute $loss(s_{R},x_{R})$ as in Equation (3.2)

Using backpropagation, update the weights of the NN architecture

end for

return $x_{0}$ and $(\pi_{i})_{i\in[R]}$

###  3.3 Numerical Results

Here we present the numerical results obtained using Algorithm 1. For
calculations, we used the PyTorch library from Python. The hidden dimension is
set to 512, batch size to 256, and the learning rate to 0.0005. Unless stated
otherwise, we set time maturity $T=1$. To assess the results, an evaluation
set of size 10000 is used. The programming code can be found on the GitHub
repository: https://github.com/janrems/DeepLearningQHedging.

We examine four market models: the standard Black-Scholes (BS) model, the BS
model with multiple independent Brownian motions, the Merton model, and the
Kou double exponential jump diffusion model. Our analysis focuses on the
convergence of losses in all these models. For the first three models, we
explore the convergence of initial values toward the specified option price as
well. Additionally, we compare the predicted hedging portfolio to its
analytical counterpart for both variations of the BS model. Furthermore, we
assess the algorithm’s performance as the input dimension increases. In the
continuous case, we achieve this by comparing results from the two studied BS
variations. In jump models, we test scalability concerning the input dimension
in a Merton case. Finally, we investigate the scalability of the algorithm in
the temporal dimension, specifically when adjusting the timestep parameter $R$
and maturity $T$.

####  3.3.1 Black Scholes market model

For the BS model, we take the number of time steps $R=80$. Recall, the BS
model is a case of the complete market model, which means that there exist
initial wealth and optimal hedging portfolio that, at least in a continuous
case, assures $loss=0.$ Apart from that, we have nice analytic solutions for
both option price and hedging portfolio.
We consider a case where in Equation (3.1) we set $\alpha_{0}=0.3,$
$\sigma_{0}=0.2,$ $\gamma_{0}=0$, $\lambda=0$ while we put $s_{0}=1$ and
$K=0.5.$ Different choices of listed parameters result in different training
times but do not affect the model’s accuracy. For this particular choice of
parameters, we compute 6000 epochs.

(a) Loss

(b) Initial values

(c) Market realization

Figure 5: Convergence of loss and initial values and one market realization
for the BS model

All the results presented here and in the other models are obtained in the
following way. At the epoch where the minimal loss is achieved, we save the
network weights. Then, using those weights, we evaluate our model on a larger
evaluation set of size 10000.

The first two outputs of the algorithm are the optimal initial wealth
$\hat{x}_{0}$ and the minimal obtained loss $loss_{min}$. Both of them can be
found in Table 1. In figures 5(a) and 5(b), we can see how loss and initial
value $x_{0}$ both converge towards 0 and theoretical BS option price 0.5,
respectively.

Now, let us take a look at our algorithm’s other output, namely hedging
portfolio $\pi.$ Unlike initial value $x_{0}$ which is deterministic,
portfolio $\pi$ differs over different market realisations. We select
portfolio realizations obtained on the evaluation set and denote them with
$\hat{\pi}.$ We also wish to determine how close $\hat{\pi}$ is to theoretical
portfolio given by the BS model. To obtain its discrete version from our
discrete stock process $s_{i\in[R]}$ we define

$d_{i}=\frac{1}{\sigma_{0}\sqrt{t_{R}-t_{i}}}\Big{(}\log\big{(}\frac{s_{i}}{K}\big{)}+\frac{\sigma_{0}^{2}}{2}(t_{R}-t_{i})\Big{)},$

and put

$\varphi_{i}=\Phi(d_{i})\frac{s_{i}}{x_{i}},$

where $\Phi$ stands for the standard normal cumulative distribution function.

We define and estimate the distance between $\hat{\pi}$ and $\varphi$ as a
discrete $L^{2}$ distance estimate, namely

$d(\hat{\pi},\varphi)=E\big{[}\lVert\hat{\pi}-\varphi\rVert_{L^{2}}\big{]}\approx\frac{1}{M}\sum_{j\in[M]}\sqrt{\sum_{i\in[R]}(\hat{\pi}_{i}^{(j)}-\varphi_{i}^{(j)})^{2}\Delta_{t}}.$
(3.4)

We can see one instance of market realization, namely the stock process $s$,
state process $x$, computed portfolio $\hat{\pi}$ theoretical portfolio
$\varphi$ and option payoff at terminal time $F$ in Figure 5(c).

The estimated portfolio follows the theoretical one closely. Moreover, as
expected, we manage for the terminal wealth to be almost exact to the option
payoff.

Model |  BS price |  $\hat{x}_{0}$ |  $d(\hat{\pi},\varphi)$ |  $loss_{min}$
---|---|---|---|---
BS |  0.500 |  0.501 |  0.312 |  5.84e-6
Table 1: Comparison of theoretical and numerical results in BS model

####  3.3.2 Black Scholes market with multiple independent Brownian motions

In contrast to the classical Black-Scholes market with one Brownian motion,
here the noise in the model comes from multiple independent Brownian options.
In turn, this results in a non-complete market model. Let us recall that this
means there is no guarantee that the terminal wealth $X_{T}$ equals the option
payoff $F$ almost surely. Besides, there is, in general, a continuum of
possible option prices. However, as it was discussed in Section 2.4.1, in the
case of the European call option, following the minimal-variance hedging
approach, we end up with the price and the optimal hedging strategy coming
from the classical Black-Scholes market with single Brownian motion and
corresponding volatility coefficient equal to the norm of volatility
coefficients of multi-dimensional Brownian motion.

This gives us a nice opportunity to study how our algorithm scales in the
presence of multidimensional noisy input. We choose the same parameters as in
the previous section, the only difference being the volatility coefficient,
which is now, of course, multidimensional and given by
$\sigma_{0}=[0.11,0.16,0.05].$ Note that we have $\lVert\sigma_{0}\rVert=0.2$.

(a) Loss

(b) Initial values

(c) Market realisation

Figure 6: Convergence of loss and initial values and one market realization in
the BS model with multiple Brownian motions

Comparing both results in Table 2 and graphs in Figure 6 with their
counterparts in Section 3.3.1, we can see that our algorithm scales well with
the increased size of the input. Even though the results in both tables differ
a bit, it is only natural because of the random nature of our machine learning
algorithm.

Model |  BS price |  $\hat{x}_{0}$ |  $d(\hat{\pi},\varphi)$ |  $loss_{min}$
---|---|---|---|---
BSMB |  0.500 |  0.502 |  0.226 |  6.14e-6
Table 2: Comparison of theoretical and numerical results for the BS model with
multiple Brownian motions (BSMB)

####  3.3.3 Merton market model

In our particular example of the Merton model, we follow the setting from
Section 2.5 and again use Algorithm 1, where we set parameters to
$\alpha_{0}=0.2,$ $\sigma_{0}=0.2,$ $\gamma_{0}=1,$ $s_{0}=1$ and $K=0.5.$ For
the log-normally distributed jumps, we set expected value $\mu=-0.2$ and
standard deviation $\delta=0.05,$ while jump intensity $\lambda=5.$ Because of
the downward jumps, we need to work with finer discretization to avoid
negative wealth and stock values. Hence, we choose $R=150$, which in turn
leads to longer training time since we now have 150 times 2 LSTM cells in our
NN. As the stock process is noisier than the one in the BS case, so is the
portfolio, which in turn also means a harder learning task. We use 7,000
epochs and disregard the training rounds where negative jumps occur.

In Figure 7(c), we see one market realization at the epoch at which the lowest
loss was obtained.

The portfolio is much more dynamic, which, as discussed, results in slower
learning. Furthermore, we see a more significant gap between terminal wealth
and option payoff.

(a) Loss

(b) Initial values

(c) Market realization

Figure 7: Convergence of loss and initial values and one market realization
for the Merton model

As we can see in the figures 7(a) and 7(b), both losses and initial values
converge toward desired quantities. However, we can see that convergence
becomes slow after just 3000 epochs. Tackling the plateau problem with, for
instance, scheduling the learning rate is one of the tasks for future work.
Notice there are a few spikes in both of these graphs. We believe they are due
to computational errors that occur when the wealth process takes values close
to zero, which is more often than in the previous two cases due to downward
jumps.

The option price for the case of the Merton model is given by Equation (2.28).
Using the Monte Carlo approach, we get an estimation $\hat{z}=0.519$. Here, we
again emphasize that this price differs from the one presented by Merton in
Equation (2.26), which equals 0.515 for our particular choice of parameters.
This means that the difference between the option price proposed in (Merton,
1976) and the minimal variance price is nontrivial in practice. In Figure 8,
we can see how the distribution of the differences between terminal wealth and
option payoff in Merton’s and our approach differ. We can see that using
Merton’s portfolio, one usually observes small gains, which are balanced with
occasional large losses. On the other hand, the distribution is symmetric
around zero using our approach. Apart from that, we are able to avoid large
losses entirely, which is an advantage. Consequently, using a minimal variance
hedging strategy is more costly, as discussed in Section 2.5.

Figure 8: Distribution of the difference between terminal wealth and option
payoff under Merton’s and our approach in 1000 realizations

The performance of our deep learning algorithm for the Merton model is
presented in Table 3. Compared to the performance in the BS case, the error of
price prediction is around ten times higher, which is due to the increased
volatility in the model.

Model |  $\hat{z}$ |  $\hat{x}_{0}$ |  $loss_{min}$
---|---|---|---
Merton |  0.519 |  0.521 |  8.13e-5
Table 3: Comparison of theoretical and numerical results in Merton model

In the case of both complete and incomplete markets, we showed that the deep
learning algorithm performs well by comparing the results with cases where
explicit solutions or at least Monte Carlo estimates are available.

To analyze the scalability of the algorithm concerning an increase in input
dimension, similar to the continuous case, we examine the Merton model jump
diffusion process driven by a multi-dimensional compound Poisson process.
Specifically, we compare the outcomes with a one-dimensional version guided by
a modified compensated Poisson process using mixed distributions. We make use
of the following well-known result.

###### Proposition 3.2

Let $X_{1}(t)=\sum_{i=1}^{N_{1}(t)}Y^{1}_{i}$ and
$X_{2}(t)=\sum_{i=1}^{N_{2}(t)}Y^{2}_{i}$ be two compound Poisson processes
where $N_{1}(t)$ and $N_{2}(t)$ are independent homogenous Poisson processes
with intensities $\lambda_{1}$ and $\lambda_{2}$ respectively. Furthermore,
jumps $Y_{i}^{1}$ and $Y_{i}^{2}$ are independent, coming from possibly
different distributions. Consider now $N(t)$, a homogenous Poisson process
with intensity $\lambda=\lambda_{1}+\lambda_{2}$ and random variables

$\displaystyle Y_{i}\sim\begin{pmatrix}&Y_{i}^{1}\qquad&Y_{i}^{2}\\\
&\frac{\lambda_{1}}{\lambda_{1}+\lambda_{2}}\qquad&\frac{\lambda_{2}}{\lambda_{1}+\lambda_{2}}\end{pmatrix}.$

Then processes $X(t)=\sum_{i=1}^{N(t)}Y_{i}$ and $X_{1}(t)+X_{2}(t)$ have the
same distribution.

This result easily extends to the sum of an arbitrary number of compensated
Poisson processes. For simulation purposes, we select a three-dimensional
compound Poisson process with intensities $[3,5,2]$. The lognormal jumps have
means $[0.1,0.1,0.05]$ and standard deviations $[0.05,0.02,0.01]$. The
obtained results are presented in Table 4. Once again, the similarity in the
results indicates that our algorithm scales effectively even in cases
involving discontinuous multi-dimensional input.

Model |  $\hat{z}$ |  $\hat{x}_{0}$ |  $loss_{min}$
---|---|---|---
Merton 3-D |  0.507 |  0.506 |  7.01e-5
Merton mixed |  0.507 |  0.506 |  7.54e-5
Table 4: Comparison of theoretical and numerical results in Merton model

####  3.3.4 Kou market model

Here, we present numerical results for a market model introduced in (Kou,
2002). Kou proposed a model where the stock’s behaviour is governed by a jump
diffusion process where the jumps have a double exponential distribution. This
means we can use the same type of discretization as for the Merton case
described in equations (3.1) and (3.1), where the jump variables $Y_{i}^{l}$
now follow a double exponential distribution

$\displaystyle
Y_{i}^{l}\sim\begin{pmatrix}&\text{Exp}(\eta^{l}_{1})\qquad&-\text{Exp}(\eta^{l}_{2})\\\
&p^{l}\qquad&1-p^{l}\end{pmatrix},$

where $p^{l}\in[0,1]$ represents the probability of exponentially distributed
jump being either upward or downward. To ensure the existence of moments, we
need additional conditions on intensities of the positive jump part, hence
$\eta^{l}_{1}>1$ while $\eta_{2}^{l}>0$. The compensator of the drift in
Equation (3.1) changes accordingly and has components

$k^{l}=p^{l}\frac{\eta^{l}_{1}}{\eta^{l}_{1}-1}+(1-p^{l})\frac{\eta_{2}^{l}}{\eta_{2}^{l}+1}-1.$

Unfortunately, the property of jumps being both positive and negative and
having heavier tails makes the Kou model impractical when one wants to obtain
option price through equivalent martingale measures and Mont-Carlo approach as
described in Section 2.5. The reason is that the condition
$G(e^{Y_{i}^{l}}-1)>-1$, for $G$ as in Equation (2.27), is not satisfied for
general choice of parameters $\eta_{1}^{l}$, $\eta_{2}^{l}$ and $p^{l}$.

Nonetheless, the option price can still be obtained through the use of the
deep learning algorithm. Again, to avoid negative wealth and stock values, we
select $R=150.$ Due to the heaviness of the tails in jump distribution, one
needs to be careful when choosing parameters. In particular, $\eta_{2}$
shouldn’t be too small. Here we choose $s_{0}=1,K=0.5,$
$\alpha_{0}=0.15,\sigma_{0}=0.2,\gamma_{0}=1,\lambda=10,\eta_{1}=50,\eta_{2}=25$
and $p=0.3$. According to (Kou, 2002), these parameters should reflect the
ones observed on the US stock market. Graphs of the loss and initial value
convergence and a graph of one market realization can be found in Figure 9.
The minimal obtained loss is 1.54e-5, lower than the losses recorded for the
Merton model due to the choice of less noisy parameters. The algorithm
predicts the initial wealth value $x_{0}=0.4987$. We know that the true option
price should be higher than the BS price of 0.5 but not by much due to more
conservative jump parameters. Hence, the predicted result does not seem too
far off. The reason for the undershoot in the predicted initial value is the
inclusion of the positive jumps in the Kou model.

(a) Loss

(b) Initial values

(c) Market realization

Figure 9: Convergence of loss and initial values and one market realization
for the Kou model

A possible additional direction for advancing the current study is to explore
the utilization of deep learning algorithms in pricing American and path-
dependent options within the context of the double exponential jump diffusion
model, drawing inspiration from the research conducted in (Kou and Wang,
2004).

####  3.3.5 Scalability in temporal dimension

In this section, we explore how the algorithm’s performance is influenced by
variations in the temporal dimension. Specifically, we consider different time
maturities $T$ and varying numbers of timesteps $R$.

We examined the algorithm’s performance within the context of both the
complete BS market model and the incomplete Merton market model. Various
combinations of parameters $T$ and $R$ were scrutinized. The financial model
parameters remain consistent with those outlined in Sections 3.3.1 and 3.3.3.

Concerning machine learning parameters, we maintained a batch size of 256, a
hidden dimension of 512, and a learning rate of 0.0005. For practical
considerations, we conducted the learning process for a fixed 3000 epochs in
each studied case. In both market models, the algorithm’s performance was
evaluated by comparing the loss and the absolute error of predicted option
prices. For the BS model, the estimated expected $L^{2}$ loss defined in
Equation (3.4) was also included in the assessment.

Results are presented in Tables 5 and 6 for the BS and Merton models,
respectively.

R/T |  0.5 |  1.0 |  2.0
---|---|---|---
40 |  6.058e-5 |  2.917e-5 |  8.198e-5
80 |  4.563e-5 |  1.716e-5 |  8.246e-5
160 |  4.122e-5 |  2.164e-5 |  3.41e-5
(a) Loss

R/T |  0.5 |  1.0 |  2.0
---|---|---|---
40 |  0.0026 |  0.0103 |  0.0323
80 |  0.0020 |  0.0094 |  0.0329
160 |  1.580e-5 |  0.0080 |  0.0286
(b) Absolute error of option price

R/T |  0.5 |  1.0 |  2.0
---|---|---|---
40 |  0.3171 |  0.4369 |  0.7221
80 |  0.4161 |  0.5951 |  1.1797
160 |  0.4399 |  1.0670 |  1.3513
(c) Expected $L^{2}$ distance

Table 5: Comparison of algorithm performance for different choices of
timesteps $R$ and maturities $T$ \- BS model

It should be stated that due to the random nature of both inputs and the
algorithm presented results are noisy in nature as well.

Nonetheless, we may with large certainty conclude that the algorithm’s
performance decreases with increasing maturity. This is particularly evident
in the absolute error of option prices (Table 5(b)) and the expected $L^{2}$
distance (Table 5(c)). The use of discrete Euler-Maruyama approximation to the
continuous solution likely contributes to this trend, where errors accumulate
over time. We believe enhanced computing power may contribute to improved
performance through finer discretization, extended training time, and higher
hidden dimensions.

In examining performance concerning the discretization parameter $R$, it is
harder to draw strong conclusions since the difference in results is less
significant. It seems that finer discretization positively impacts the
algorithm’s performance in terms of loss and absolute error in option prices.
Conversely, an inverse trend is observed in the expected $L^{2}$ distance.
This discrepancy may arise from increased variation in the discrete
theoretical portfolio as the number of timesteps grows. Nevertheless, focusing
on loss and option price, which are the key outputs of our algorithm, we find
a positive response to an increased number of discretization points. As
discussed earlier, this may prove beneficial when considering longer
maturities.

R/T |  0.5 |  1.0 |  2.0
---|---|---|---
40 |  1.976e-5 |  8.527e-5 |  0.0003
80 |  3.156e-5 |  4.914e-5 |  0.0002
160 |  1.482e-5 |  3.728e-5 |  0.0002
(a) Loss

R/T |  0.5 |  1.0 |  2.0
---|---|---|---
40 |  0.0004 |  0.0085 |  0.0290
80 |  0.0011 |  0.0072 |  0.0150
160 |  0.0024 |  0.0063 |  0.0127
(b) Option price absolute error

Table 6: Comparison of algorithm performance for different choices of
timesteps $R$ and maturities $T$ \- Merton model

Before presenting the results for the Merton case, we must first address the
problem regarding the use of discretizations with a small number of timesteps,
which was already mentioned in Section 3.3.3. To circumvent the problem of
negative wealth and stock values, we present an alternative discretization
approach. Instead of directly working with the discrete wealth process $x$
given by the update rule in Equation (3.1), we take its logarithm. Using Itô’s
formula and Euler-Maruyama scheme we obtain its discrete version through an
update rule

$\displaystyle
y_{i+1}=y_{i}+[(\alpha_{0}-\lambda^{\top}k)\pi_{t}-\frac{\pi_{t}^{2}\sigma_{0}^{\top}\sigma_{0}}{2}]\Delta_{t}+\pi_{t}\sqrt{\Delta_{t}}B_{i}\sigma_{0}+\log(\gamma_{0}\pi_{t}J_{i}+1),$

where $y_{0}=\log(x_{0})$ and then put $x_{i}=\exp(y_{i})$. The same is done
for the discrete stock process $s$ where we additionally put $\pi=1$.

Let us now present the results regarding the Merton case. In tables 6(a) and
6(b), we see that the algorithm responds to changes in maturity and the number
of timesteps similarly as in the BS case. The algorithm’s worse performance is
due to the increased randomness in the model, as it was explained in Section
3.3.3. Additionally, it must be mentioned that the algorithm’s performance
slightly decreases when the logarithmic approach outlined above is used. This
decline could be attributed to the non-linear dependence of the process $y$ on
the control variable $\pi$. Hence, for the jump diffusion models, we advise
the use of the original update rule with the number of timesteps at least 150.

## Data availability

The author confirms that all data generated or analysed during this study are
included in this published article. Furthermore, primary and secondary sources
and data supporting the findings of this study were all publicly available at
the time of submission.

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## Competing interests

The author declares no competing interests.