Option Pricing and Hedging in Jump Diffusion Models

Home , Jump diffusion

U.U.D.M. Project Report 2010:7

Option pricing and hedging in jump diffusion models

Yu Zhou

Examensarbete i matematik, 30 hp Handledare och examinator: Johan Tysk

Maj 2010

Department of Mathematics Uppsala University

Master thesis of Mathematics

Specialization in Financial Mathematics

Option Pricing and hedging in Jump-diffusion Models

Yu Zhou

Advising professor: Johan Tysk

Department of Mathematics, Uppsala University

May, 2010

Abstract

The aim of this article is to solve European Option pricing and hedging in a jump-diffusion framework. To better describe the reality, some major events, for example, may lead to dramatic change in stock price; we impose a jump model to classic Black

Scholes model.

Under the assumption that the underlying asset is driven by a simple jump-diffusion process, along the fact that risk free rate and volatility are deterministic functions of time, by changing the probability measure to risk-neutral measure Q, we obtain the pricing formula of European options. The hedging strategy is defined in order to minimize the risk of hedging under a risk neutral measure.

Key Words: Black-Scholes model Jump-diffusion model option pricing

Contents

1. Introduction

1.1 Jump-Diffusion Model………………………………………………………… 3

1.2 Incomplete Market ...... …...3

1.3 Mathematical Tools……………………………………………………………..4

1.4 About this article……………………………………………………………….4

2. Option Pricing and Hedging in the Black-Scholes Model

2.1 Introduction to Option Pricing…………………………………...…………….5

2.2 Option Pricing in Black-Scholes Model………………………………………….6

2.3 Pricing………………………………………………………………………..7

2.4 Hedging…………………………………………………………...………….8

3. Option Pricing and Hedging in the Jump-Diffusion Model

3.1Dynamics of the Underlying Asset……………………………………………… 9

3.2 Conditions for St to be a Martingale………………………………………….10 � 3.3 European Options Pricing…………………………………………………….12

3.3.1 Admissible Portfolios…………………………………………………...12

3.3.2 European Options Pricing………………………………………………14

3.4 Hedging European Options…………………………………………………...16

4. Reference……………………………………………………………………….20

Chapter 1 Introduction

1.1 Jump-Diffusion models

In the Black-Scholes Model, the stock price is driven by Brownian motion and is based on a continuous function of time. However, in reality this is not often the case, since certain important events can lead to dramatic change in the stock price.

To model such phenomenon, many studies introduced discontinuous stochastic

processes by adding jump-diffusion to the classic model. To price options, we must notice that a jump diffusion model leads to an incomplete market. Therefore, the classic hedging methods are not applicable here.

There are rather extensive studies in the jump-diffusion model. As early as 1976,

Robert Merton noted that when major events happen, stock price change discontinuously, or, jump. He also noted that the stock price is driven by Brownian motion (1976) as well as Poisson process. Having taken that into consideration,

Merton created the Jump-diffusion model. [4]

1.2 Incomplete market

In an arbitrage-free market, there is an equivalent probability measure such that the underlying asset is a martingale under risk neutral measure, which implies that the asset is a semi-martingale under the objective measure P. Mathematically, a complete market means that any contingent claim can be replicated as a stochastic integral of a sequence of semi-martingales. The integrant in such replications provides a sequence of hedging strategies which are self-financing or, super hedging.

Whilst in Jump-diffusion models, more random variables are added to the market, making it Incomplete. In this case, a general claim is not necessarily a stochastic integral of the underlying asset. Economically speaking, this claim has an

intrinsic risk that we only hope to reduce to this minimal component. Thus the problem is to find and characterize these strategies which minimize the risk.

1.3 Mathematical tools

The main mathematical tools to apply to this field are (1) Stochastic Differential

Equations, (2) Monte Carlo (simulation) and (3) Martingale methods. As the first two tools are rather often used in previous papers in our department, this article would try to derive the same results with Martingale methods.

In the more general martingale approach, one specifies a stochastic process for the underlying asset. Then one can choose an equivalent probability measure turning the discounted underlying into a (possibly local) martingale and computes the derivative's value as the conditional expectation of its discounted payoff under this risk-neutral measure. If the model has a Markovian structure, then this value turns out to be some function u, say, of the state variables. In the PDE approach, one can describe the state variables by a stochastic differential equation (SDE) and then derives for the function u based on the underlying martingale valuation a PDE involving the coefficients of the given SDE. [5]

1.4 About this article

Our study will be divided into three parts.

(1) The first chapter is the introduction, in which we present the background of the jump-diffusion models and hedging in incomplete market.

(2) In the second chapter we begin with the basic theory of option pricing,

Black-Scholes framework and hedging, where r is constant.

(3) The third chapter we assume that the jump follows a Poisson distribution, and that the interest rate is time deterministic. We firstly find the sufficient and necessary conditions for the underlying asset to be a martingale. Under such conditions, by changing the measure, we can obtain the pricing formula under risk-neutral probability measure Q, and then be able to derive the hedging strategy by minimizing the risk.

Overall, this article presents an alternative way to solve option pricing and hedging problem, in the Jump-diffusions model and incomplete market.

Chapter 2 Option Pricing and Hedging in

the Black-Scholes Model

2.1 Introduction to Option Pricing

2.1.1 Options

An option is a contract between a buyer and a seller that gives the buyer the right, (but not the obligation), to buy or to sell a particular asset on or before the options expiration time, at an agreed price (the strike price). In return for granting

the option, the seller collects a payment (the premium) from the buyer.

A call option gives the buyer the right to buy the underlying asset and a put option gives the buyer of the option the right to sell the underlying asset. If the buyer chooses to exercise this right, the seller is obliged to sell or buy the asset at the agreed price.

The buyer of a call option wants the price of the underlying instrument to rise in the future; the seller either expects that it will not, or is willing to give up some of the upside (profit) from a price rise in return for the premium and retaining the opportunity to make a gain up to the strike price.

In case of a put option, buyer acquires a short position by purchasing the right to sell the underlying asset to the seller of the option for specified price during a specified period of time. If the option buyer exercises their right, the seller is obligated to buy the underlying instrument from them at the agreed upon strike price, regardless of the current market price. [Wiki]

2.1.2 Option Pricing

In exchange for having an option, the buyer pays the seller or option writer a fee, known as the option premium. The option premium is the maximum loss in an option exchange, of the party taking long position of the underlying asset, which depend on

the price change of the underlying asset.

Since the underlying asset is usually a risky asset (e.g. a stock), the price of underlying asset is usually a stochastic process. But when the underlying asset price is settled down, the options written on that is also settled down. Mathematically, for

option price Ft, there exists a binomial function F(t, S) such that the price of the option Ft = F(t, S). At the examining date, Ft is deterministic.

+ (ST K) (call) FT = + (K ST) (put) − � Option pricing problem is about− to derive the above Ft, hence it is an inverse problem.

2.2 Black Scholes Model option pricing and hedging

2.2.1 The Model

Black Scholes is a model based on continuous time. There are two different types of assets on the market:

0 The risk free asset, which has price St at time t, and satisfies the equation:

0 0 dSt = rSt dt

0 0 rt where r 0 is the risk free rate; thereforeSt = S0 e , from now on we assume

0 ≥ S0 = 1 for convenience. Then,

0 rt St = e , t 0.

∀ ≥ And the risky asset, which has price St at time t, and satisfies the equation:

dSt = μStdt + σStdBt

where μ and σ > 0 are constant, and that Bt is a standard Brownian motion.

Now define the present value of St as St, i.e.,

� rt St = e St, − � By Ito formula,

rt rt dSt = re Stdt + e dSt = St (μ− r)dt + −σdBt = StσdWt � − t ( t)t where Wt = Bt + , t = . � � − � � θ μ− σ θ σ T 1 T We define L = exp ds 2ds , dQ = L dP T 0 s 2 0 s T �− ∫ θ − ∫ θ � Then Q and P are equivalent, then we know from Girsanov Theorem that Wt is

a Standard Brownian Motion under Q.

2 By Ito formula, S = S exp ( W t) , thus S is a martingale under Q. t 0 t 2 t σ � � ⁡σ − � Therefore Q is so called the risk neutral measure or equivalent martingale measure.

2.2 Self financing Portfolio

0 0 Consider a portfolio = ( t)0 t T = (ht , ht), where ht and ht represent the

ϕ ϕ ≤ ≤ number of shares of risk-free asset and risky asset, respectively. Thus this portfolio is,

at time t, worth

0 0 Vt( ) = ht S1 + htSt

Furthermore, its discounted valueϕ is

rt 0 Vt( ) = e Vt( ) = ht + htSt. − rt � ϕ ϕ � where St = e St is the underlying asset price under measure Q that we are going to − � discuss later.

S 0 Definition 2.2.1 An t adapted portfolio = ( t)0 t T = (ht , ht) is called

ℱ ϕ ϕ ≤ ≤ self-financing if the value process Vt satisfies the conditions

0 0 dVt( ) = ht dS1 + htdSt T T 0 2 |ϕht |dt + ht dt < ∞ (2.1) 0 0 � � where the latter condition makes the portfolio admissible, which is known as the

integrable condition.

0 Definition 2.2.2 Portfolio = ( t)0 t T = (ht , ht) is called attainable if it is

ϕ ϕ ≤ ≤ self-financing, as well as its discounted value Vt( ) is non-negative and

� ϕ square-integrable.

2.3 European options pricing

To price a European option, we define its contingent claim h = f(ST), where f(S) = (S K)+ for a call option and f(S) = (K S)+ for a put option.

− − A portfolio is called can be replicated if for its contingent claim h there exists a

strategy ϕ such that the value of the portfolio strategy is equal to h, i.e. Vt(ϕ) = h.

This portfolio strategy is thus called the replicate strategy.

Now we denote F(t, S) be the price of the option at time t. so,

r(T t) S r(T t) S F(t, S) = E e h t = E e f(ST) t ∗ − − ∗ − − For European call: � �ℱ � � �ℱ �

r(T t) C(t, S) = SN(d1) Ke N(d2) S 2 ln + r+σ (T t) − − K 2 where d2 = , d2 = d1 σ T− t, and N(x) is standard normal �σ T �t − distribution √ − − √ −

For European put:

P(t, S) = ke r(T t) S + C(t, S) − − where it has the same notations as in a call option− .

Results of price expressions in this section refer to source [1].

2.4 Hedging

With the previous definition for replication, suppose our claim h admits an Ito

representation for the form

T h = h0 + htdSt (2.2) 0 � where h satisfies equality (2.1).

T Clearly, strategy Vt(ϕ) = h0 + 0 htdSt is admissible. Moreover, it is ∫ self-financing. Thus, Ito representation (2.2) leads to a strategy which produces h

from initial value of h0 with no risk involved.

Now, we introduce the standard arbitrage-free assumptions, namely the

risk-neutral martingale measure, or the equivalent martingale measure. We assume a

martingale measure Q such that

Q 2( , , P) P ∂ ∈ ℒ Ω ℱ and St is a martingale under Q. ∂

And ht is obtained in a Randon-Nikodym derivative [6] F ht = (t, St) Δ(t, St) St ∂ ≜ where Δ = Fs(t, s(t)) is called its Δ∂ hedging.

Note that in the Black Scholes Model ht is the partial derivative of the value of

the option at time t with respect to the underlying asset price at time t.

So far we have summarized the mathematical construction of hedging in a

complete financial market, where every claim is attainable. In such case, hedging

allows total elimination of risk in handling an option.

Chapter 3 Option Pricing and Hedging in

the Jump-Diffusion Model

To introduce “jump”, we have to consider a striking feature that distinguishes it from the standard Black-Scholes model: this makes the market incomplete, and there is no perfect hedging of options in this case. It is no longer possible to price options using a replicating portfolio, and the set of probability measures under which the discounted stock price is a martingale is infinite.

In this chapter, we will discuss the computation of European option prices and examine hedging strategies that minimize the quadratic risk under the pricing measure that we choose.

3.1 Dynamics of the Underlying Asset

Let us consider a financial market without any transaction fee, in which there are two types of assets, the risk free asset and the risky asset; and the time interval

[0, T], with the maturity time T. We assume the risk-free asset value is Pt; short rate is a time-deterministic function rt; risky asset value is St.

Risk-free asset value Pt follows the stochastic derivative function

dPt = rtdt, P0 = 1 The risky asset consists two parts, a continuous part modeled by a geometric

Brownian motion, and a jump part, with the jump size modeled by a jump function

Ut and the jump time modeled by a Poisson process, i.e.,

dSt = μt Stdt + σtStdWt + UtStdNt (3.1) where μt and σt are time deterministic functions; Wt is a standard Brownian motion; Nt is a Poisson process with parameter λ, which is independent of Wt; Ut is the jump function; and τj is the j-th stay time. So, Ut is the relative jumping

U = S S /S strength of S, i.e., j τj τj τj . Here the risky asset price is obviously left − − � − � continuous, hence τj is used to distinguish that. −

Note that {Uj} is a sequence of independent identical distributions. We define the distribution of {Uj} follows a function .

υ We take integration of (3.1) and get the dynamics of the asset,

t t St = S0 + Su [ udu + udWu] + Su Uu dNu 0 0 � μ σ � The last term can be further transformed as,

N N N t t t t t S U dN = S U d(u ) = S U = [S S ] u u u u u τj τj j τj τj 0 j=1 0 j=1 − j=1 − � � � − � � − Hence,

N t t S = S + S [ du + dW ] + [S S ] (3.2) t 0 u u u u τj τj 0 j=1 − � μ σ � − Using the fact that Ut is the relative jumping size, there’s another version for

this equation

Nt t 1 t S = S (1 + U ) exp μ 2 du + σ dW (3.3) t 0 j u 2 u u u j=1 0 0 �� − σ � � ��

3.2 Conditions for St to be a Martingale

�

In this section we will provide a formal proof to the Sufficient and Necessary

Conditions for St to be a martingale under Q, which enables us to change measure. � Lemma 3.2.1 For all s > 0, denote by s the -algebras σ UNs +1, UNs +2, UNs +3, … 𝒢𝒢 σ � � are independent to s.

ℱ Lemma 3.2.2 Let (y, z) be a measurable function from Rd × R to R, such that for

φ d any real number z the function y (y, z) is continuous on R , and let(Yt)t 0 be a

≤ ↦ φ d left-continuous process, taking values in R , adapted to the filtration ( t)t 0.

ℱ ≥ Assume that, for all t > 0,

t 2 E ds (dz) (Ys, z) < ∞ ∗ 0 �� υ φ � Then the process Mt defined by

N t t ( ) Mt = (Y j, Uj) ds (dz) Ys, z j=1 0 � φ τ − λ � � υ φ is a square-integrable martingale and

t 2 Mt ds (dz) (Ys, z) 0 − λ � �0υ φ is a martingale. Note that by convention j=1 = 1.

∑ Lemma 3.2.3 We keep the hypotheses and notations of Lemma 3.2.3. Let (At)t 0 be

≥ t t 2 < ∞ > = an adapted process such that E 0 As ds for all t 0. Let Lt 0 AsdsdWs. ∗ �∫ � ∫ Then the product (LtMt)t 0 is a martingale.

≥ Proofs for Lemma 3.2.1, 3.2.2, 3.2.3 refer to source [2].

Consider that Ft is a deterministic function that satisfies the condition

1 t F2 dt < .∞ Now let dW = dW + F dt , we can find a non-negative random 2 0 t t t t − ∫ � variable Lt in the following expression,

t 1 t L = exp [ F dW F2du] . t 0 u u 2 0 u ⁡− ∫ − ∫ Given this expression, Lt is naturally the solution of the SDE,

dLt = FtLtdWt

By Girsanov Theorem we know that that Wt is a standard Brownian motion � under risk neutral measure Q. [2]

Thus, for 0 < s < t T we have the risk-neutral E St s as ∗ ∀ ≤Nt �� ℱ � t t t r du 1 E S = E e 0 t S 1 + U exp 2 du + dW | t s 0 j u 2 u u u s ∗ ∗ −∫ j=1 0 0 �� ℱ � � �� μ − σ � � σ �� ℱ �

Nt t 1 t = E S 1 + U exp r 2 du + (dW F du) | 0 j u u 2 u u u u s ∗ j=1 0 0 � �� μ − − σ � � σ � − � ℱ � Nt t t 1 t = E S 1 + U exp ( r F ) du 2du + dW | 0 j u u u u 2 u u u s ∗ j=1 0 0 0 � �� μ − − σ − � σ � σ � �� ℱ � Nt t 1 = SsE 1 + Uj exp u ru uFu du s 2 ∗ j=Ns+1 � � � � � �� μ − − σ � �� t 1 = S e EU1(t s) exp r F du s u u 2 u u λ − s � t �� μ1 − − σ � � = Ss exp u ru uFu + EU1 du s 2 � �� μ − − σ λ � �

Hence, St is a martingale under Q if and only if � t 1 u ru uFu + EU1 du = 0 s 2 � �μ − − σ λ � or,

t 1 r F du = EU (t s) s u u 2 u u 1 ∫ �μ − − σ � −λ − or,

1 t 1 r F du = EU . t s s u u 2 u u 1 − ∫ �μ − − σ � −λ Consider the last equation, 0 < s𝑡𝑡 t T, let t s, we have that

t ∀ rt ≤tFt≤= EU1 →

where, μ − − σ −λ

t rt + EU1 Ft = (3.4) t μ − λ σ

Clearly, St is a martingale under Q if and only if (3.4) is satisfied, sufficiently � and necessarily, given the integrable condition of course,

T 2 Ft dt < ∞ 0 � Under such conditions, we have

Nt t 1 t S = S 1 + U exp EU + 2 du + dW t 0 j j 2 u u u j=1 0 0 � �� ⁡�− � �λ σ � � σ � We are going to use these results in the following sections.

3.3 European Option Pricing

3.3.1 Admissible Portfolios

In this section we will define a trading strategy, as in the Black-Scholes model,

0 0 by a portfolio = (ht , ht), 0 t T , that has amount ht and ht in two assets at

ϕ ≤ ≤ time t, respectively, taking values in R2, representing the amounts of assets held over

0 time. But, to take the jump into account, we will constrain the processes ht and ht to be left-continuous. The value at time t of the strategy is given in the following

ϕ definition.

S 0 Definition 3.3.1 The t adapted portfolio = (ht , ht), 0 t T is called

ℱ ϕ ≤ ≤ self-financing if,

t 0 rs ds dVt = ht rte 0 dt + htdSt ∫ s 0 rs ds Taking into account dVt = ht rte 0 dt + htSt( tdt + tdWt) between the ∫ μ σ

V V = h S U jump times and, at a jump time j, t jumps by an amount j j j j. − τ Δ τ τ τ Precisely, the self-financing condition is

N t t t V = V + h0dB + h S ( ds + dW ) + h S U t 0 s s s s s s s j j j 0 0 j=1 − � � μ σ � τ τ or,

t s t 0 ru du Vt = V0 + hs rse 0 ds + hsSs ( s Fs s)ds + sdWs 0 ∫ 0 � Nt � � μ − σ σ � � + h S U (3.5) j j j j=1 − � τ τ

0 Definition 3.3.2 A portfolio = (ht , ht), 0 t T is called admissible if it satisfies

ϕ ≤ ≤ the equality (3.5) and satisfies the conditions,

T T 0 2 2 |hs |ds < . . , hs Ss ds < 0 0 � ∞ 𝑎𝑎 𝑠𝑠 𝐸𝐸 �� ∞

Actually, in order to make sure the value of a hedging strategy square-integrable, we have to impose the previous conditions, which are stronger than the class of self-financing strategies. [2]

0 Proposition 3.3.3 In an admissible portfolio = (ht , ht), 0 t T with initial value

ϕ 0 ≤ ≤ V0, for any ht, 0 t T, there exists a unique ht that keeps the portfolio admissible.

≤ ≤ The discounted value for this portfolio is given by

N t t t V == V + h S dW + h S U ds h S (dz)z. t 0 s s s s j j j s s 0 j=1 − 0 � � � σ � � τ τ − λ � � � υ

Proof of Proposition 3.3.3 The value of the portfolio at time t can be written as

ϕ the decomposition Vt = Yt + Zt, in which

t s t 0 ru du Yt = V0 + hs rse 0 ds + hsSs ( s Fs s)ds + sdWs 0 ∫ 0 � Nt� � μ − σ σ � � Z = h S U t j j j j=1 − � τ τ Then, differentiating the discounted product Yt, we have

t t � t ru du ru du ru du dYt = de 0 Yt = rte 0 Ytdt + e 0 dYt − ∫ − ∫ − ∫ � Moreover, it can be written as −

t s t s ru du ru du Yt = V0 + rse 0 Ysds + e 0 dYs 0 −∫ 0 − ∫ t � s � − ru du � = V0 + rse 0 Ysds 0 −∫ t s t � − ru du ru du + e 0 hsrse 0 ds + hsSs ( s Fs s)ds + sdWs 0 − ∫ ∫ t t s s � �ru du � ru du � μ − σ σ �� = V0 + rse 0 Ysds + e 0 hsSs ( s Fs s)ds + sdWs (3.6) 0 −∫ 0 − ∫ � − � � μ − σ σ � �

Continue by doing the same transform to Zt, we have

N N � t t s t t j r du r du r du r du 0 u Z = e 0 u Z = e 0 u h S U = e 0 u + e j h S U t t j j j τ j j j − ∫τ − ∫ j=1 − ∫ − j=1 − ∫ − � τ τ τ τ N� �N � � t t s j t r du 0 ru du = e 0 u h S U + e j r ds h S U τ j j j s j j j − ∫τ j=1 − ∫ − j=1 j − τ τ τ τ � � ��τ − � where in the last term, the integration would only take place on the points of s j ,

� ≥ τ � thus we have,

Nt Nt j t s r du r du Z = e 0 u h S U + e 0 u h S U ds t τ j j j j j j j=1 − ∫ − 0 − ∫ j=1 − � � τ τ � �− � τ τ � Nt j t s r du r du = e 0 u h S U + e 0 u r Z ds (3.7) τ j j j s s j=1 − ∫ − 0 − ∫ � τ τ � −

We get Vt = Yt + Zt by combining (3.6) and (3.7). After rearrangement, we � � � obtain,

t t t 0 Vt = V0 + rs(Ys + Zs)ds + hs rsds + hsSs ( s Fs s)ds + sdWs 0 0 0 � � − �Nt � � � � � μ − σ σ � � + h S U j j j j=1 − � τ �τ 0 To simplify the upper formula, we substitute Vt by ht + htSt in the right hand � � side of the equality and the terms with h0 cancels out, which yields

N t t V = V + h S [( F )ds + dW ] + h S U t 0 s s s s s s s j j j 0 j=1 − � � � μ − σ σ � � τ �τ Finally we take into account the equality (3.4) and get the following one that

we need to prove. We will continue to use this in the next sections.

N t t t V = V EU h S ds + h S dW + h S U t 0 1 s s s s s s j j j 0 0 j=1 − � � � � τ �τ − � λ N� σ � t t t = V(0) + h S dW + h S U ds h S (dz)z (3.8) s s s s j j j s s 0 j=1 − 0 � � σ � � τ �τ − λ � � � υ ∎

0 Remark 3.3.4 By Proposition 3.3.3 it is clear that the processes ht and ht , together

with the initial value V0, we can uniquely determine one given the other two.

0 Remark 3.3.5 Note that there is no ht term in the right hand side of equality (3.8),

0 this indicates that the admissible portfolio = (ht , ht) with initial value V0 is

ϕ completely determined by the process (ht)0 t T representing the amount of the risky

≤ ≤ asset.

3.3.2 European Options Pricing

Let us consider a European option with maturity T; its contingent claim be h; it

is T-measurable and square-integrable. The seller of this option receives V0 at time

ℱ 0 and holds an admissible portfolio between times 0 and T. If Vt represents the

value of this portfolio at time t, the payoff for the seller at maturity is given by

VT h, which is also the hedging mismatch. If this quantity is positive, the seller of

− the option earns money, otherwise loses some. Its present value is then VT

t � r du − e 0 u h − ∫ In order to better replicate the contingent claim of European option, we need to

minimize the payoff VT h, namely the risk. One way to measure the risk is to define

− the expected square, i.e.,

t T rdu 2 R0 = E (VT e 0 h) ∗ − ∫ � Applying the identity E(X2) = E2(X)−+ E(X E(X))2 to the variable X =

t rdu − e 0 (VT h) we obotain, − ∫ t t t 2 T− 2 rdu rdu rdu R0 = E VT e 0 h + E VT e 0 h E(VT e 0 h) − ∫ ∗ − ∫ − ∫ �� − T� 2 2 �� − − � − � Since, the condition 0 hs Ss ds < in Definition 3.3.2 implies that the 𝐸𝐸 �∫ � ∞ discounted value Vt is a square-integrable martingale, we have that E VT = V0. ∗ � t t � rdu rdu � � Moreover, we have, E VT e 0 h = V0 E e 0 h ∗ − ∫ ∗ − ∫ �� −T � − � � By applying this identity to the R0 representation we obtain,

t 2 t t 2 T rdu rdu rdu R0 = V0 E e 0 h + E e 0 h E e 0 h + VT V0 ∗ − ∫ ∗ − ∫ ∗ − ∫ � − � �� − � � � − � According to Remark 3.3.5, the quantity VT V0 depends only on the process

T � − ht. Then it is obvious that when minimizing R0, the seller will expect a premiumV0 =

T rdu E e 0 h . This appears to be the initial value of any strategy designed to − ∫ � � minimize the risk at maturity. For the same argument, when selling the option at

T T rdu 2 time t > 0, one wants to minimize the quantity Rt = E (VT e t h| t) will ∗ − ∫ t rdu � − ℱ expect for a premium V(t) = E(e 0 h| t). − ∫ ℱ Hence, we will use this quantity to define the price of the option, i.e.,

T rdu V0 = E e 0 h is the price of the European option at time 0 and Vt = − ∫ t rdu� � E(e 0 h| t) is that at time t. − ∫ ℱ

Now, we will give an explicit expression for the price of the European call or put.

By the definition of European options, the contingent claim h can be specified as

follows,

h = f(ST), call: f(x) = (x k)+ , put: f(x) = (k x)+ T rdu V(t) = E e t f(ST) t − − − ∫ NT Nt � T �ℱ � T 1 T rdu 2 F du + dW(u) t − t 2 u t = E e f St 1 + UNt +j e t ∗ − ∫ j=1 ∫ �μ− σ −σ � ∫ σ � � � � � � � �ℱ � = F(t, St)

With Lemma 3.2.1 (UNs )s t and t are independent, we can rewrite the ≥ ℱ equality as,

NT Nt T T 1 T rdu 2 F(u)du + dW(u) ( ) t − t 2 t F t, x = E e f x 1 + UNt +j e ∗ − ∫ j=1 ∫ �μ− σ −σ � ∫ σ � � � � � � ��

Then we plug in the option price for the Black-Scholes model,

T 1 T T 2 ( ) rdu t F u du + t dW F0(t, x) = E e t f xe 2 ∗ − ∫ ∫ �μ− σ −σ � ∫ σ � � � �� getting,

N(T t)

EUj (T t) F(t, x) = E F0 t, xe − 1 + Uj ∗ −λ − j=1 � � � � �� n e (T t) n(T t)n = E F t, xe EUj (T t) 1 + U (3.9) ∞ 0 j −λ − n! n=0 ∗ −λ − j=1 λ − � � � �� For different distributions of jumping size, we can compute the corresponding

EUj numerically and then can derive the expression for F(t, x) explicitly.

3.4 Hedging European options

We continue to use the notations that we defined in the last section.

T t r(u)du 2 h = f(ST), V0 = F(0, S0), R0 = E (e 0 f(ST VT)) ∗ − ∫ � In section (3.3.1) we have already reached a conclusion that− given two out of the

0 three variables ht, ht , and V0, we can determine the third one explicitly. In section

(3.3.2) we have seen that the initial value of any admissible strategy aiming at

T T rdu minimizing the risk R0 at maturity is gien by V0 = E e 0 h . − ∫ � � Now we will determine the process (ht)0 t T representing the amount of the

≤ ≤ T risky asset over the period [0, T], so as to minimize R0. Specifically, we consider an

0 admissible portfolio = (ht , ht), 0 t T, with the value Vt at time t and that

T ϕ ≤ rdu≤ the initial value satisfies V0 = E e 0 h . Then we try to give the interpretation − ∫ T� � for quadratic risk at maturity R0.

Firstly, take into account the equality (3.8):

N t t t V = F(0, S ) + h S dW ) + h S U ds h S (dz)z t 0 s s s s j j j s s 0 j=1 − 0 � � σ � � τ �τ − λ � � � υ T T r(u)du rdu We also have that h = e 0 f(ST) = e 0 F(T, ST). Let us define a function − ∫ − ∫ T T rdu � rdu F(t, x) = e 0 F(t, xe 0 ), so that − ∫ ∫ T t T rdu rdu rdu F t, St = e 0 F(t, St) = e 0 E (e 0 h| t) − ∫ − ∫ ∗ −∫ �� ℱ Hence, F t, St is a martingale under Q.

�� Let Yt = St and (Yt, z) = F t, Yt(1 + z) F(t, Yt) − � φ �� − � In our case,

Y , U = F , S 1 + U F , S j j j j j − − φ � τ � � �τ �τ � �� − � �τ �τ � = F , S F , S j j j j − � �τ �τ � − � �τ �τ � We denote

t M = Nt Y , U ds (dz) (Y , z) t j=1 j j 0 s

N τ t ∑ φ � � − λ ∫ ∫ ν t φ = F , S F , S ds (dz) F s, S (1 + z) F s, S j j j j s s 0 j=1 − − − � �� τ �τ � − � �τ �τ �� − λ � � ν �� − �� Nt = [F , S F , S ] j j j j j=1 − τ τ � � �τ �t � − � �τ � � ds (dz) F s, Ss(1 + z) F s, Ss (3.10) 0 − λ � � ν �� − �� in which, according to Lemma (3.2.2), we know that the process Mt is a

square-integrable martingale.

2 + Now let us consider equality (3.9), F t, St is C [0, T) × R on we can apply

�� Ito formula in each fragment between the jump times.

Apply Ito formula to F t, St ,

t F�� t F F t, S = F(0, S ) + s, S 2S2ds + s, S [( r F )ds + dW ] t 0 S s s s x s s s s s s s 0 ∂� 0 ∂� �� σ �Nt � � � μ − − σ σ � 1 t 2∂F ∂ + s, S 2S2ds + F , S F , S 2 x2 s s s j j j j 0 ∂ � j=1 − � � � �σ � � �τ �τ � − � �τ �τ � ∂ Together with the equality (3.10) , we get,

t F F t, S M = F(0, S ) + s, S 2S2dW t t 0 x s s s s 0 ∂� �� − t F � � � �σ �F 1 2F + s, S∂ EU S + s, S + s, S 2S2 x s j s x s 2 x2 s s s 0 ∂� ∂� ∂ � � �t � � ��−λ �� σ + ∂ds (dz) F s, Ss(1 +∂z) F s, Ss∂ ds 0 λ � � ν �� − �� We know that F t, St and Mt are martingales under Q; therefore the process

�� F t, St Mt is also a martingale, which implies that the third term of the right hand

�� − side in the above equality should be zero.

Therefore,

t F F t, S M = F(0, S ) + s, S 2S2dW t t 0 x s s s s 0 ∂� �� − � � � �σ � Then, ∂

h VT = F T, ST VT = F T, ST MT + MT VT T F � − � ��= �F(�0,−S �) + �� s�, S− S dW − � 0 x s s s s 0 ∂� � � � �σ � � Nt T ∂ T + V + dsh S (dz)z h S dW h S U 0 s s s s s s j j j 0 0 j=1 − �− λ � � � ν − � � σ � − � τ �τ � Nt + F , S F , S j j j j j=1 − � �τ � �τ � T� �τ � − �τ �� ds (dz) F s, Ss(1 + z) F s, Ss 0 − λ � � ν �� N−t �� T F = s, S h S dW + F , S F , S h S U x s s s s s j j j j j j j 0 ∂� j=1 − − � � � � �τ � �τ τ �τ � T� � � − � σ � � �τ � − �τ � − � ∂ (1) (2) ds (dz) F s, Ss(1 + z) F s, Ss zhsSs MT + MT 0 − λ � � ν �� − �� − � � ≜ where,

T F M(1) = s, S h S dW T x s s s s s 0 ∂� �Nt � � � � − � � σ � ∂ M(2) = F , S F , S h S U T j j j j j j j j=1 − − τ τ τ τ � �� τ � �T− � �τ � � − � � ds (dz) F s, Ss(1 + z) F s, Ss zhsSs 0 − λ � � ν �� − �� − � �

(1) (2) According to Lemma 3.2.3 Mt Mt is a martingale under the risk neutral

∙ measure Q, and it follows,

(1) (2) (1) (2) E Mt Mt = M0 M0 = 0 ∗ � ∙ � ∙ Whence,

2 2 T 2 (1) (2) R0 = E (h VT) = E MT + E MT ∗ ∗ ∗ � � where, − � � � �

2 T F E M(1) = E ( [ s, S h ]S ds) T x s s s s ∗ ∗ 0 ∂� 2 T 2 (2) � � � � � � − � σ E MT = E { ds (dz) F∂ s, Ss(1 + z) F s, Ss zhsSs } ∗ ∗ 0 � � λ � � ν �� − �� − � �

The risk at the maturity is then given by,

T F RT = E s, S h S ds 0 x s s s s ∗ 0 ∂� �� −T � � σ � 2 ∂+ E ds (dz) F s, Ss(1 + z) F s, Ss zhsSs ∗ 0 �λ � � ν �� − �� − � � � Using the identities (which are obvious),

s F F r du s, S = (s, S ), F s, S (1 + z) = e 0 u F s, S (1 + z) , x s x s s s ∂� ∂ − ∫ � � � s � � � ru�du � � ∂ ∂ F s, Ss = e 0 F(s, Ss) −2∫ T s r du F �� = E e 0 u (s, S ) h S2 2 x s s s s ∗ 0 − ∫ ∂T �� − � σ 2 + ∂ ds (dz) F s, Ss(1 + z) F(s, Ss) zhsSs ds 0 λ � � ν �T � � − − � � � It follows that the minimal risk R0 is obtained by finding a hs that gives minimal value for the function in brackets [ ] above.

∗

Take the derivative of [ ] with respect to hs and let it equal to zero,

2 2 F ∙ [ ] = 2Ss s (s, Ss) hs 2 z Ss F s, Ss(1 + z) F(s, Ss) zhsSs d (z) hs x ∂ ∂ ∗ − σ 0� − � − λ � � � � − − � ν ∂ ∂ Solving this≜ equation for hs yields, since (ht)0 t T is left-continuous,

≤ ≤ hs = (s, Ss ) − where, Δ 1 F F s, x(1 + z) F(s, x) ( ) 2 ( ) s, x = 2 2 [ s s, x + (dz)z ] s + (dz)z x x ∂ � � − Δ σ λ � ν By using thisσ hλs∫, weν obtain a ∂process which determines an admissible portfolio minimizing the risk at maturity. Note that when there is no jump, i.e. = 0, this

λ recovers hedging of the Black-Scholes Model.

Δ Results in this section have been checked with source [2].

T Remark 3.4.1 In the Black-Scholes framework, the hedging is perfect, i.e. R0 = 0.

However, in the jump-diffusion case, where the market is incomplete, the minimized risk is generally positive.

Reference

[1] Tomas Björk Arbitrage theory in Continuous Time, second edition Oxford

University Press 2004

[2] Damien Lamberton & Bernard Lapeyre Introduction to Stochastic Calculus

Applied to Finance, second edition Chapman & Hall 2007

[3] S. G. Kou A Jump Diffusion Model for Option Pricing with Three Properties:

Leptokurtic Feature, Volatility Smile, and Analytical Tractability Columbia

University 1999

[4] R. C. Merton Option Pricing When Underlying Stock Returns Are Discontinuous

Journal of Finance Economics 1976

[5] David Heath Martingale versus PDEs in Finance: An Equivalence Result with

Examples Univ. of Tech., Sydney

[6] Hans Föllmer, Martin Schweizer Hedging of Contingent Claims under Incomplete

Information 1990