barrier option

7/31/2019 Barrier Option

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Contents

1 Introduction 1

1.1 Problem overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.1 Uncertain Volatility Model . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.2 Stochastic exit time control problem . . . . . . . . . . . . . . . . . . 7

1.2.3 Stochastic control and dominating strategy . . . . . . . . . . . . . . . 8

1.3 Probability Spaces and Filtrations . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4 Brownian Motion and Wiener Process . . . . . . . . . . . . . . . . . . . . . . 10

1.5 Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Single-asset Barrier Option 15

2.1 Pricing and Hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 Practical Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3 Numerical simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3 Multi-asset Barrier Options 23

3.1 Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2 Hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.3 The replicating strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.4 Two-asset barrier option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.5 Optimization problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4 Numerical solutions 35

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.2 Numerical Scheme and algorithm . . . . . . . . . . . . . . . . . . . . . . . . 36

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CONTENTS CONTENTS

4.2.1 Single-asset case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.2.2 Two-asset case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

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Chapter 1

Introduction

1.1 Problem overview

The year of 1973 was a special year for financial mathematics. In this year, the most out-

standing paper on option pricing by Black and Scholes [14] appeared. The model is known

as the Black-Scholes formula. Since then, the model has been very popular among practi-

tioners and researchers, because it is easy to use and open for development. An important

characteristic of this model is the assumption that the volatility of the underlying asset is

constant. This assumption has been argued by researchers or practitioners, especially after

the identification of the so-called volatility smile by Rubinstein and Reiner [64] who claim

that the volatility is stochastic in the real market. The constant volatility can not explain

the observed market price for options. As a result, hedging strategy using a constant value

of volatility can result in a problem.

Realizing this problem, one may try to find another model that employs volatility which

depends on time and stock prices. However, such a model may result in a misspecification

of the volatility. El-Karoui, Jeanblanc-Picque, and Shreve [33] provided conditions under

which the Black-Scholes formula is robust with respect to a misspecification of volatility.

Romagnoli and Vargiolu in [63], Gozzi and Vargiolu [39], proposed a new method for the

study of robustness of the Black-Scholes formulae for several assets. Avellaneda, Levy, and

Paras [2] and Avellaneda and Paras [3], considered the volatility following stochastic process

but lying on an interval band [min, max]. Then bounds on the option prices are obtained

by setting the volatility equal to min and max depending on the convexity or concavity of

the option price functions.

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1.1 Problem overview Introduction

In the case of barrier option or other options where the payoff function are non-convex,

the method proposed by El-Karoui, Jeanblanc-Picque, and Shreve [33], is not applicable.

This is due to the fact that the barrier option price may not increase monotonically with

volatility. Moreover, the value function of the option is neither convex nor concave. To

sell barrier options, one generally trades them above their theoretical Black-Scholes price.

Another method used to hedge and price barrier options is by static hedging. This strategy

does not involve continuous rebalancing as in dynamic hedging. Such static hedging normally

involves setting up a portfolio at the beginning of the contract that is guaranteed to match

the payout of the options to be hedged. This method of hedging is firstly discussed by

Derman, Ergener, and Kani [30]. This paper describes a numerical algorithm for singlebarrier options in the context of a binomial tree representing the evolution of a stock with

time and level dependent volatility. In a related paper, Carr, Ellis and Gupta [19] discuss

the static replication of barrier option under the Black-Scholes model. Similarly, Brown,

Hobson, and Rogers [17] demonstrate how to set up model-free overhedges and underhedges

for certain simple classes of single barrier options using a probability approach.

In this thesis, we analysis the robustness of European multi-asset barrier option. Our

work is motivated by Gozzi and Vargiolu in [39], but we discuss hedging strategy of a multi-

asset barrier option, an option governed by a multidimensional diffusion process. Consider

a riskless asset M whose price is assumed to be constantly 1 for t [0, t) and d risky asset

whose vector price St = (S1, , Sdt ) follows the dynamic

dSt = SttdWt, (1.1)

where Wt is an d-dimensional Brownian motion under risk-neutral measure Q. Our main

assumption in this model is that the volatilities are stochastic, presented in a matrix process

(t)t taking the values in a closed bounded set M(d,n,R). We define here, M(d,n,R)

is a space of d n real matrices and St is a diagonal matrix who elements are (S1t , , S

dt ).

Consider a payoff function for a barrier option h(ST)1{>T}, where is the first moment of

time where the stock price hits a bounded domain O Rd prespecified level of barrier. The

payoff function h in this case is discontinous and the value function is

v(t, x) = EQ[

h(St,x,Tt )1{>(Tt)}|Ft]

which is not a convex function. Since we work on the assumption of stochastic volatility,

then the market could be incomplete. As a result, the agent can not perfectly hedge the

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Introduction 1.1 Problem overview

volatility. Then he uses the superprice to hedge the option. Following Avellaneda, Levy, and

Paras [2] and Avellaneda and Paras [3], Romagnoli and Vargiolu [63], Gozzi and Vargiolu

[39], we fix the price of the option as vt = v(t, St) and set a self-financing portfolio consisting

of quantity t of the risky asset St in the hedging portfolio Xt as

it =v

Sit(t, St), 0 t T.

Here v is the solution of following nonlinear PDE,

tv(t, x) + H(x, D2xv(t, x)) = 0, t [0, T), x R

d

v(T, x) = h(x) x Rd

v(t, x) = 0, x O

(1.2)

where H is given by

H(x, D2

x

v) =1

2supt

Tr(D2

x

v(t, x)(xt)(xt)).

This is known as the Black-Scholes-Barenblatt equation, a version of the Hamilton-Jacobi-

Bellman equation. From the financial point of view, in order to obtain the superprice, the

payoff function h

EQ[

h(St,x,tTt )1{>(Tt)}|Ft]

.

is maximized with respect to . By this strategy, the agent is able to protect himself against

the worst possible case.

The results of this thesis are Theorem ?? in Chapter 3. We show, by a probability

approach, that the value function of the exit control problem v is continuous with respect

to time t and space of price x, and is regular enough to apply the Ito formula. Using this

regularity we show that the pair (v, ) is a superstrategy, see Chapter 4, Theorem 3.3. In

the case of a single-asset barrier option, the exit control problem is a bang-bang solution,

see Theorem 2.1 in Chapter 5. In Chapter 3, Theorem ??, we show that it =

Siv(t, St)

is bounded. Therefore, choosing as a superstrategy is valid and makes sense. We also

demonstrate that the value function of the exit control problem can be approximated by a

sequence of functions (v). We show that v satisfies the dynamic programming principle

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1.2 Literature review Introduction

(DPP), and v v as 0. We also demonstrate that v is also a viscosity solution of the

HJB equation.

This kind of approach to stochastic volatility models was initiated by Avellaneda Levy,

and Paras [2], Avellaneda and Paras [3]. They considered the case of single-asset European

with different maturity time in the portfolio. The recent paper by El-Kouri, Jeanblanc-

Picque, and Shreve [33] discuss a similar problem, single-asset case, but they assume that

the payoff is a convex function. Under this assumption, they succeed in showing that the

BS equation is robust. This is obtained by dominating the stochastic volatility with a

deterministic function of the stock price. Romagnoli and Vargiolu [63] extend this problem

into a multi-asset European derivative. A more detail and deep discussion in this area isgiven by Gozzi and Vargiolu [39].

1.2 Literature review

Our interest is in hedging strategy using super-replication method, known also as a dominat-

ing strategy, in a multi-asset barrier option of which the volatilities are unspecified. We need

to solve the Hamilton-Jacobi-Bellman equation, which is a version stochastic exit control

problem. Therefore, we need to review some topics related to this thesis.

1.2.1 Uncertain Volatility Model

According to Arbitrage Pricing Theory, if the market presents no arbitrage opportunities,

there exists a probability measure on future scenario such that the price of any security

is the expectation of its discounted cash-flows, Duffie [28]. Such a probability is known

as a martingale measure, Harisson and Pliska [40]. It is true that pricing measure is often

difficult to calculate precisely and there may exist more than one measure which is consistent

with a given market 1, Avellaneda, Levy, and Paras [2]. Based on this fact, it is useful to

view incomplete markets as they are reflecting the many choices for derivatives asset prices

that can exist in an uncertainty market. The source of the uncertainty mainly comes from

unpredictable volatility. Avellaneda, Levy, and Paras [2] and Avellaneda and Paras [3],1Uniqueness of the martingale is equivalent to market completeness. A model for security market is said

to be complete if the volatility matrix is full rank in the sense that the number of underlying asset equals to

the number of source of randomness.

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Introduction 1.2 Literature review

assume that the underlying asset St follows a diffusion process with non-constant interest

rate and volatility

dSt = rtStdt + tStdWt. (1.3)

The volatility process (t) fluctuates within an interval

0 < min t max. (1.4)

The volatility process (t) that satisfies (1.4) induces a unique probability measure Q = Q

on the space of prices St. Let denote the set of all measures that can be induced within

the constraint (1.4). Now consider a portfolio X ofd options with expiration dates t1 t2 td1 and payoff functions h1(St1), h2(St2), , hd(Std). Avellaneda, Levy, and Paras [2]

show that the present day worst-case scenario estimate for the buyer side is the value v(t, St)

of the function

v(t, St) = supt

EQ

d

i:tit

etit rsdshi(Sti)

(1.5)

where EQ is the expectation operator with respect to the measure Q and the dynamic

price process (1.3). Then, by Itos theorem, the problem can be converted into a nonlinear

partial differential equation (DPP), which is a version of the Black-Scholes-Barenblatt (BSB)

equation.

Now we consider the multi-asset case. Consider a riskless asset M whose price is assumed

to be constantly 1 for t [0, t) and d risky assets whose vector price St = (S1, , Sdt ) follows

the dynamic

dSt = SttdWt. (1.6)

Here W is an d-dimensional Brownian motion under risk-neutral measure Q. The matrix

process () takes the values in a closed bounded set M(d,n,R), and St is a diagonal

matrix who elements are (S1t , , Sdt ). Such a problem has been considered by Lyons [55],

Romagnoli and Vargiolu [63], Gozzi and Vargiolu [39]. The fair price v(, ) of the multi-asset

contingent claim at time t 0 can be found via the non-arbitrage principle, Duffie [28]:

v(t, x) = supt

EQ[

h(St,x,tTt )]

. (1.7)

Then v must satisfy the associated Black-Scholes-Barenblatt (BSB) equation which is a

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version of the Hamilton-Jacobi-Bellman (HJB):

tv(t, x) +

1

2supt

Tr(D2xv(t, x)(xt)(xt)) = 0, t [0, T), x Rd

v(T, x) = h(x) x Rd

(1.8)

where D2xv(t, x) =

2

xixjv(t, x)

ij

.

The situation becomes more complex if the multi-asset European options and barrier

options are combined in a portfolio. Let be the first moment of time when the stock price

hits the prespecified level of barrier. Then, the price of the barrier options is given by

v(t, x) = sup

EQ[

h(St,x,Tt )1{>(Tt)}]

. (1.9)

The standard approach to hedge this option is to model the behaviour of the underlying

asset and from the model, a fair price then derived. This price is expected to be perfectly

hedging the contingent claim. The approach is to maintain an ever-changing position in

the underlying assets. This method is known as dynamic hedging which needs continuous

rebalancing: maintaining the tracking error, that is the difference between the actual value

and the theoretical value of a self-financing portfolio, to be always positive. In works by

Avellaneda Levy, and Paras [2], Avellaneda and Paras [3], and El-Kouri, Jeanblanc-Picque,

and Shreve [33], the maximum volatility always overestimates the claim. This does not

happen if our claim is a barrier option. The maximum volatility sometimes underestimates

the contingent claim price. This is due to the fact that the option price function can not be

convex or concave for all prices x and all times t. Therefore, the option price is not increasing

monotonically as the volatility increases.

The problem now comes from the fact that the value function might not be smooth

enough to satisfy the boundary condition v = 0 in the classical sense. To solve this problem,

Crandall and Lions [21], Crandall, Ishii, and Lions [22], Lions [52],[53], [54], Fleming and

Soner [34] introduces the notion of a viscosity solution. This notion is not only giving a sense

of a continuous solution for the value function v, but also a sense of a discontinuous viscosity

solution. Equation (1.8) is a version of a stochastic exit control problem which is extensively

studied, see for example Barles and Burdeau [6], Barles and Rouy [7], Barles and Perthame

[8],[9], Barles and Souganidis [10]. For the non-degenerate case of (1.12), one may refer to

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Introduction 1.2 Literature review

Bensoussan [11], Fleming and Rishel [35], Fleming and Soner [34], Krylov [49], where the

classical PDE approaches for stochastic control problem are discussed.

1.2.2 Stochastic exit time control problem

For the exit control problem the main difficulty comes from the treatment of the boundary

condition. As suggested by Lions [52] or Barles and Rouy [7], it is rather simple to create

examples in which the value function is continuous in an open set O and can be extended

continuously to O, but where its extension does not satisfy the boundary condition. Problems

may also occur when the diffusion degenerates along the normal direction to the boundary.

Following Barles and Rouy [7], it is necessary to relax the boundary condition which has to

be read as

min

tv(t, x) + H(x, D2xv(t, x), v(t, x)

0 on O

and

max

tv(t, x) + H(x, D2xv(t, x), v(t, x)

0 on O.

These inequalities have to be understood in the viscosity sense. The viscosity notion has been

well presented in the Users Guide of Crandall, Ishii and Lions [22], including the presentation

of these boundary conditions. This type of boundary condition was first considered by Lions

[52], Barles and Perthame [8],[9], Barles and Souganidis [10]. A more general definition

of viscosity solution, including equations with discontinuous Hamiltonian is discussed in

Fleming and Soner [34], Bardi and Capuzzo-Dolcetta [4]

There are two strategies to solve the this type of problem: the continuous approach

and the discontinuous one. Lions [52],[53], [54], uses the continuous approach to solve this

problem. He approximates the exit control problem by a continuous function. Then he

solves the problem analytically using PDE theory. The discontinuous approach is more

likely a control problem rather than a PDEs one, where one needs to apply the techniques

of variational inequalities, relaxed control and weak convergence. See for examples Barles

and Burdeau [6], Barles and Rouy [7], Barles and Perthame [8], or Flaming and Soner [34].

The continuous approach discussed by Lions [52],[53], [54] suggests that one first prove

that the value function satisfies the Dynamic Programming Principle (DPP), [52]. Then

from DPP, he derives the Hamilton-Jacobi-Bellman equation to which the value function

is a viscosity solution. When the non-degenerate condition is assumed on the boundary,

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one may prove that the value function is the unique continuous solution to the exit control

problem by using a Strong Comparison Theorem ( Barles and Burdeau [6], Barles and Rouy

[7]), or one may refer to Users Guide of Crandall, Ishii and Lions [22]. Once the strong

comparison theorem is established, one can obtain the existence theorem by the Perrons

method known as Ishii Lemma, see Fleming and Soner [34].

The most recent method to solve this problem uses the notion of the Lp-viscosity solution.

One reason for using Lp-viscosity solutions is that H does not need to be continuous in x as

in [22]. It has been shown by Caffarelli, Crandall, Kocan, and Swiech in [25] that when the

equations are degenerate, uniqueness for viscosity solutions fails even for the one dimensional

case, see example 2.4 in [25]. However, it has been noted by Caffarelli et. al [25] that viscositysolutions are Lp-viscosity solutions whenever H is continuous, see Proposition 2.9 in [25].

Note that a comprehensive treatment of the Lp-theory of fully nonlinear parabolic equations

is presented in a recent paper by Crandall, Kocan, and Swiech [24],[25].

1.2.3 Stochastic control and dominating strategy

The application of the stochastic control theory to financial markets is relatively new in

applied mathematics. Intensive research in this area has been initiated by Merton [58].

In his paper, the optimal wealth and the optimal consumption-rate process are formulated

as a stochastic control model with the logarithmic value function. Later, Cvitanic and

Karatzas [20] published a paper dealing with the hedging contingent claim using the theory

of stochastic control. Hedging contingent claims based on the idea of dominating strategy in

the case of incomplete market, was discussed by Cvitanic and Karatzas [26], and El-Karoui

and Quenez [32]. The price of a contingent claim obtained by this method is called the

upper-hedging price or super price. Cvitanic and Karatzas [26] and El Karoui and Quenez

[32] demonstrate that the upper hedging price, which is a minimization problem, can be

transformed to a dual maximization problem. Similar transformation has also been done by

Schmock, Shreve, and Wystup [65], but in a different problem.

The solution of the BSB equation gives rise to an optimal dominating strategy for deriva-

tive securities. This result implies that the BSB equation is the dynamic programmingequation for the following control problems:

v(t, x) = inf tA()

EQ[

h(St,x,tTt )]

(1.10)

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Introduction 1.3 Probability Spaces and Filtrations

and

v+(t, x) = suptA()

EQ[

h(St,x,tTt )]

. (1.11)

The BSB equation corresponding to the function v+ in (1.11) can be written in the form

tv+(t, x) +

1

2sup

Tr(D2xv+(t, x)(xt)(xt)

) = 0, t [0, T), x Rd

v+(T, x) = h(x) x Rd

. (1.12)

1.3 Probability Spaces and Filtrations

In this subsection, we will define Brownian motion and the associated mathematical model

known as the Wiener process. Our discussion closely follows Durrett [29], and the book

written by Krylov [50], for the basic theory of diffusion processes. We also refer to Karatzas

and Shreve [48], and Oksendal [60] for an introduction to stochastic differential equations.

A probability space is a triple (, F,P) where is the space of elementary events or

outcomes, F is a -algebra of subsets of and P : F [0, 1] is a probability measure. A

collection of -algebras F = (Ft)t0, satisfying

Fs Ft F

for all s t is called a filtration.

Definition 1.1. A filtration F = {Ft} is said to satisfy the usual conditions if it is right-

continuous and F0 contains all the P-negligible events in F.

A random variable X is a real-valued function defined on , such that for every Borel

set B B(R), we have X1(B) = { : X() B} F. Let us recall that the Borel

-algebra B(R) is the smallest -algebra containing the open sets ofR. A stochastic process

{Xt; 0 t T} is a family of Rd-valued random variables defined on (, F,P). The

stochastic process {Xt; 0 t T} is called measurable if, for every B B(Rd), the set

{(t, ); Xt() B} belongs to the product -algebra B([0, )) F, in other words, if the

mapping

(t, ) Xt() : ([0, ) , B[0, ]) F) (Rd, B(Rd))

is measurable.

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1.4 Brownian Motion and Wiener Process Introduction

We will denote by FXt = (Xs; s t) the smallest -algebra such that Xs is FXt -

measurable for all s t. We say that a stochastic process (Xt, 0 t T) is adapted to the

filtration (Ft) if Xt is a Ft-measurable random variable for each t 0. A stochastic process

(Xt, 0 t T) is called progressively measurable with respect to the filtration (Ft) if, for

each t 0 and B B(Rd) , the set {(s, ); 0 s t, , Xs() B} belongs to the

product -algebra B([0, t)) Ft, in other words, if the mapping

(s, ) Xs() ; ([0, t] , B[0, t]) Ft) (Rd, B(Rd))

is measurable, for each t 0.

Definition 1.2. A random variable X is square integrable ifEX2

< . A process (Xt, t 0)

is square integrable if supt0 EX2t < . If (Xt, t 0) is considered on a finite time interval

0 t T, then it is square integrable if sup0tT

EX2t < .

1.4 Brownian Motion and Wiener Process

Brownian motion is one of the most important objects in the theory of stochastic processes.

It has been applied in many branches of science and engineering. Recently, it has also been

widely applied in the field of mathematical finance. One may refer to [50] or [48] for a

construction of Brownian motion. There are many ways to define the Brownian motion. We

adopt the point of view that Brownian motion is defined on the space = C([0, ),Rd)

equipped with a Wiener measure P, see Krylov [50] for details. This version is known as

canonicalBrownian motion. The mathematical formulation of Brownian motion is known as

the Wiener process2

. The mathematical definition of the Wiener process is given as follows:

Definition 1.3. An F-adapted process {Wt : 0 t T} defined on a probability space

(, F,P) is called a Wiener process on [0, T] if it has the following four properties:

(i) W0 = 0.

(ii) The increments of Wt are independent; that is for any finite set of times, the random

variable

Wt Ws

is independent of Fs for any s, t such that 0 s t T.

2In honor to N. Wiener who firstly constructed a mathematical model of Brownian motion

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Introduction 1.5 Martingales

(iii) For any 0 s t T the increment Wt Ws has a Gaussian distribution with mean

0 and variance t s.

(iv) For all in a set of probability one, Wt() is a continuous function of t.

1.5 Martingales

The aim of this subsection is to summarize the basic theory of martingales which is related

to our discussion. Our discussion in this section follows closely Steele [69]. We also refer to

Musiela M and M. Rutkowski, [59] for theorems which are relevant to Non-arbitrage pricing.

Definition 1.4. A stochastic process {Xt, t 0} which is adapted to a filtration F is a

martingale with respect to F = (Ft) if

1. E(|Xt|) < for all 0 t T,

2. E(Xt|Fs) = Xs, a.s. for all 0 s t T.

Definition 1.5. A stochastic process {Xt, t 0}, adapted to a filtration F is a supermartin-

gale (submartingale) if it is integrable, and for any t and s, 0 s t T,

E(Xt|Fs) ()Xs a.s.

Remark 1.6. If (Xt) is a supermartingale, then (Xt) is a submartingale.

Remark 1.7. The Wiener process {Wt, 0 t T} is a square integrable martingale, since

EW2t = t

Definition 1.8. A stochastic process {Xt, t 0} is said to be uniformly integrable, if[|Xt|a]

|Xt| 0 uniformly in t, as a

Teorema 1.1. (Doobs martingale)

LetY be an integrable random variable, that isE|Y| < , and define

Mt = E(Y|Ft).

Then Mt is uniformly integrable.

The next result is an obvious consequence of the Burkholder-Davis-Gundy theorem.

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1.5 Martingales Introduction

Teorema 1.2. Burkholder-Davis-Gundy

For every p 0, there exist two constants cp and Cp such that, for all continuous local

martingales M vanishing at zero,

cpE[

M, Mp/2]

E [(M)p] CpE

[M, Mp/2

]where Mt = supst |Ms|.

Corollary 1.9. Let

Mt =

tT0

FsdWs.

Then

cpE

T

0

|Ft|2dt

p/2 E sup

tT|Mt|

p CpE

T

0

|Ft|2dt

p/2.

Definition 1.10. A random variable 0 is a stopping time , with respect to the filtration

F, if the event { t} belongs to the -algebra Ft for every t 0. A stopping time is an

optional time of the filtration if { < t} Ft for every t 0.

Proposition 1.11. Let X = {Xt, 0 t < } be a progressively measurable process, and

let be a stopping time of the filtrationF

. Then the random variable X defined on theset { < } F, is F-measurable, and the stopped process {Xt, Ft, 0 t < } is

progressively measurable.

The proof of Proposition 1.11 is given in Karatzas and Shreve [48] page 9.

Teorema 1.3. Optional Sampling Theorem

Let (Xt) be a martingale and a stopping time. Then the stopped process (Xt) is a

martingale. In particular, for any t, EXt = EX0.

Definition 1.12. An adapted process (Xt) is called a local martingale if there exists a

sequence of stopping times n, such that n as n and for each n, (Xtn) is a

uniformly integrable martingale.

Definition 1.13. Quadratic Variation of Martingales

Let n = maxi

(tni+1 tni ) as n . The Quadratic variation of a process (Xt) is

defined as a limit in probability

Xt = lim

ni=1

(Xtni

Xtni1

)2. (1.13)

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Introduction 1.5 Martingales

If (Xt) is a martingale, then (X2t ) is a submartingale. By compensating X

2t by an

increasing process, it is possible to make it into a martingale. The process which compensates

X2t to form a martingale turns out to be the quadratic variation of process Xt.

Teorema 1.4. If (Xt) is a local martingale, then X, Xt exists. Moreover X2t X, Xt is

a local martingale.

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1.5 Martingales Introduction

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Chapter 2

Single-asset Barrier Option

In this chapter we discuss the single asset case, which is another important result of this

thesis. My result here is original and it might be one of some contributions of this thesis.

2.1 Pricing and Hedging

Let us consider a very common example of barrier option, that is the knock out and up call

of the European type. If 0 t T, St = x and the call has not knocked out prior to time

t, then the price process for this option is given by an adapted process, {vt; 0 t T},

satisfying

vT = (ST K)1{>T}.

Here K is the strike price of the option and is the first moment of time when the process

St hits the barrier H, defined by

= inf{t 0; St H}. (2.1)

Assuming that P is already the risk neutral measure and 0 < K < H , the value of the

knock-out barrier option at time t with initial stock price x is given by

J(t, x; ) = E

[(St,x,Tt K)

+1{>(Tt)}

], 0 t T. (2.2)

For a constant volatility, t = , the explicit solution of (2.2) can be derived by the method

of reflection principle of Brownian motion, Rich [62]. One may refer to Rich [62] for the

closed form solution of (2.2).

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2.1 Pricing and Hedging Single-asset Barrier Option

As is shown in the previous chapter, Theorem 3.1, (2.2) is also the solution of partial

differential equation

tv(t, x) +

1

22x2

2

x2v(t, x) = 0, 0 t < T, 0 x < H (2.3)

with terminal and boundary conditions

v(T, x) = (x K)+, 0 x < H (2.4)

v(t, x) = 0, x H, 0 t T. (2.5)

Now assume that the true volatility is limited to move in a certain interval, i.e.

t I = [min, max].

The assumption that (t) is adapted to F makes it functional of the Brownian paths {Wt, 0

t T}, so that it is dependent on the past of the Brownian motion or stock price. This

volatility can be interpreted as a control to find the worst and the best case price of the

barrier option. Since the seller does not know the true volatility, he will estimate the fair

price of the claim within an interval of prices, which is known as the interval of admissible

prices. Therefore, we expect that the price of the claim lies in the interval

v(t, St,x,t ) vt v

+(t, St,x,t ), (2.6)

where

v(t, St,x,t ) = infI

E[

(St,x,Tt K)+1{>(Tt)}

](2.7)

and

v+(t, St,x,t ) = supI

E

[(St,x,Tt K)

+1{>(Tt)}

]. (2.8)

As already discussed in the previous chapter, in order to have superstrategy, we fix the price

vt of the option and the quantities t of the risky asset St in the hedging portfolio Xx,,vt as

vt = v(t, St,x,t ), t =

xv(t, St,x,t ), 0 t T (2.9)

where v is the solution of the HJB equation

tv(t, x) +

1

2supI

2t x2

2

x2v(t, x) = 0, 0 t < T, 0 x < H (2.10)


v(T, x) = (x K)+, 0 x < H (2.11)

v(t, x) = 0, x H, 0 t T. (2.12)

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Single-asset Barrier Option 2.1 Pricing and Hedging

Therefore, the portfolio process satisfies

d(Xx,,vt ) = tSt,x,t dWt.

Initially, at t = 0, takeXx,,v0 = v(0, S0).

Then

Xx,,vT = v( T, St,x,T )


v(T, x) = (x K)+, if > T (2.13)

v(t, H) = 0, if T. (2.14)

Remark 2.1. In Theorem ??, we have shown that in order to have delta hedging admissible,

we have to impose the condition

E

T0

2t dt < .

Moreover, in the case when

t =

x

v(t, St,x,t ), 0 t T,

then Xx,,vt is a supermartingale.

Proposition 2.2. Suppose that v is a solution of (2.10)-(2.12) for any convex payoff function.

Then v is not convex or concave in x for any t > 0.

Proof. We prove by contradiction. Suppose that v is convex or concave for all t and x. Note

that v is positive for x < H and v approaches zero when x H and x 0 for every fixed

time t. Therefore, it must be concave. However, ift approaches T, v(T, x) = h(x) which is

a convex function. This produces a contradiction.

Teorema 2.1. Letv be a solution of the HJB equation (2.10) with terminal condition (2.11)

and boundary condition (2.12), and define

t(x) =

max if2

x2v(t, x) > 0

min if2

x2v(t, x) < 0.

(2.15)

Then t is an optimal bang-bang control, v is the superprice and is the superstrategy.

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2.2 Practical Issues Single-asset Barrier Option

Proof. Since v is a unique solution of the HJB equation (2.10)-(2.12), then v is the optimal

price; see Chapter 4. Then, clearly t is an optimal control, because HJB is computed with

supremum.

2.2 Practical Issues

The contingent claim h(x) = (x K)+1{>T} is discontinuous at the barrier H. This results

in an unbounded delta hedging at the maturity of the barrier option. The large delta hedging

may cause instability in the hedging strategy (See figure 2.1). The delta hedging becomes

14 15 16 17 18 19 20 21 22 23 24 25 260

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Stock Price (S)

O

ptionPricev(t,S

)

Call and Up Barrier Option

3 weeks to maturity

2 weeks to maturity

1 week to maturity

Figure 2.1: The barrier option price given by (2.10)- (2.12) with K=20, H=23, = 0.20

very negative near the barrier, t =x

v(t, x) as t T.

If our portfolio consists of a non-risky asset invested in a money market and risky assets

in a stock, then in the case where the stock price does not cross the barrier, the seller covers

this short position with funds shares in the stock. If the stock price hits the barrier and the

option is knocked out, the hedging strategy is in the region where t is large and negative.

In this case, the seller covers his short position with the money market.

To avoid the large delta being taken, one can put a constraint on the hedging portfolio

and then use this constraint to bound the super-replication strategy. This approach has been

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Single-asset Barrier Option 2.3 Numerical simulation

suggested by Schmock, Shreve, and Wystup [65]. They impose constraints on the delta and

show that the cheapest super-replicating claim that satisfies this constraint can be found as

the solution of a dual problem of a stochastic control problem. Another method to avoid

instability in the hedging strategy is proposed by Shreve [68], Chap.20, p.218. He imposes

the boundary condition

v(t, x) + H

xv(t, x) = 0, x H, 0 t T,

instead of

v(t, H) = 0 x H, 0 t T

where is a tolerance parameter. This approach guarantees that the Ht remains bounded

and the value of the portfolio is always sufficient to cover a hedging error within Ht of

the short position.

2.3 Numerical simulation

In this subsection, we consider a numerical example which illustrates the previous discussion.

In particular, we generate a Call and Up barrier option of European type with strike price

K =$20 and barrier H =$23. Since the true volatility is not known, we expect the volatility

to be moving within interval [min, max] = [0.10, 0.20] and option expiration at T = 0.25

year. Then we use the HJB equation (2.10)- (2.12) to calculate the superprice. We also

assume that we initially can buy or sell the option at the mid volatility, (max + min)/2.

Here we report in Figure 2.2 the subprice and superprice barrier option computed using

explicit schemes (the algorithm is given in the next chapter).

Figure 2.2 illustrates a comparison between the extreme prices that are obtained by

pricing with a constant volatility, linear PDEs and those obtained from the BSB equation.

Since the extreme prices for options are obtained by using the two extreme volatilities, one

might believe that the extreme price for the portfolios would be given by the Black-Scholes

prices with some constant volatility in the range min max. As shown in Figure

2.2, the theoretical price calculated by the Black-Scholes formula is too low to enter into

a delta-hedging strategy that protects against the worst case situation. The superhedging

strategy obtained from the BSB equation would protect the hedger against the movement

of the volatilities within the band.

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2.3 Numerical simulation Single-asset Barrier Option

14 16 18 20 22 24 260

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Stock Price (S)

OptionPricev(t,S

)


20%

18%

12%

10%

Superprice

Subprice

Figure 2.2: The dotted lines represent the superprice and subprice of the barrier option computed by

(2.10)- (2.12) and the solid lines represent the extreme value of the option computed by the linear equation

(2.3)-(2.5).

The following figure shows superstrategy with 1-3 months to maturity, computed using

equation (2.9). It shows how delta of the portfolio superstrategy varies as the option gets

closer to maturity.

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Single-asset Barrier Option 2.3 Numerical simulation

15 16 17 18 19 20 21 22 231

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

Stock Price (S)

S

uperstrategy,

Delta


1 month to maturity

3 months to maturity

2 months to maturity

Figure 2.3: Delta superstrategy computed from the superprice given in figure 2.2

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2.3 Numerical simulation Single-asset Barrier Option

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Chapter 3

Multi-asset Barrier Options

3.1 Pricing

Based on the position of the barrier, we categorize the multi-asset barrier options into three

different types. The first one is the external barrier option. The value of the option depends

on the value of another asset. If the tradeable stock hits a certain level of barrier then the

value of such an option is zero. The second one is the basket barrier option. The value

of this option depends on whether the underlying assets in the basket hit a certain level of

barrier or not. The third one termed the max/min barrier option is a barrier option where

the value of the option depends on whether the maximum of the underlying assets hits a

certain level of barrier or not. To begin with let us define a price process for the multi-asset

barrier option.

Definition 3.1. A price process for a barrier option is any adapted process {vt; 0 t T}

satisfying

vT = h(St,x,tT )1{>T}, a.s.

where h : Rd+ [0, ) is a given function and is the first moment of time when St hits

the barrier, defined as

= inf{t > 0; St O}. (3.1)

Here St is the solution of (??), O Rd+1

and O is the boundary of O

As an illustration, we firstly, discuss the class of multi-asset barrier options whose prices

depend on an external barrier variable.

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3.1 Pricing Multi-asset Barrier Options

The external barrier determines whether the option is knocked out when the stock price

breaches the prespecified level, or stays alive until the expiry time T. The valuation of

multi-asset barrier options with a single-sided external barrier has been discussed in several

papers, for example in Heynen and Kat [42], who have presented analytic valuation formulae

for European-style barrier options with a single barrier. A similar problem, but with the

external barrier following an exponential function, has also been discussed by Kwok, Wu,

and Yu [51]. They employ the method of images to find the Green function of the governing

differential equation. In this subsection, we follow closely the discussion by Wong and Kwok

[73], but we are not interested in the analytic valuation of the option as discussed in this

working paper.We propose here a payoff function for a multi-asset barrier option with an external barrier

in which the terminal payoff is characterized by

h(ST) = (max(S2T, , S

dT) K)

+1{>T}.

We adopt the usual Black-Scholes assumptions on the capital market and we assume that

the volatilities are fixed. In the risk-neutral assumption, the stock price Sit, i = 1, , d

follow the lognormal diffusion processes. Let ij denote the correlation coefficients between

dWi and dWj which are constant. We define

xi =1

iln

SitSi

and i = i/2, i = 1, 2, , d.

Let H denote the upper barrier. The call option will be knocked out when S1t H at any

time before expiry time T. We define

H =1

1ln

H

S1.

The value of the multi-asset barrier option with barrier level H is given by

v(t, x) = E[

h(Sx,Tt)1{>(Tt)}]

=

H0

Dd1

(x,T, )(max(S2Te2x2 , , SdTe

dxd) K)+dxd, , dx1

0 t T. (3.2)

Here Dd1 is the domain in the (d 1)-dimensional (x1, , xd) - plane in which

max(S2Te2x2, , SdTe

dxd) > K is attained. Let Dd1i denote the domain in which SiTe

ixi

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Multi-asset Barrier Options 3.1 Pricing

is the maximum of the (d 1) -quantities S2e2x2, , Sde

dxd. Then Dd1i is given by

Dd1i =

{(x2, , xd) : xi

1

iln

K

Si

xi

j

i xj 1i ln S

i

Sj , j = 2, , d , j = i}

, i = 2, , d.

Following Wong and Kwok [73], (x,T, ) is the fundamental solution to the d-dimensional

Fokker-Plank equation

t=

1

2

di=1

dj=1

ij

xixj

dj=1

j

xj,

x1 < H, < xj < , j = 2, , d, t > 0. (3.3)

This formulation gives rise to the analytical evaluation of the expectation integral in many

dimensions. This is beyond our discussion. Instead we convert the problem into the partial

differential equation given by the following theorem.

Teorema 3.1. Suppose that v is a solution of the partial differential equation

tv(t, St) +

1

2tr(D2xv(t, St)(St)(St)

) = 0, 0 t T, S1t H, (3.4)


v(T, S1T) = h(S1T), S

1t < H, 0 t T (3.5)

v(t, S1t ) = 0, S1t H, 0 t T. (3.6)

Then v is given by (3.2).

Proof. The proof is standard in terms of its application to the exit control problems. How-

ever, we here is dealing with its application to the pricing of barrier option. So, in view of its

application to the financial problem, the proof is original. To simplify notation, we denote

the operator

L =1

2tr[( )D2xv(t, St)].

By the Markov property, the stochastic process er(T)v( T, ST) is a martingale under

Q. Without loss of generality, take r = 0, by applying Itos formula to (3.2) then integrating

from t to T, we have

v( T, ST) = v(t, St) +

Tt

tv(r, Sr) + Lv(r, Sr)

dr

+

Tt

xv(r, Sr)(Sr)

dWr (3.7)

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3.2 Hedging Multi-asset Barrier Options

Since v(T, ST) and the last term of (3.7) is also a martingale, then the Reimann integralTt

tv(r, Sr)) + Lv(r, Sr)(St)

dWr

dr (3.8)

is also a martingale. Therefore

tv(r, Sr) + Lv(r, Sr) = 0, 0 r T, S

1r < H, (3.9)

which also satisfy the Black-Scholes equation with final and boundary conditions (3.5)-

(3.6).

3.2 Hedging

The barrier option sellers objective is to find a strategy, i.e. an amount , which enables

him to make a good commitment to hedge the contingent claim h(x) at time t = T. He is

expecting that his starting wealth, x will increase such that he can cover his obligation

Xx,T h(Sx,T )1{>T} a.s.

where Xx,T is the solution of the linear stochastic differential equation

dXx,t =n

i=0

itdSit , t T. (3.10)

which is

Xx,t = x +

t0

ni=1

iudSiu, 0 t T. (3.11)

In other words, given a contingent claim h(S

x,

T )1{>T}, we consider the smallest price thatthe seller can accept from the buyer at the initial contract which will enable the seller to

cover his obligation at the final contract T without any risk, in the sense of (3.11). This is

denoted by

v+(t, x) = inf

x 0 | such that Xx,T h(Sx,Tt)1{>(Tt)}; P a.e.

; (3.12)

This price v is the smallest amount such that Xx,T super-replicates h(Sx,T )1{T}. Any such

satisfying the above condition (3.12) is also called a superstrategy or a superprice.

The buyers objective is to find a portfolio strategy so that the payment that he receives

at time T makes it possible for him to cover the debt he incurred at time t = 0. The largest

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Multi-asset Barrier Options 3.3 The replicating strategy

amount x 0 that enables the buyer to achieve this is called the subhedging price for the

contingent claim h(Sx,T )1{>T}, and denoted by

v

(t, x) = sup

x 0 | such that X

x,

T h(S

x,

Tt)1{>(Tt)};P

a.e.

. (3.13)

The price v(t, x) is the largest amount that the buyer can afford to pay at initial contract,

which guarantees that this amount can cover the debt at final contract without any risk.

Any such satisfying the above condition (3.13) is also called a substrategy or a subprice.

In order to ensure that the portfolio is self-financed, we should set Xx,t = v(t, Sx,t ) for

all t [0, T]. Then by Itos formula, we have

dXt = Xt,dWr + t

v(t, Sx,t ) + Lv(t, Sx,t ). (3.14)

We also know, from the definition of the self-financing value of the portfolio , that

dXx,t =d

i=1

itdSit .

This gives

ir =

xiv(r, St,x,), 0 r T, i = 0, , d;

then

Xx,T = v( T, Sx,T) (3.15)

subject to

v(T, S) = h(Sx,T ) if x O

v(t, x) = 0 if x / O.

(3.16)

3.3 The replicating strategy

We assume that the volatilities are stochastic, but restricted to move within an admissible

set A(). In the real situation the agent does not know the true volatilities, instead he uses

another model, that is,

dSit = Sit

nj=1

ijt dWjt , i = 1, , d (3.17)

to hedge the contingent claim, where A() is a certain admissible volatility, Sit = xi is

the initial condition and x = (xi) is a vector in Rd. These volatilities can be interpreted as a

control to find the worst or best case price of the multi-asset barrier option. In this model,

there are two sources of uncertainty, that is Wt and the volatility . Since the agent does

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3.3 The replicating strategy Multi-asset Barrier Options

not know these two objects, he will estimate the fair price of the claim within the interval

price, which is known as the interval of admissible prices. The arbitrage free price of the

barrier option is given by

vt = EQ[


. (3.18)

Since we do not know yet whether our contingent claim is attainable or not. Therefore, we

expect that the arbitrage free price of the claim lies in the interval

v(t, x) vt v+(t, x), 0 t T, (3.19)

where

v+(t, x) = supA()

E[

h(St,x,Tt )1{>(Tt)}

](3.20)

and

v(t, x) = inf A()

E[


. (3.21)

Now we adopt the definition of a replicating strategy for unspecified volatilities given in

Touzi [70], Karatzas [47], or Frey [37].

Definition 3.2. A super-replicating price for a contingent claim h at time t is given by

v(t, x) = inf {x 0 | A() admissible, such that

Xt,x,T h(St,x,Tt )1{>(Tt)};P a.e.,

. (3.22)

This is the minimum price that the agent can accept in order to super-replicate the claim.

If the set is empty, then is zero. Any such process , which may depend on , is called a

super-replicating strategy or superstrategy.

Definition 3.3. A sub-replicating price for the contingent claim h at time t is given by

v(t, x) = sup {x 0 | A() admissible, such that

Xt,x,T h(St,x,Tt )1{>(Tt)};P a.e.,

. (3.23)

Any such process , which may depend on , is called a sub-replicating strategy or sub-

strategy.

Remark 3.4. Another version of super-replicating and sub-replicating strategy is also given

by El-Kouri et al. [33] or Romagnoli and Vargiolu [63]. If v = v = vt, then vt is the arbitrage

free price of the contingent claim h.

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Multi-asset Barrier Options 3.3 The replicating strategy

Teorema 3.2. The process

Xt,x,t Xt = supA()

E

[h(St,x,Tt )1{>(Tt)}

](3.24)

is a supermartingale

Proof. The portfolio is self-financing, and Xt is bounded from below, hence by Theorem 3.5

in Krylov [49], p.149, Xt is a supermartingale.

As we noticed in Definition 3.2 the super/substrategy depends on the choice of volatility

process (t). This choice can create arbitrage opportunities. Therefore, v+t (respectively v

t )

may be considered as a stochastic control problem where the lower bound and the upper

bound of its solution can be interpreted as a sub arbitrage price and super arbitrage

price, respectively. Before we convert our problem into a stochastic control problem in the

HJB equation, the following theorem gives an idea that with a superstrategy, one can have

Xt,x,T h(St,x,Tt )1{>(Tt)}. This means the portfolio overhedges the contingent claim.

Teorema 3.3. Letv be a price process for a contingent claim and let be a portfolio process.

If v is the super-hedging price as defined by Definition 3.2, then there exists a pair (v, )

such that

v(t, x) = v(t, x) = supA()

E[


.

In particular,

v(T, x) h(St,x,Tt ) a.s. (3.25)

Proof. The following proof is original. This might be a significant contribution of the thesis

in hedging barrier option. Take a superstrategy associated with an upper hedging price

as defined in Definition 3.2. Then

Xx,,vT E(St,x,Tt )1{>(Tt)}

for every admissible control , but by Proposition 4.4.2 in Krylov [49], Xx,T is a super-

martingale. This implies that

v(t, x) = Xx,t E

Xt,x,T |Ft

E

[h(St,x,Tt )1{>(Tt)}

] admissible . (3.26)

Hence, v(t, x) v(t, x).

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3.3 The replicating strategy Multi-asset Barrier Options

To prove that v(t, x) v(t, x), we apply Itos formula to the process Sx,T, giving

v( T, Sx,T) = v(t, x) +

Tt

tv(r, St,x,r ) + Lv(t,r,S

x,r )

dr

+

T

t

x x

v(r, St,x,r )dWr. (3.27)

Taking expectation of both sides, we have

Eh(St,x,T )1{>T} = v(t, x) + E

Tt

rv(r, Sx,r ) + Lv(r, S

x,r )

dr. (3.28)

Now we take the supremum of both sides, giving

v(t, x) v(t, x) + supA()

E

T

t

rv(r, Sr) + Lv(r, S

x,r )

dr. (3.29)

Since the expectation in (3.29) is zero, we have

v(t, x) v(t, x). (3.30)

Therefore,

v(t, x) = v(t, x) = supA()

E[

(St,x,Tt )1{>(Tt)}]

.

Before we discuss the more specific example of the multi-asset barrier option, let us write

the HJB equation (??)-(??) in sense of multi-asset barrier option version

Teorema 3.4.

tv(t, St) +

1

2sup

Tr(D2xv(t, Sx,t )(x)(x)

) = 0, t [0, T), x Rd

v(T, x) = h(x) x Rd

v(t, x) = 0, x O

(3.31)

This equation is also known as the Black-Scholes-Barenblatt (BSB) equation. To give a

better idea about this problem, in the following section we discuss two-asset barrier option.

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Multi-asset Barrier Options 3.4 Two-asset barrier option

3.4 Two-asset barrier option

In standard European options, there are several types of payoff functions of two-asset barrier

options. For examples:

1. Max/min of two-asset barrier option:

h(T, S1T, S2T) = (max(S

1T, S

2T) K)

+1{>T},

where

= {t 0, max(S1t , S2t ) H}.

2. Basket barrier option :

h(T, S1T, S2T) = (aS

1T + (1 a)S

2T K)

+1{>T},

where a is a portion of asset S1 and

= {t 0, max(S1t , S2t ) H}.

3. External barrier option :

h(T, S1T, S2T) = (S1T K)+1{>T},

where

= {t 0, S2t H}.

We choose the max/min of two-asset barrier option as an example. Let the prices of the

stocks at time t be S1t and S2t . The risk neutral price processes for the two assets S

1t and S

2t

follow the stochastic differential equations

dS1t = 1t S

1t dW

1t (3.32)

dS2t = 2t S

2t dW

2t (3.33)

S10 = x1, S20 = x2. (3.34)

Let denote the correlation coefficient of the Brownian motion dW1t and dW2t . Assume that

the interest rate is zero. We can write the dynamics of the two assets in a more compact

vectorial notation:

d

S

1t

S2t

=

S

1t 0

0 S2t

(

1t )

2 1t 2t

1t 2t (

2t )

2

d

W

1t

W2t

(3.35)

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Multi-asset Barrier Options 3.5 Optimization problem

Hence we have

At,x = x D2xv x =

x212v

x21x1x2

2v

x1x2

x1x22v

x1x2x22

2v

x22

.

Now we assume that the true volatilities are not known and limited to move in a certain

interval. We write the set of admissible volatilities as follows:

=

R

22

=

1t 1t

2t 2t

, 1 1t +1 , 2 2t +2 , +

.

Then we have

=

(1t )2 1t

2t

1t 2t (

2t )

2

.

Let At,x = A be

A =

a b

b c

.

Then the corresponding Hamilton-Jacobi-Bellman equation can be written as:

v

t+

1

2max

A()tr(A) = 0, (3.43)

with terminal and boundary condition

v(T, x1, x2) = (min(x1, x2) K)+, 0 max(x1, x2) < H (3.44)

v(t, S1T, S2T) = 0, , max(x1, x2) H, 0 t T. (3.45)

3.5 Optimization problem

In this subsection we discuss the optimization problem appearing in the HJB equation (3.43).

First, we write the function to be maximized as follows:

f(1, 2, ) = tr(A) = a21 + 2b12 + c

22, (3.46)

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3.5 Optimization problem Multi-asset Barrier Options

where 1 1t

+1 ,

2

2t

+2 ,

+. The standard form is given by

QP1

minimize f(1, 2, )

subject to 1 1t +1 ,

2 2t

+2 ,

+.

(3.47)

We present the variables as a vector x and its lower bound and upper bound as l and u,

respectively:

l =

1

2

and u =

+

1

+2

+

. (3.48)

The problem QP1 is a special form of bound constraint of the optimization problem. This

form leads to considerable simplification of the optimality conditions. The Lagrange mul-

tiplier for an active bound on xi, = 1, , 3 is given by gi(x) if xi = li, and by gi(x

) if

xi = ui ( since the constraint xi ui can be written as xi ui). Then the sufficient

condition for x to be a local minimizer off subject to bound (3.48) is given by the following

theorem.

Teorema 3.5. First order sufficient conditions for optimality(Gill, Murray, and

Wright [38])

If x is feasible for QP1 and there exists x such that x satisfies the following conditions

1. l x u

2. x = l if f(x) > 0

3. gL > 0 and gU < 0

then x is a local minimizer of f.

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Chapter 4

Numerical solutions

The following algorithm for solving the HJB equation is original. Our result here is a

significant contribution of the thesis.

4.1 Introduction

In Chapter 4, we have shown that the stochastic exit control problem for pricing barrier

options with unspecified volatilities can be converted into the nonlinear PDE (HJB equation).

We have also shown that the nonlinear PDE of the stochastic exit control problems converges

to the viscosity solutions (see Crandall et al. [22]). Barles [5] gives a detailed discussion about

stability, consistence, and convergence to the viscosity solution.

One may refer to [75] for the implicit method or the Crank-Nicolson method, used to

solve linear PDEs for barrier options. Boyle [16], also developed an explicit finite difference

method for pricing barrier options based on a lattice or trinomial approach. The linear PDE

for pricing barrier options has been discussed in Zvan et al. [75], [76]. For the nonlinear PDE,

one may refer to Pooley, Forsyth and Vetzal [61]. The authors use the implicit discretization

method to find the best and the worst price of the uncertain volatility. They also demonstrate

the numerical convergence properties of the nonlinear PDE in the case of a single asset.

In this chapter, we focus on computational and implementation issues of the finite dif-

ference methods for solving the HJB equation. We not only discuss the single asset case but

also the two-asset-case. We also present here the algorithm of how to solve the HJB equation

in two-asset case. In the algorithm, we demonstrate how we can make use MATLAB func-

tion, FMINCON, to find the optimal price. In this discussion we are not interested in the

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4.2 Numerical Scheme and algorithm Numerical solutions

efficiency or the accuracy of the approximation. Such a problem is beyond our discussion.

One may refer to Pooley, Forsyth and Vetzal [61] or [75] for the efficiency or the accuracy

for the implicit method or the Crank-Nicolson method

4.2 Numerical Scheme and algorithm

4.2.1 Single-asset case

In the case of a single-asset, we use explicit schemes to approximate the HJB equation.

The reason for choosing these schemes is that we want to demonstrate the condition under

which the approximation converges to the closed form solution. Other schemes, such as the

implicit and Crank-Nicolson schemes, are unconditionally stable. Now we rewrite again the

HJB equation:

rv(r, Sr) + sup

minmax

1

22S2r

2

S2v(r, Sr) = 0, r < T, Smin Sr Smax (4.1)

with terminal condition

v(T, ST) = (ST K)+ Sr < H, 0 r T (4.2)

and boundary condition

v(r, Sr) = 0 Sr H, 0 r T. (4.3)

To discuss the discretization of (4.1), we firstly introduce the new variables x and t defined

by

x = log(S/K), t = T /1

22, S = Kex+u(, x).

Then (4.1) can be written as

u

=

2u

x2, xmin < x < xmax, 0 < H. (4.7)

Here

u(, x) = 1Ke1

2 (k1)x+1

4 (k+1)2

v(, S), 0 < < 122T

and k = r/0.52, xmin = log(Smin/K), xmax = log(Smax/K). The choice of volatility

depends on the sign of the2v

x2. To discuss the discretization, we divide the x axis into

equally spaced nodes with interval length x and the t time axis into equally spaced nodes

with interval t. Now we choose positive integers N and M as the number of intervals in

the x axis and = t axis, respectively. We define grid points along the space and time axes,

xi = xmin + ix for i = 1, N,

tj = jt for j = 1, , M,

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where x = (xmax xmin)/N, t =12

2T /M. We denote the midpoint of the j-th time

interval by

tj 12

=1

2(tj1 + tj) = (j

1

2)t

and we write

U1

2

i =1

2(Uji + U

j1i ) = u(xi, tj 1

2

).

Now we approximateu

tby

u

t

u(tj, xi) u(tj t, xi)

t=

Uji Uj1i

t(4.8)

and approximate2u

x2

by

2u

x2

u(tj 12

, xi + x) 2u(tj 12

, xi) + u(t1 12

, xi x)

t

U

j 12

i+1 2Uj 1

2

i + Uj 1

2

i1

(x)2.

We can write (4.4) in general form as follows

vji vj1i

t=

vji+1 2vji + v

ji1

(x)2, (4.9)

where [0, 1] is a temporal weighting factor determining the type of scheme being used:

fully implicit when = 1, Crank-Nicolson when = 12

and fully explicit when = 0. Now

consider the case when = 0, that is explicit scheme.

Because the scheme is explicit, it is stable under some condition. The stability depends

on the step lengths t and x. From (4.9), we have

vji =

1

t

(x)2

vji +

1

2

t

(x)2

(

vj+1i+1 + vj+1i1

). (4.10)

In order to prevent any oscillation, the following condition must be satisfied

(1 t

(x)2) 0 (4.11)

The PDE is solved using an explicit scheme which is conditionally convergent. Based on

this scheme, we propose the following algorithm for a numerical solution of (4.1) and (4.2):

(j+1)2 =

2

max if

j+1

0

2min if j+1 0

(4.12)

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Numerical solutions 4.2 Numerical Scheme and algorithm

where

=

vj+1i+1 2v

j+1i + v

j+1i1

(x)2

for i = 1, , M and j = 1, , N.

Algorithm :

Initialize {vj0}N1j=0 using the initial and boundary conditions (4.5).

For i = 0 to M 1 do

v0i = 0

For j = 1 to N 2 do

gradj+1i = vj+2i1 2v

j+1i1 + v

ji1

Ifgradj+1i 0

= min

= 0.52t/x2

Else

= max

= 0.52t/x2

endif

vj+1i = (1 2)vj+1i1 + v

j+2i1 + v

ji1

For k = h to N 1 do

vki = 0

end do j

end do i

Here h is the point at which the scheme hits the barrier.

In the algorithm above, the barrier is assumed to be observed continuously. In other

words, the barrier is checked at every time during the life of the option. In practice, the

barrier is often observed at discrete-time intervals such as daily or weekly. The algorithm

above can easily be adjusted to compute prices and to hedge the discrete-time barrier option.

Moreover, it also can be used to handle double discrete barrier options.

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4.2.2 Two-asset case

Consider the HJB equation:

v

s + maxA()1

2 21S21

2v

S21+

1

222S22

2v

S22+ 12S1S2

2v

S1S2

= 0,

Smin S1, S2 Smax, (4.13)

with initial conditions

v(s, Smin, S2) = 0 (4.14)

v(s, S1, Smin) = 0; (4.15)

terminal condition

v(T, S1, S2) = (max(ST1 , S

T2 ) K)

+, (4.16)

and the boundary condition

v(s, St1, St2) = 0., max(S

t1, S

t2) H. (4.17)

Here () is a two-dimensional matrix such that A(). To simplify computation we

convert the HJB equation (4.13) into a dimensionless form. We consider the following changeof variables

S1 = Kex,

S2 = Key,

s = T t

12

12,

v = Ku(x,y,s).

Then the first derivative of v with respect to time t and price S is given by

v

s=

1

212K

u

t(4.18)

2v

S21=

S1

K

S1

u

S1

=

K

S21

u

x+

K

S21

2u

x2(4.19)

2v

S22=

S2

K

S2

u

S2

=

K

S22

u

y+

K

S22

2u

y2(4.20)

2v

S1S2=

S1K

S2

u

S2 = K

S1S2

2u

xy(4.21)

By substituting these into (4.13), then the problem can be formulated as follows;

v(s, S1, S2) = Ku(t,x,y) (4.22)

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where u is the solution of following dynamic programming equation

u

t= max

A()

12

2u

x2+

21

2u

y2+ 2

2u

xy

(4.23)

with initial conditions

u(t, xmin, y) = 0 (4.24)

u(t,x,ymin) = 0, (4.25)

terminal condition

u(T, x , y) = (max(ex, ey) 1)+, (4.26)

and the boundary condition

u(t,x,y) = 0, max(St1, St2) H. (4.27)

Here xmax = ymax = log(Smax/K) andxmin = ymin = log(Smin/K).

In finite difference mesh, we divide the x and y axes into J equally spaced nodes with

interval x and y, and the t axis into N equally spaced nodes with interval t. We define

grid points along the space and time axis as follows;

t =1

212T/J,

tj = jt for 0 j J,

xm = mx for N m N+,

yn = nx for N n N+,

where x = (xmax xmin)/N = (ymax ymin)/N, and N and N+ are positive integer with

N+ N = N. Now we introduce grid function

Ujm,n u(xm, yn, tj)

u0m,n u0(xm, yn)

tj = (tj1 + tj) = (j )t

Now we approximateu

tby

u

t

u(xm, yn, tj) u(xm, ym, tj t)

t

Ujmn Uj1mn

t, (4.28)

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and approximate2u

x2,2u

y2and

2u

xyby second central difference, we have

2u

x2

u(xm + x, yn, tj) 2u(xm, yn, tj) + u(xm x, yn, t1)

(x)2

Uim+1,n 2U

jm,n + U

jm1,n

(x)2.

2u

y2

u(xm, yn + y, tj) 2u(xm, yn, tj) + u(xm, yn y, t1)

(y)2

Uim,n+1 2U

jm,n + U

jm,n1

(y)2.

2u

xy

1

2x

u(xm + x, yn + y, tj) u(xm x, yn + y, tj)

2x

u(xm + x, yn y, tj) u(xm x, yn y, tj)2x

Uim+1,n+1 Ujm1,n+1U

jm+1,n1 + U

jm1,n1

(2x)2.

Substituting these into (4.23), we obtain

Ujmn Uj1mn

t= max

A()

12

Uim+1,n 2Ujm,n + U

jm1,n

(x)221

Uim+1,n 2Ujm,n + U

jm1,n

(x)2

2U

i

m+1,n+1 U

j

m1,n+1U

j

m+1,n1 + U

j

m1,n1(2x)2

for N + 1 m, n N+ 1 a n d 1 j J,

with U0m,n = u0m,n for N

m, n N+.

We will use the implicit method solve the problem, so we choose the temporal weighting

= 1. The algorithm is given as follows

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Algorithm :

input N, J, T, r

Calculate k = 0.5 1 2 T/J, h = (xmax xmin)/N,a = 2r/(12), b = 1/2

Initialize v0(1 : N + 1, 1 : N + 1) using the initial value v0 = (max(ex, ey) 1)+

Search for the barrier

If found H = k then

v0(k : N + 1, :) = 0

v0(:, k : N + 1) = 0

EndIf

x0 = [x10, x

20, x

30] (starting point)

lb = [1 , 2 ,

] (lower bound)

ub = [+1 , +2 ,

+] (upper bound)

For i = 2 to N do

For j = 2 to N do

a1 = vi+1,j 2vi,j + vi1,j

a2 = vi+1,j+1 vi,j+1 ui+1,j1 + vi1,j1

a2 = vi,j+1 2vi,j + vi,j1

f(sg1, sg2, rh) = a1sg21 + a3sg

22 + 2a2sg1sg2rh

CALL MATLAB function fmincon(f,x0,[],[],[],[],lb,ub)

1 = sg1, 2 = sg2, = rh

aa = 2 r/(1 2), bb = 1/2

cc = k b/(2 h2), dd = k/(2 b h2), ee = k/(4 h2), ss = 1 dd cc

ui,j = ss vi,j + cc(vi+1,j + vi1,j) + dd(vi,j+1 + vi,j1) + ee(vi+1,j+1

vi,j+1 vi+1,j1 + vi1,j1)

end do j

end do i

Solve the system equation of size (N 1)2 (N 1)2

43


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44


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