Pricing Some Path-dependent Discretely Sampled Options Using a Partial Differential Equation Approach
Daniel Eskilsson
U.U.D.M. Project Report 2003:11
Examensarbete i matematik, 20 poäng Handledare och examinator: Johan Tysk
September 2003
Department of Mathematics
Uppsala University
Abstract
Using a partial differential equation approach we in this thesis develop a method for the pricing of a certain class of path-dependent discretely sampled exotic options. By introducing auxiliary state variables the problems with the path-dependency are overcome. The resulting partial differential equation to be solved is the Black-Scholes equation with the path- dependent quantities treated as parameters and with jump conditions across sampling dates.
This can be implemented into a numerical scheme. We also investigate the distribution of the
path-dependent quantity for a particular option from the class of options dealt with.
“The greatest of all gifts is the power to estimate things at their true worth.”
− La Rochefoucauld, Reflexions; ou sentences
et maximes morales.
“Nowadays people know the price of everything and the value of nothing.”
− Lord Henry Wotton,
in Oscar Wilde’s,
The Picture of Dorian Gray.
Acknowledgements
I would like to thank Johan Tysk for his work as my advisor. I am also grateful to Jonas
Persson for providing important comments on the numerical computations. I am greatly
indebted to Lars Luthman for helping me turning the mathematics into Matlab-code. Special
thanks to Daniel Pérez for encouraging discussions and for making the work during the
summer more fun. Finally I would never have succeeded in finishing this thesis without the
support and patience of Anna Ahlin.
Table of Contents
1 INTRODUCTION ... 7
1.1 D ISPOSITION ... 8
2 ARBITRAGE THEORY IN CONTINUOUS TIME... 9
2.1 B ACKGROUND ... 9
2.2 A SSUMPTIONS AND A SSET D YNAMICS ... 9
2.3 T RADING S TRATEGIES ... 13
2.4 C OMPLETENESS AND H EDGING ... 15
3 GENERALISED MODEL FOR PRICING OF THE CONTINGENT CLAIMS... 18
3.1 S ETUP ... 18
3.2 S AMPLING S TRUCTURE ... 19
3.2.1 Jump Condition ... 20
3.3 D ERIVATION OF THE P ARTIAL D IFFERENTIAL E QUATION ... 21
3.4 P ROPOSITION 3.1 (P RICING EQUATION 1)... 26
3.5 S IMILARITY R EDUCTION ... 27
3.6 P ROPOSITION 3.2 (P RICING EQUATION 2)... 29
4 NUMERICAL ANALYSIS... 30
4.1 O UTLINE ... 30
4.2 T HE G REEKS ... 31
4.3 D IFFERENTIATION U SING A G RID ... 32
4.4 A PPROXIMATING THE D ERIVATIVES ... 33
4.5 F INAL C ONDITIONS AND P AYOFFS ... 34
4.6 B OUNDARY C ONDITIONS ... 34
4.7 J UMP C ONDITIONS ... 35
4.8 T HE E XPLICIT F INITE -D IFFERENCE M ETHOD ... 36
5 ANALYSIS OF THE DISTRIBUTION OF A PATH-DEPENDENT QUANTITY ... 38
5.1 A P RACTICAL E XAMPLE ... 38
5.2 M ONTE C ARLO S IMULATION ... 39
5.4 E STIMATION OF THE P ROBABILITY D ENSITY F UNCTION ... 44
5.4.1 Estimated Probability Density Functions from Monte Carlo Simulation 1... 45
5.4.2 Estimated Probability Density Functions from Monte Carlo Simulation 2... 51
6 COMPARISON OF PRICES ... 52
7 CONCLUDING REMARKS ... 56
8 APPENDIX A ... 58
8.1 H ISTOGRAMS F ROM S IMULATION 1 ... 58
9 APPENDIX B... 63
9.1 M ATLAB C ODE ... 63
10 REFERENCES ... 64
1 Introduction
This thesis is written to fulfil the requirements of the Master of Science degree in Mathematics at Uppsala University.
When I for the first time came across Relax Sverige 1 * a real world example from the class of European path-dependent discretely sampled exotic options, which this thesis mainly will focus on I became very puzzled, because I found the payoff structure very complex to grasp with my “financial common sense”. My first thought was that this is very confusing and I started to ponder about how one should price such a contract. According to Taleb [TAL] bad investment banks use to trick customers by taking advantage of the customers’ statistical misperception and he also claims that:
“It is easier to fool someone with a distributional confusion. […] Until customers gained in sophistication, covered writes were the best game in town.
As customers started understanding a little more statistics, the game moved to the fancier exotic options payoffs. The distributional confusion moved to the notion of path dependency.” [TAL, p. 351]
Therefore it is important to investigate how the valuation of such a contract can be done. If one really understands the underlying logic and the tools used, one has a powerful knowledge applicable to many different situations, even outside the world of finance.
A path-dependent option has a payoff at maturity that depends on the history of the
underlying asset price as well as its price at maturity [WDH, p. 148]. In this thesis I will
develop a mathematical model for pricing of some discretely sampled path-dependent
European exotic options. In order to accomplish this I will apply a partial differential equation
approach, based on the original Black and Scholes analysis [BSC]. The way I deal with the
problem contains no approximations other than inherent in the final numerical solution. I will
also investigate the probability density function of the path-dependent quantity in the case of
Relax Sverige 1, and also compare the “fair” price I derive for this particular contract with the
issue price and the price on the market. The path-dependent quantity refers to the extra state variable carrying past information, which is necessary in order to keep track of the potential payoff at expiry.
1.1 Disposition
Chapter two presents in a brief manner arbitrage pricing theory and its mathematical background. With this as a foundation, we develop in Chapter three a generalised mathematical model for pricing of some European path-dependent discretely exotic option in the standard Black-Scholes setting. In Chapter four we discuss the numerical solution of the problem using a finite difference approach. Since this is a thesis in mathematics we will use an intuitively understandable numerical method and not the most advanced and efficient one.
Establishing the fact that the problem is solvable and that many techniques are available will
satisfy us. We will, indeed, find a value numerically, but we will use a basic method with a lot
more to wish, but it is enough for our purposes and has hopefully the advantage of
transparency. Chapter five contains an analysis of the probability distribution function of the
path-dependent quantity in the case of Relax Sverige 1. In Chapter six we compare our
derived price with the actual price of the contract Relax Sverige 1. Finally we sum up in
Chapter seven with some concluding remarks.
2 Arbitrage Theory in Continuous Time
2.1 Background
There are two main approaches to the valuation of derivative instruments. One of them is the
“martingale method”, initiated by Cox and Ross (1976) and Harrison and Kreps (1979). This idea is based on writing the value of the claim as the expected value under a risk-adjusted measure of the discounted payoff. Using probabilistic methods the expectation can be calculated. The other approach is the “partial differential equation method”, due to Black and Scholes (1973) and Merton (1973). This technique consists of constructing for the price of the derivative instrument a partial differential equation together with appropriate boundary conditions. The solution to the partial differential equation can then be found using analytical or numerical methods [DKR].
Dewynne and Wilmott [DEW] advocate approaching the pricing of exotic options from a partial differential equation point of view. They claim this technique is more sophisticated and they encourage the mathematical modelling aspects and the numerical solving of such problems. In valuation of options they prefer this method to the search for explicit solutions because that inevitably leads to compromises in the underlying modelling [DEW]. Moreover, when the problem has been formulated in the form of a partial differential equation form we may consider ourselves to be on well-known territory, because we then have more than a century’s worth of theory which to rely on [WDH, p. iii]. Inspired by these ideas, we have chosen to utilise a partial differential approach when we try to price Relax Sverige 1 and other similar contracts. Before we start with the pricing we introduce some relevant theory in the next section.
2.2 Assumptions and Asset Dynamics
All the results henceforth will be derived in the following frameworkunless otherwise
statedwhich is an arbitrage-free market model with continuous trading. The foundations of
of the important fundamental building blocks, which mainly is based upon [MUR] and [BJÖ], compare also [∅KS, pp. 247].
As in Black and Scholes seminal paper (1973), we, throughout this, thesis make the following assumptions regarding the conditions in the market for the underlying asset as well as for the derivative instrument:
• It is possible to borrow and lend money at the short-term interest rate, which is known and constant through time;
• The underlying asset pays no dividends;
• The derivative instrument can only be exercised at maturity;
• There are no costs associated with buying or selling the derivative instrument or the underlying asset;
• All securities are perfectly liquid and divisible. However, in contrast to Black and Scholes we do not assume that the market for the derivative instrument is liquid and exist a priori;
• There are no restrictions on short selling or borrowing.
Before we go on we need to know what a standard Brownian motion is.
Definition 2.1 A stochastic process W defined on a probability space ( , is called standard Brownian motion if the following conditions hold.
)
P F, Ω ,
1. W(0) = 0.
2. The process W has independent increments, i.e. if r < s < t < u then W(u) − W(t) and W(s) − W(r) are independent stochastic variables.
3. For s < t the stochastic variable W(t) − W(s) has the Gaussian distribution with mean 0 and variance t − s.
4. W has continuous trajectories. [BJÖ, p. 27]
In this model the a priori existing market consists of two assets. The first asset in this model
we refer to as the underlying risky asset, S t , which is assumed to follow a geometric Brownian
motion [BSC]. This means that the dynamics of the asset price process is the following linear stochastic differential equation (SDE) of the form
t t t
t S S dW
dS = α + σ . (2.1)
Equation (2.1) should be interpreted as the shorthand version of the following integral equation, where S 0 ∈ R + is the initial asset price,
∫
∫ +
+
= t u t u u
t S S du S dW
S
0 0
0 α σ , ∀ t ∈ [ ] 0 , T .
The drift and the volatility of S are denoted by α ∈ R and σ > 0 , respectively. For t ∈ [ ] 0 , T , W t is defined to be a standard Brownian motion defined on a probability space (Ω, F, P). Here P denotes the objective probability measure and F = { } F t t ≥ 0 denotes the filtration, where F t
can be said to encode all the information generated on the time interval [ . If a stochastic process Y is such that we have Y
]
t , 0 F t
t ) =
( for all then we claim that Y is adapted to the filtration F. The first integral above is an ordinary Riemann integral; the second one is understood in the Itô sense. To fully understand (2.1) we need Itô’s formula.
≥ 0 t
Theorem 2.1 (Itô’s formula) Let the process X t have a stochastic differential given by
t
t dt dW
dX = α + σ .
Let g ( ) t , x ∈ C 2 ( [ 0 , ∞ ) × R ) . Then Y t = g , ( t X t ) has a stochastic differential given by
( ) ( ) 2 2 ( , ) ( 2
2 , 1
, t t t t t
t t X dX )
x dX g
X x t dt g X t t
dY g ⋅
∂ + ∂
∂ + ∂
∂
= ∂ ,
where ( ) ( )( ) dX t 2 = dX t dX t is computed according to the rules
.
,
0 dW dW dt
dt dW dW dt dt
dt ⋅ = ⋅ = ⋅ = ⋅ =
Proof. A sketch of the proof can be found in [ ∅ KS].
The process X t is called an Itô process. This class of processes are continuous semimartingales and
u ,
t t
t X du dW
X = + ∫ + ∫
0 '
0
0 α β ∀ t ∈ [ ] 0 , T ,
gives what is called its canonical decomposition, which we will need later. As Björk puts it, every stochastic integral is a martingale, modulo an integrability condition [BJÖ, p. 24].
Starting from S 0 at time 0, the solution to (2.1) is easily found using Itô’s formula to ln ( ) S t ,
( )
( ) ( ) ( )
2 . 1
2 1 1
2 1 ln 1
2
2 2 2
2
dt dW
dt
dt S S
dW S dt S S
S dS S dS
S d
t
t t t t t
t t
t t t t
σ σ
α
σ σ
α
− +
=
− +
=
−
=
Integration from 0 to t of both sides yields
( ) S t ( ) S W t t , t [ ] 0 , T 2
ln 1
ln 0 2 ∀ ∈
− + +
= σ α σ ,
which leads to
[ T t t W
S
S t t , 0 ,
2
exp 1 2
0 ∀ ∈
− +
= σ µ σ ] . (2.2)
The second asset the a priori market consists of is risk free and B t denotes its price process.
The dynamics is given by
dt rB
dB t = t . (2.3)
Let us emphasise that the fact that it has no dW-term is a defining property of a risk free asset.
By convention B 0 = 1, thus, B t = e rt , for t ∈ [ ] 0 , T , is a solution of (2.3). In this particular case we can interpret B t as the price of a bond [BJÖ, p. 76].
2.3 Trading Strateg es i
Definition 2.2 A trading strategy φ = ( φ 1 , φ 2 ) which is an F-adapted stochastic process over the time interval [ 0 , T ] is self-financing if its wealth process V ( ) φ , which is set to equal
( ) S B t [ ] T
V t φ = φ t 1 t + φ t 2 t , ∀ ∈ 0 , ,
satisfies the following condition
( ) = ( ) + ∫ t u u + ∫ t u u ∀ ∈ [ ]
t V dS dB t T
V
0 2
0 1
0 φ φ φ , 0 ,
φ
The class of all self-financing trading strategies is denoted by Φ .
The discounted wealth process is denoted by * ( ) ( ) 1 * 2
t t t t t
t S
B
V φ φ
V φ = φ = + , where
t t
t B
= S S *
refers to the discounted underlying asset price. If φ is a self-financing strategy
( ) ( ) , [ ] 0 , ,
0
* 1
* 0
* V dS t T
V
t u u
t φ = φ + ∫ φ ∀ ∈
is satisfied.
We need to exclude arbitrage opportunities from Φ . An arbitrage possibility is equivalent to
the possibility of making a positive amount of money out of nothing with probability 1, a so-
called free lunch on the financial market. We now define this central concept.
Definition 2.3 An arbitrage possibility in a financial market is a trading strategy φ such that
( ) 0 , { ( ) 0 } 1 { ( ) 0 } 0
0 φ = P V T φ ≥ = and P V T φ > >
V .
We say that the market is arbitrage free if there are no arbitrage possibilities [BJÖ, p. 80, and MUR, p. 232].
In order to restrict trading strategies to absence of arbitrage opportunities we need to define the notion of a P * -admissible trading strategy. Before that we must introduce what a martingale measure respectively a spot martingale measure is.
Definition 2.4 A probability measure Q on ( Ω , F ) , equivalent to P, is called a martingale measure for the process S * if S * is a local martingale under Q. Similarly, a probability measure P * is said to be a spot martingale measure if the discounted wealth of any self- financing trading strategy follows a local martingale under P * [MUR, p. 113].
Lemma 2.1 A probability measure is a spot martingale measure if and only if it is a martingale measure for the discounted stock price S * .
Proof. For a proof, please see [MUR, pp. 113 − 114].
In the Black-Scholes framework, the martingale measure for the discounted stock price process is unique, and is explicitly known, as the following result shows.
Lemma 2.2 The unique measure Q for the discounted stock price process S * is given by the Radon-Nikodým derivative
− − −
= r T
r W d
d
T 2
) 2
( 2 exp 1
σ α σ
α P
Q , P − a.s.
Under the martingale measure Q, the discounted stock price S * satisfies
*
*
*
t t
t S dW
dS = σ ,
and the process W * which equals
[ ] T t
r t W
W t * t − , ∀ ∈ 0 ,
−
= σ
α ,
follows a standard Brownian motion on the probability space ( Ω , F, Q ) .
Proof. See [MUR, p. 114] for an idea of the proof.
We are now ready to make the following definitions.
Definition 2.5 A trading strategy φ ∈ Φ is called P * -admissible if the discounted wealth process
( ) B V ( ) t [ ] T
V t * φ = t − 1 t φ , ∀ ∈ 0 , ,
follows a martingale under P * . We write Φ ( ) P * to denote the class of all P * -admissible trading strategies.
2.4 Completeness and Hedging
Definition 2.6 We say that a claim χ can be replicated, alternatively that it is reachable or hedgeable, if there exist a P * -admissible self-financing trading strategy φ such that
( ) a.s.
V T φ = χ , P −
In this case we say that φ is a hedge against χ . Alternatively, φ is called a replicating or
hedging P * -admissible self-financing trading strategy. If every contingent claim is reachable
we say that the market is complete.
If a claim is hedgeable we thus have a natural price process ∏ ; ( ) t χ = V t ( ) φ where φ is consistent with Definition 2.6. By just accepting P * -admissible strategies we have excluded all arbitrage opportunities, consequently we can uniquely define a contingent claims arbitrage free price.
Proposition 2.1 Suppose that the claim χ can be hedged using the P * -admissible self- financing trading strategy φ . Then the only price process ∏ ( ) t ; χ that is consistent with no arbitrage is given by ∏ ; ( ) t χ = V t ( ) φ .
Proof. If at some time t we have ∏ ( ) t ; χ > V t ( ) φ then we can make an arbitrage by buying the portfolio and selling the claim short, and vice versa if ∏ ; ( ) t χ < V t ( ) φ .
Later on in this thesis we will need this extremely important theorem.
Theorem 2.2 The Black-Scholes model given by
, ,
t t t
t t t
dW S dt S dS
dt rB dB
σ
α +
=
=
where we assume that r, α and σ are deterministic constants and σ > 0 , is complete.
Proof. The proof requires fairly deep results from probability theory and is thus outside the scope of this thesis.
We will refer to Μ = ( S , B , Φ ( P *) ) as the arbitrage-free Black-Scholes model of a financial market.
Theorem 2.3 (Formula of Risk Neutral Valuation)
Let χ be a contingent claim that is attainable in M and matures at time T. The arbitrage-free
price, ∏ ; ( ) t χ = V t ( ) φ , at time t ∈ [ 0 , T ] , is given by
( ) ; * 1 , ∀ t ∈ [ ] 0, T
=
∏ − t
T P
t F
E B B
t χ χ
Proof. χ ( ) φ ( ) φ ( ) φ ( ) χ
* ;
* V t
B B V B F
E V B B F
E
B t
t t t t T T P t t T P
t = = = ∏
=
.
3 Generalised Model for Pricing of the Contingent Claims
3.1 Setup
The payoff at the date of maturity for the contingent claim is defined by
−
+
= ∑
= −
N −
i i
i i
S S , S
χ
1 1
, 1
0 min
max λ β ,
where λ , are positive real constants. Our main problem is to determine the arbitrage free β price of the claim. We will use the standard notation ∏ ( ) t ; χ for the price process of the claim χ. In contrast to ordinary vanilla options the function of the value will not just depend on t and S(t). In this particular case we have to introduce two new variables Z(t) and S ( ) t . Thus
( ) t ; = F ( S ( t ), t , S ( t ), Z ( t ) )
∏ χ
where
( ) t
S = the value of S at the previous sampling = S i .
and
( ) i i
j j
j
j Z
S S t S
Z =
−
+
= ∑
= −
−
1 1
, 1
0 min ,
max λ β
Here the index i refers to the sampling just prior to time t. Exactly what all this means will
hopefully be clear in the next section.
3.2 Sampling Structure
The term to maturity is the fixed time interval [0, T N ]. This interval has N number of deterministic equidistantly distributed time points, the sampling dates, denoted by T i , where i = 1, …, N and
O < T 1 < … < T N − 1 < T N .
On the sampling dates the path-dependent quantity Z(t) is measured, and takes the constant value Z i for T i ≤ t < T i+1 . Z is updated on the sampling date T i according to the following rule
−
+
= −
0, S min ,
max 1 S S
Z
Z i λ i . (3.1)
On the time interval [0, T 1 ) Z 0 takes the value β .
We define T i to mean the time immediately after a sampling has occurred. When regarding the model on an infinitesimal time scale we have the interpretation that the sampling happens, not at the time point T i , but rather at T i −, which is equivalent to T i − dt.
t S
S i
S 3
T i
T 3
Figure 3.1. Illustration of sampling structure .
3.2.1 Jump Condition
From (3.1) it is obvious that Z might be discontinuous at T i . A very important observation, though, is that the value of the option immediately before the sampling time and directly after the sampling takes place must be the same. This can easily be understood from an absence of arbitrage point of view; if the option behaved discontinuously across a known sampling date it would provide an arbitrage opportunity. Hence, across a sampling date the value of the contract is continuous. Expressed more compact in mathematical form as
( i ) ( i )
T
t F S S Z F S S Z
i
