Mathematics in Finance

232 Pages·2011·0.75 MB·English

Mathematics in Finance
June 12, 2011
2
Contents
0 Introduction 7
0.1 The Different Asset Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
0.2 The Correct Price for Futures and Forwards . . . . . . . . . . . . . . . . . . 8
1 Discrete Models 15
1.1 The Arrow-Debreu Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.2 The State-Price Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.3 The Up-Down and Log-Binomial Model . . . . . . . . . . . . . . . . . . . . . 30
1.4 Hedging in the Log-Binomial Model . . . . . . . . . . . . . . . . . . . . . . . 35
1.5 The Approach of Cox, Ross and Rubinstein . . . . . . . . . . . . . . . . . . 43
1.6 The Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
1.7 Introduction to the Theory of Bonds . . . . . . . . . . . . . . . . . . . . . . 43
1.8 Numerical Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2 Stochastic Calculus, Brownian Motion 45
2.1 Introduction of the Brownian Motion . . . . . . . . . . . . . . . . . . . . . . 46
2.2 Some Properties of the Brownian Motion . . . . . . . . . . . . . . . . . . . . 54
2.3 Stochastic Integrals with Respect to the Brownian Motion . . . . . . . . . . 61
2.4 Stochastic Calculus, the Ito Formula . . . . . . . . . . . . . . . . . . . . . . 77
3 The Black-Scholes Model 89
3.1 The Black-Scholes Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3
4 CONTENTS
3.2 Solution of the Black-Scholes Equation . . . . . . . . . . . . . . . . . . . . . 95
3.3 Discussion of the Black and Scholes Formula . . . . . . . . . . . . . . . . . . 103
3.4 Black-Scholes Formula for Dividend Paying Assets . . . . . . . . . . . . . . . 108
4 Interest Derivatives 111
4.1 Term Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.2 Continuous Models of Interest Derivative . . . . . . . . . . . . . . . . . . . . 111
4.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5 Martingales, Stopping Times and American Options 113
5.1 Martingales and Option Pricing . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.2 Stopping Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.3 Valuation of American Style Options . . . . . . . . . . . . . . . . . . . . . . 134
5.4 American and European Options, a Comparison . . . . . . . . . . . . . . . . 145
6 Path Dependent Options 149
6.1 Introduction of Path Dependent Options . . . . . . . . . . . . . . . . . . . . 149
6.2 The Distribution of Continuous Processes . . . . . . . . . . . . . . . . . . . . 155
6.3 Barrier Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
6.4 Asian Style Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
Appendix 178
A Linear Analysis 179
A.1 Basics of Linear Algebra and Topology in Rn . . . . . . . . . . . . . . . . . . 179
A.2 The Theorem of Farkas and Consequences . . . . . . . . . . . . . . . . . . . 184
B Probability Theory 189
B.1 An example: The Binomial and Log–Binomial Process . . . . . . . . . . . . . 191
B.2 Some Basic Notions from Probability Theory . . . . . . . . . . . . . . . . . . 203
B.3 Conditional Expectations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
CONTENTS 5
B.4 Distances and Convergence of Random Variables . . . . . . . . . . . . . . . . 223
6 CONTENTS
Chapter 0
Introduction
0.1 The Different Asset Classes
............
7
8 CHAPTER 0. INTRODUCTION
0.2 The Correct Price for Futures and Forwards
A future contract can be seen as a standardized forward agreement. Futures are for instance
only offered with certain maturities and contract sizes, whereas forwards are more or less
customized. However, from a mathematical point of view, futures and forwards can be con-
sideredtobeidenticalandthereforewewillonlyconcentrateonthefirstinourconsiderations
throughout this chapter. A future contract, or simply future, is the following agreement:
Two parties enter into a contract whereby one party agrees to give the other one
an underlying asset (for example the share of a a stock) at some agreed time T
in the future in exchange for an amount K agreed on now.
Usually K is chosen such that no cash flow, i.e. no exchange of money is necessary at the
time of the agreement. Let us assume the underlying asset was a stock then we can introduce
the following notation :
S : Price of a share of the underlying stock at time 0 (present time).
0
S : Price of a share of the stock at maturity T. This value is not known at time 0 and hence
T
considered to be a random variable.
S −K: Value of the future contract at time T seen from the point of view of the buyer.
T
The crucial problem and the repeating theme of these notes will be questions of the
following kind:
What is the value or fair price of such a future at time 0? How should K be
chosen so that no exchange of money is necessary at time 0?
Game theoretical approach: pricing by expectation
One way to look at this problem, is to consider the future contract to be a game having
the following rule: at time T player 1 (long position) receives from player 2 (short position)
the amount of S − K in case this amount is positive. Otherwise he has to pay player 2
T
0.2. THE CORRECT PRICE FOR FUTURES AND FORWARDS 9
the amount of K − S . What is a “fair price” V for player 1 to participate in this game?
T
Since the amount V is due at time 0 but the possible payoff occurs at time T we also have
to consider the time value of money or simply interest. If r is the annual rate of return,
compounded continuously, the value of the cash outflow V paid by player 1 at time 0 will
be worth erT ·V at time T.
Gametheoreticallythisgameissaidtobefairiftheexpected amount of exchanged moneyis0.
Theorem 0.2.1 (Kolmogorov’s strong law of large numbers).
Suppose X ,X ,X ,... are i.i.d random variables, i.e. they are all independently sampled
1 2 3
from the same distribution, which has mean (= expectation) µ. Let S be the arithmetical
n
average of X ,X ,...,X , i.e.
1 2 n
n
1 (cid:88)
S = X .
n i
n
i=1
Then, with probability 1, S tends to µ as n gets larger, i.e. lim S = µ a.s.
n n→∞ n
Thus, if the expected amount of exchanged money is 0, and if our two players play their
game over and over again, the average amount of money exchanged per game would converge
to 0.
Since the exchanged money has the value −VerT +S −K at time T, we need:
T
E(−V ·erT +(S −K)) = 0,
T
or
(1) V = e−rT(E(S )−K).
T
Here E(S ) denotes the expected value of the random variable S .
T T
Conclusion: In order to participate in the game player 1 should pay player 2 the amount
of e−rT(E(S ) − K) at time 0, if this amount is positive. Otherwise player 2 should pay
T
player 1 the amount of e−rT(K−E(S )). Moreover, in order to make an exchange of money
T
unnecessary at time 0, we have to choose K = E(S ).
T
10 CHAPTER 0. INTRODUCTION
This approach seems quite reasonable. Nevertheless, there are the following two objec-
tions. The second one is fatal.
1) V depends on E(S ). Or, if we choose K so that V = 0 then K depends on E(S ).
T T
But, usually E(S ) is not known to investors. Thus, the two players can only agree to
T
play the game if they agree on E(S ), at least for player 1 E(S ) should seem to be
T T
higher than for player 2.
2) ChoosingK = E(S )canlead toarbitragepossibilitiesasthe following example shows.
T
Example: Assume E(S ) = S , and choose the “game theoretically correct” value K = S .
T 0 0
Thus, noexchangeofmoneyisnecessaryattime0. Nowaninvestorcouldproceedasfollows:
At time 0 she sells short n shares of the stock, and invests the received amount (namely
nS ) into riskless bonds. In order to cover her short position at the same time she enters
0
into a future contract in order to buy n shares of the stock for the price at S = E(S ).
0 T
At time T her bond account is worth nerTS . So she can buy the n shares of the stock
0
for nS , close the short position and end up with a profit of nS (erT −1). In other words,
0 0
although there was no initial investment necessary at time 0, this strategy will lead to a
guaranteed profit of nS (erT −1). This example represents a typical arbitrage opportunity.
0
Pricing by arbitrage
The following principle is the basic axiom for valuation of financial products. Roughly it
says : “There is no free lunch”.
In order to formulate it precisely, we make the following assumption: Investors can buy
units of assets in any denomination, i.e. θ units where θ is any real number.
Suppose that an investor can take a position (choose a certain portfolio) which has no
net costs (the sum of the prices is less than or equal to zero). Secondly, it guarantees no
losses in the future but some chance of making a profit. In this (fortunate) situation we