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"Ales ?erný's new edition of Mathematical Techniques in Finance is an excellent master'slevel treatment of mathematical methods used in financial asset pricing. By updating the original edition with methods used in recent research, ?erný has once again given us an uptodate firstclass textbook treatment of the subject."—Darrell Duffie, Stanford University
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Darrell Duffie
Ales Cerný's new edition of Mathematical Techniques in Finance is an excellent master'slevel treatment of mathematical methods used in financial asset pricing. By updating the original edition with methods used in recent research, Cerný has once again given us an uptodate firstclass textbook treatment of the subject.— Darrell Duffie, Stanford University
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Ales Cerný is professor of finance at the Cass Business School, City University London.
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Mathematical Techniques in Finance
Tools for Incomplete MarketsBy Ales Cerny
Princeton University Press
Copyright © 2009 Princeton University PressAll right reserved.
ISBN: 9780691141213
Chapter One
The Simplest Model of Financial MarketsThe main goal of the first chapter is to introduce the oneperiod finite state model of financial markets with elementary financial concepts such as basis assets, focus assets, portfolio, ArrowDebreu securities, hedging and replication. Alongside the financial topics we will encounter mathematical toolslinear algebra and matricesessential for formulating and solving basic investment problems. The chapter explains vector and matrix notation and important concepts such as linear independence.
After reading the first two chapters you should understand the meaning of and be able to solve questions of the following type.
Example 1.1 (replication of securities). Suppose that there is a risky security (call it stock) with tomorrow's value S = 3, 2 or 1 depending on the state of the market tomorrow. The first state (first scenario) happens with probability 1/2, the second with probability 1/6 and the third with probability 1/3. There is also a riskfree security (bond) which pays 1 no matter what happens tomorrow. We are interested in replicating two call options written on the stock, one with strike 1.5 and the second with strike 1.
1. Find a portfolio of the stock, bond and the first call option that replicates the second call option (socalled gamma hedging).
2. If the initial stock price is 2 and the riskfree rate of return is 5%, what is the noarbitrage price of the second option?
3. Find the portfolio of the bond and stock which is the best hedge to the first option in terms of the expected squared replication error (socalled delta hedging).
This chapter is important for two reasons. Firstly, the oneperiod model of financial markets is the main building block of a dynamic multiperiod model which will be discussed later and which represents the main tool of any financial analyst. Secondly, matrices provide an effective way of describing the relationships among several variables, random or deterministic, and as such they are used with great advantage throughout the book.
1.1 OnePeriod Finite State Model
It is a statement of the obvious that the returns in financial markets are uncertain. The question is how to model this uncertainty. The simplest model assumes that there are only two dates, which we will call today and tomorrow, but which could equally well be called this week and next week, this year and next year, or now and in 10 min. The essential feature of our twodate, oneperiod model is that no investment decisions are taken between the two dates. One should be thinking of a world which is at a standstill apart from at 12 noon each day when all economic activity (work, consumption, trading, etc.) is carried out in a split second.
It is assumed that we do not know today what the market prices will be tomorrow, in other words the state of tomorrow's world is uncertain. However, we assume that there is only a finite number of scenarios that can take place, each of which is known today down to the smallest detail. One of these scenarios is drawn at random, using a controlled experiment whereby the probability of each scenario being drawn is known today. The result of the draw is made public at noon tomorrow and all events take place as prescribed by the chosen scenario (see Table 1.1 for illustration).
Let us stop for a moment and reflect how realistic the finite state model is. First of all, how many scenarios are necessary? In the above table we have four random variables: the value of the FTSE index, the level of UK base interest rate, UK weather and the result of the ChelseaWimbledon football game. Assuming that each of these variables has five different outcomes and that any combination of individual outcomes is possible we would require [5.sup.4] = 625 different scenarios. Given that in finance one usually works with two or three scenarios, 625 seems more than sufficient. And yet if you realize that this only allows five values for each random variable (only five different results of the football match!), then 625 scenarios do not appear overly exigent.
Next, do we know the probability of each of the 625 scenarios? Well, we might have a subjective opinion on how much these probabilities are but since the weather, football match or development in financial markets can hardly be thought of as controlled random experiments we do not know what the objective probabilities of those scenarios are. There is even a school of thought stating that objective probabilities do not exist; see the notes at the end of the chapter.
Hence the finite state model departs from reality in two ways: firstly, with a small number of scenarios (states of the world) it provides only a patchy coverage of the actual outcomes, and secondly we do not know the objective probabilities of each scenario, we only have our subjective opinion of how much they might be.
1.2 Securities and Their Payoffs
Security is a legal entitlement to receive (or an obligation to pay) an amount of money. A security is characterized by its known price today and its generally uncertain payoff tomorrow. What constitutes the payoff depends to some extent on the given security. For example, consider a model with just two scenarios and one security, a share in publicly traded company TRADEWELL Inc. Let us assume that the initial price of the share is 1, and tomorrow it can either increase to 1.2 or drop to 0.9. Assume further that the shareholders will receive a dividend of 0.1 per share tomorrow, no matter what happens to the share price.
The security payoff is the amount of money one receives after selling the security tomorrow plus any additional payment such as the dividend, coupon or rebate one is entitled to by virtue of holding the security. In our case the payoff of one TRADEWELL share is 1.3 or 1 depending on the state of the world tomorrow.
Security price plays a dual role. The stock price today is just thata price. The stock price tomorrow is part of the stock's uncertain payoff.
Throughout this chapter and for a large part of the next chapter we will ignore today's prices and will only talk about the security payoffs. We will come back to pricing in Chapter 2, Section 2.5. Throughout this book we assume frictionless trading, meaning that one can buy or sell any amount of any security at the market price without transaction costs. This assumption is justified in liquid markets.
Example 1.2. Suppose S is the stock price at maturity. A call option with strike K is a derivative security paying
S  K if S > K, 0 if S [less than or equal to] K.
The payoffs of options in Example 1.1 are in Table 1.2.
1.3 Securities as Vectors
An ntuple of real numbers is called an ndimensional vector. For
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
we write x, y [member of] [R.sup.n]. Each ndimensional vector refers to a point in ndimensional space. The above is a representation of such a point as a column vector, which is nothing other than an n x 1 matrix, since it has n rows and 1 column. Of course, the same point can be written as a row vector instead. Whether to use columns or rows is a matter of personal taste, but it is important to be consistent.
Example 1.3. Consider the four securities from the introductory example. Let us write the payoffs of each security in the three states (scenarios) as a threedimensional column vector:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
These securities are depicted graphically in Figure 1.1.
In MATLAB one would write
a1 = [1;1;1]; a2 = [3;2;1]; a3 = [1.5;0.5;0]; a4 = [2;1;0];
1.4 Operations on Securities
We can multiply vectors by a scalar. For any [alpha] [member of] R we define
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
This operation represents [alpha] units of security x.
Example 1.4. Two units of the third security will have the payoff
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
If we buy two units of the third security today, tomorrow we will collect 3 pounds (dollars, euros) in the first scenario, 1 in the second scenario and nothing in the third scenario. In MATLAB one would type
2*a3;
If we issued (wrote, sold) 1 unit of the fourth security, then our payoff tomorrow would be
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
In other words, we would have to pay the holder of this security 2 in the first scenario, 1 in the second scenario and nothing in the third scenario. In MATLAB one types
a4;
Various amounts of securities [a.sub.3] and [a.sub.4] are represented graphically in Figure 1.2.
One can also add vectors together:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
With this operation we can calculate portfolio payoffs. A portfolio is a combination of existing securities, which tells us how many units of each security have to be bought or sold to create the portfolio. Naturally, portfolio payoff is what the name suggests: the payoff of the combination of securities. The word 'portfolio' is sometimes used as an abbreviation of 'portfolio payoff', creating a degree of ambiguity in the terminology.
Example 1.5. A portfolio in which we hold two units of the first option and issue one unit of the second option will have the payoff
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Graphically, this situation is depicted in Figure 1.3. In MATLAB the portfolio payoff is
2*a3a4;
1.5 The Matrix as a Collection of Securities
Often we need to work with a collection of securities (vectors). It is then convenient to stack the column vectors next to each other to form a matrix.
Example 1.6. The vectors [a.sub.1], [a.sub.2], [a.sub.3], [a.sub.4] from Example 1.3 form a 3 x 4 payoff matrix, which we denote A,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
The market scenarios (states of the world) are in rows, securities are in columns. In MATLAB
A = [a1 a2 a3 a4];
1.6 Transposition
Sometimes we need a row vector rather than a column vector. This is achieved by transposition of a column vector:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Note that x* (transpose of x) is a 1 x n matrix. Conversely, transposition of a row vector gives a column vector. Should we perform the transposition twice, we will end up with the original vector:
(x*)* = x.
Example 1.7.
[a.sup.*.sub.1] = [1 1 1], [a.sup.*.sub.2] = [3 2 1], [a.sup.*.sub.3] = [1.5 0.5 0], [a.sup.*.sub.4] = [2 1 0].
In MATLAB transposition is achieved by attaching a prime to the matrix name. For example, [a.sup.*.sub.1] would be written as
a1';
The vectors [a.sup.*.sub.1], [a.sup.*.sub.2], [a.sup.*.sub.3, [a.sup.*.sub.4]] stacked under each other form a 4 x 3 matrix B
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)
in MATLAB
B = [a1'; a2'; a3'; a4'] (1.2)
Matrix B from equation (1.1) is in fact the transpose of matrix A
B = A*,
thus instead of (1.2) in MATLAB one would simply write
B = A';
In general, we can have an m x n matrix M (denoted M [member of] [R.sup.m x n]), where m is the number of rows and n is the number of columns. The element in the ith row and jth column is denoted [M.sub.ij]. The entire jth column is denoted [M.sub.j] while the entire ith row is denoted [M.sub.i]]. According to our needs we can think of the matrix M as if it were composed of m row vectors or n column vectors:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
The transpose of a matrix is obtained by changing the columns of the original matrix into the rows of the transposed matrix:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Hence, for example, [M.sup.*.sub.1] = [([M.sub.1]).sup.*] and [M.sup.*.sub.1] = [([M.sub.1]).sup.*], which in words says that the first row of the transposed matrix is the transpose of the first column of the original matrix.
Example 1.8. Suppose a 3 x 4 payoff matrix A is given. To extract the payoff of the third security in all states, in MATLAB one would simply write
A(:,3);
On the other hand, if one wanted to know the payoff of all four securities in the first market scenario, one would look at the row
A(1,:);
1.7 Matrix Multiplication and Portfolios
The basic building block of matrix multiplication is the multiplication of a row vector by a column vector. Let A [member of] [R.sup.1xk] and B [member of] [R.sup.kx1]:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
In this simple case the matrix multiplication AB is defined as follows:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Note that A is a 1 x k matrix, B is k x 1 matrix and the result is a 1 x 1 matrix. One often thinks of a 1 x 1 matrix as a number.
Example 1.9. Suppose that we have a portfolio of the four securities from the introductory example which consists of [x.sub.1], [x.sub.2], [x.sub.3], [x.sub.4] units of the first, second, third and fourth security, respectively. In the third state the individual securities pay 1, 1, 0, 0 in turn. The payoff of the portfolio in the third state will be
[x.sub.1] x 1 + [x.sub.2] x 1 + [x.sub.3] x 0 + [x.sub.4] x 0.
If we take
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],
then the portfolio payoff can be written in matrix notation as [A.sub.3]x.
In general one can multiply a matrix U (m x k) with a matrix V (k x n), regarding the former as m row vectors in [R.sup.k] and the latter as n column vectors in [R.sup.k]. One multiplies each of the m row vectors in U with each of the n column vectors in V using the simple multiplication rule (1.3):
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
(Continues...)
Table of Contents
Preface to the Second Edition xiii
From the Preface to the First Edition xix
Chapter 1: The Simplest Model of Financial Markets 1
1.1 OnePeriod Finite State Model 1
1.2 Securities and Their Payoffs 3
1.3 Securities as Vectors 3
1.4 Operations on Securities 4
1.5 The Matrix as a Collection of Securities 6
1.6 Transposition 6
1.7 Matrix Multiplication and Portfolios 8
1.8 Systems of Equations and Hedging 10
1.9 Linear Independence and Redundant Securities 12
1.10 The Structure of the Marketed Subspace 14
1.11 The Identity Matrix and ArrowDebreu Securities 16
1.12 Matrix Inverse 17
1.13 Inverse Matrix and Replicating Portfolios 17
1.14 Complete Market Hedging Formula 19
1.15 Summary 20
1.16 Notes 21
1.17 Exercises 22
Chapter 2: Arbitrage and Pricing in the OnePeriod Model 25
2.1 Hedging with Redundant Securities and Incomplete Market 25
2.2 Finding the Best Approximate Hedge 29
2.3 Minimizing the Expected Squared Replication Error 32
2.4 Numerical Stability of Least Squares 34
2.5 Asset Prices, Returns and Portfolio Units 36
2.6 Arbitrage 38
2.7 NoArbitrage Pricing 40
2.8 State Prices and the Arbitrage Theorem 41
2.9 State Prices and Asset Returns 44
2.10 RiskNeutral Probabilities 45
2.11 State Prices and NoArbitrage Pricing 46
2.12 Asset Pricing Duality 47
2.13 Summary 48
2.14 Notes 49
2.15 Appendix: Least Squares with QR Decomposition 49
2.16 Exercises 52
Chapter 3: Risk and Return in the OnePeriod Model 55
3.1 Utility Functions 56
3.2 Expected Utility Maximization 59
3.3 The Existence of Optimal Portfolios 61
3.4 Reporting Expected Utility in Terms of Money 62
3.5 Normalized Utility and Investment Potential 63
3.6 Quadratic Utility 67
3.7 The Sharpe Ratio 69
3.8 ArbitrageAdjusted Sharpe Ratio 71
3.9 The Importance of Arbitrage Adjustment 75
3.10 Portfolio Choice with NearArbitrage Opportunities 77
3.11 Summary 79
3.12 Notes 81
3.13 Exercises 82
Chapter 4: Numerical Techniques for Optimal Portfolio Selection in Incomplete Markets 84
4.1 Sensitivity Analysis of Portfolio Decisions with the CRRA Utility 84
4.2 Newton's Algorithm for Optimal Investment with CRRA Utility 88
4.3 Optimal CRRA Investment Using Empirical Return Distribution 90
4.4 HARA Portfolio Optimizer 94
4.5 HARA Portfolio Optimization with Several Risky Assets 96
4.6 Quadratic Utility Maximization with Multiple Assets 99
4.7 Summary 102
4.8 Notes 102
4.9 Exercises 102
Chapter 5: Pricing in Dynamically Complete Markets 104
5.1 Options and Portfolio Insurance 104
5.2 Option Pricing 105
5.3 Dynamic Replicating Trading Strategy 108
5.4 RiskNeutral Probabilities in a MultiPeriod Model 116
5.5 The Law of Iterated Expectations 119
5.6 Summary 121
5.7 Notes 121
5.8 Exercises 121
Chapter 6: Towards Continuous Time 125
6.1 IID Returns, and the Term Structure of Volatility 125
6.2 Towards Brownian Motion 127
6.3 Towards a Poisson Jump Process 136
6.4 Central Limit Theorem and Infinitely Divisible Distributions 142
6.5 Summary 143
6.6 Notes 145
6.7 Exercises 145
Chapter 7: Fast Fourier Transform 147
7.1 Introduction to Complex Numbers and the Fourier Transform 147
7.2 Discrete Fourier Transform (DFT) 152
7.3 Fourier Transforms in Finance 153
7.4 Fast Pricing via the Fast Fourier Transform (FFT) 158
7.5 Further Applications of FFTs in Finance 162
7.6 Notes 166
7.7 Appendix 167
7.8 Exercises 169
Chapter 8: Information Management 170
8.1 Information: Too Much of a Good Thing? 170
8.2 ModelIndependent Properties of Conditional Expectation 174
8.3 Summary 178
8.4 Notes 179
8.5 Appendix: Probability Space 179
8.6 Exercises 183
Chapter 9: Martingales and Change of Measure in Finance 187
9.1 Discounted Asset Prices Are Martingales 187
9.2 Dynamic Arbitrage Theorem 192
9.3 Change of Measure 193
9.4 Dynamic Optimal Portfolio Selection in a Complete Market 198
9.5 Summary 206
9.6 Notes 208
9.7 Exercises 208
Chapter 10: Brownian Motion and Itˆo Formulae 213
10.1 ContinuousTime Brownian Motion 213
10.2 Stochastic Integration and Itˆo Processes 218
10.3 Important Itˆo Processes 220
10.4 Function of a Stochastic Process: the Itˆo Formula 222
10.5 Applications of the Itˆo Formula 223
10.6 Multivariate Itˆo Formula 225
10.7 Itˆo Processes as Martingales 228
10.8 Appendix: Proof of the Itˆo Formula 229
10.9 Summary 229
10.10 Notes 230
10.11 Exercises 231
Chapter 11: ContinuousTime Finance 233
11.1 Summary of Useful Results 233
11.2 RiskNeutral Pricing 234
11.3 The Girsanov Theorem 237
11.4 RiskNeutral Pricing and Absence of Arbitrage 241
11.5 Automatic Generation of PDEs and the FeynmanKac Formula 246
11.6 Overview of Numerical Methods 250
11.7 Summary 251
11.8 Notes 252
11.9 Appendix: Decomposition of Asset Returns into Uncorrelated Components 252
11.10 Exercises 255
Chapter 12: FiniteDifference Methods 261
12.1 Interpretation of PDEs 261
12.2 The Explicit Method 263
12.3 Instability 264
12.4 Markov Chains and Local Consistency 266
12.5 Improving Convergence by Richardson's Extrapolation 268
12.6 Oscillatory Convergence Due to Grid Positioning 269
12.7 Fully Implicit Scheme 270
12.8 CrankNicolson Scheme 273
12.9 Summary 274
12.10 Notes 276
12.11 Appendix: Efficient Gaussian Elimination for Tridiagonal Matrices 276
12.12 Appendix: Richardson's Extrapolation 277
12.13 Exercises 277
Chapter 13: Dynamic Option Hedging and Pricing in Incomplete Markets 280
13.1 The Risk in Option Hedging Strategies 280
13.2 Incomplete Market Option Price Bounds 299
13.3 Towards Continuous Time 304
13.4 Derivation of Optimal Hedging Strategy 309
13.5 Summary 318
13.6 Notes 319
13.7 Appendix: Expected Squared Hedging Error in the BlackScholes Model 320
13.8 Exercises 322
Appendix A Calculus 326
A.1 Notation 326
A.2 Differentiation 329
A.3 Real Function of Several Real Variables 332
A.4 Power Series Approximations 334
A.5 Optimization 336
A.6 Integration 338
A.7 Exercises 344
Appendix B Probability 348
B.1 Probability Space 348
B.2 Conditional Probability 348
B.3 Marginal and Joint Distribution 351
B.4 Stochastic Independence 352
B.5 Expectation Operator 354
B.6 Properties of Expectation 355
B.7 Mean and Variance 356
B.8 Covariance and Correlation 357
B.9 Continuous Random Variables 360
B.10 Normal Distribution 364
B.11 Quantiles 370
B.12 Relationships among Standard Statistical Distributions 371
B.13 Notes 372
B.14 Exercises 372
References 381
Index 385