Table Of Content

Mathematical Modeling in Finance with Stochastic Processes Steven R. Dunbar February 5, 2011 2 Contents 1 Background Ideas 7 1.1 Brief History of Mathematical Finance . . . . . . . . . . . . . 7 1.2 Options and Derivatives . . . . . . . . . . . . . . . . . . . . . 17 1.3 Speculation and Hedging . . . . . . . . . . . . . . . . . . . . . 25 1.4 Arbitrage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 1.5 Mathematical Modeling . . . . . . . . . . . . . . . . . . . . . 38 1.6 Randomness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 1.7 Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . 55 1.8 A Binomial Model of Mortgage Collateralized Debt Obliga- tions (CDOs) . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 2 Binomial Option Pricing Models 73 2.1 Single Period Binomial Models . . . . . . . . . . . . . . . . . . 73 2.2 Multiperiod Binomial Tree Models . . . . . . . . . . . . . . . 81 3 First Step Analysis for Stochastic Processes 89 3.1 A Coin Tossing Experiment . . . . . . . . . . . . . . . . . . . 89 3.2 Ruin Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . 94 3.3 Duration of the Gambler’s Ruin . . . . . . . . . . . . . . . . 106 3.4 A Stochastic Process Model of Cash Management . . . . . . . 113 4 Limit Theorems for Stochastic Processes 123 4.1 Laws of Large Numbers . . . . . . . . . . . . . . . . . . . . . 123 4.2 Moment Generating Functions . . . . . . . . . . . . . . . . . . 128 4.3 The Central Limit Theorem . . . . . . . . . . . . . . . . . . . 134 4.4 The Absolute Excess of Heads over Tails . . . . . . . . . . . . 145 3 4 CONTENTS 5 Brownian Motion 155 5.1 Intuitive Introduction to Diffusions . . . . . . . . . . . . . . . 155 5.2 The Definition of Brownian Motion and the Wiener Process . 161 5.3 Approximation of Brownian Motion by Coin-Flipping Sums . . 171 5.4 Transformations of the Wiener Process . . . . . . . . . . . . . 174 5.5 Hitting Times and Ruin Probabilities . . . . . . . . . . . . . . 179 5.6 Path Properties of Brownian Motion . . . . . . . . . . . . . . 183 5.7 Quadratic Variation of the Wiener Process . . . . . . . . . . . 186 6 Stochastic Calculus 195 6.1 StochasticDifferentialEquationsandtheEuler-MaruyamaMethod195 6.2 Itoˆ’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 6.3 Properties of Geometric Brownian Motion . . . . . . . . . . . 207 7 The Black-Scholes Model 215 7.1 Derivation of the Black-Scholes Equation . . . . . . . . . . . . 215 7.2 Solution of the Black-Scholes Equation . . . . . . . . . . . . . 223 7.3 Put-Call Parity . . . . . . . . . . . . . . . . . . . . . . . . . . 236 7.4 Derivation of the Black-Scholes Equation . . . . . . . . . . . . 244 7.5 Implied Volatility . . . . . . . . . . . . . . . . . . . . . . . . . 252 7.6 Sensitivity, Hedging and the “Greeks” . . . . . . . . . . . . . . 256 7.7 Limitations of the Black-Scholes Model . . . . . . . . . . . . . 264 List of Figures 1.1 This is not the market for options! . . . . . . . . . . . . . . . 20 1.2 Intrinsic value of a call option . . . . . . . . . . . . . . . . . . 22 1.3 A diagram of the cash flow in the gold arbitrage . . . . . . . . 35 1.4 The cycle of modeling . . . . . . . . . . . . . . . . . . . . . . 40 1.5 Initial conditions for a coin flip, from Keller . . . . . . . . . . 52 1.6 Persi Diaconis’ mechanical coin flipper . . . . . . . . . . . . . 53 1.7 The family tree of some stochastic processes . . . . . . . . . . 62 1.8 Default probabilities as a function of both the tranche number 0 to 100 and the base mortgage default probability 0.01 to 0.15 70 2.1 The single period binomial model. . . . . . . . . . . . . . . . . 76 2.2 A binomial tree . . . . . . . . . . . . . . . . . . . . . . . . . . 82 2.3 Pricing a European call . . . . . . . . . . . . . . . . . . . . . . 84 2.4 Pricing a European put . . . . . . . . . . . . . . . . . . . . . . 85 3.1 Welcome to my casino! . . . . . . . . . . . . . . . . . . . . . . 91 3.2 Welcome to my casino! . . . . . . . . . . . . . . . . . . . . . . 92 3.3 Several typical cycles in a model of the reserve requirement. . 117 4.1 Block diagram of transform methods. . . . . . . . . . . . . . . 130 4.2 Approximation of the binomial distribution with the normal distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 4.3 The half-integer correction . . . . . . . . . . . . . . . . . . . . 148 4.4 Probability of s excess heads in 500 tosses . . . . . . . . . . . 151 5.1 GraphoftheDow-JonesIndustrialAveragefromAugust, 2008 to August 2009 (blue line) and a random walk with normal increments with the same mean and variance (brown line). . . 165 5 6 LIST OF FIGURES 5.2 A standardized density histogram of daily close-to-close re- turns on the Pepsi Bottling Group, symbol NYSE:PBG, from September 16, 2003 to September 15, 2003, up to September 13, 2006 to September 12, 2006 . . . . . . . . . . . . . . . . . 165 6.1 The p.d.f. for a lognormal random variable . . . . . . . . . . . 209 7.1 Value of the call option at maturity . . . . . . . . . . . . . . . 232 7.2 Value of the call option at various times . . . . . . . . . . . . 233 7.3 Value surface from the Black-Scholes formula . . . . . . . . . . 234 7.4 Value of the put option at maturity . . . . . . . . . . . . . . . 241 7.5 Value of the put option at various times . . . . . . . . . . . . 242 7.6 Value surface from the put-call parity formula . . . . . . . . . 243 7.7 Value of the call option at various times . . . . . . . . . . . . 258 7.8 The red distribution has more probability near the mean, and a fatter tail (not visible) . . . . . . . . . . . . . . . . . . . . . 268 Chapter 1 Background Ideas 1.1 Brief History of Mathematical Finance Rating Everyone. Section Starter Question Name as many financial instruments as you can, and name or describe the market where you would buy them. Also describe the instrument as high risk or low risk. Key Concepts 1. Finance theory is the study of economic agents’ behavior allocating their resources across alternative financial instruments and in time in an uncertain environment. Mathematics provides tools to model and analyze that behavior in allocation and time, taking into account uncertainty. 2. Louis Bachelier’s 1900 math dissertation on the theory of speculation in the Paris markets marks the twin births of both the continuous time mathematicsofstochasticprocessesandthecontinuoustimeeconomics of option pricing. 7 8 CHAPTER 1. BACKGROUND IDEAS 3. The most important development in terms of impact on practice was the Black-Scholes model for option pricing published in 1973. 4. Since 1973 the growth in sophistication about mathematical models and their adoption mirrored the extraordinary growth in financial in- novation. Majordevelopmentsincomputingpowermadethenumerical solution of complex models possible. The increases in computer power size made possible the formation of many new financial markets and substantial expansions in the size of existing ones. Vocabulary 1. Finance theory is the study of economic agents’ behavior allocating their resources across alternative financial instruments and in time in an uncertain environment. 2. Aderivativeisafinancialagreementbetweentwopartiesthatdepends onsomethingthatoccursinthefuture, suchasthepriceorperformance ofanunderlyingasset. Theunderlyingassetcouldbeastock, abond, a currency, or a commodity. Derivatives have become one of the financial world’s most important risk-management tools. Derivatives can be used for hedging, or for speculation. 3. Types of derivatives: Derivatives come in many types. There are futures, agreements to trade something at a set price at a given date; options, the right but not the obligation to buy or sell at a given price; forwards, like futures but traded directly between two parties instead of on exchanges; and swaps, exchanging one lot of obligations for another. Derivatives can be based on pretty much anything as long as two parties are willing to trade risks and can agree on a price [48]. Mathematical Ideas Introduction One sometime hears that “compound interest is the eighth wonder of the world”, or the “stock market is just a big casino”. These are colorful say- ings, maybe based in happy or bitter experience, but each focuses on only one aspect of one financial instrument. The “time value of money” and 1.1. BRIEF HISTORY OF MATHEMATICAL FINANCE 9 uncertainty are the central elements that influence the value of financial instruments. When only the time aspect of finance is considered, the tools of calculus and differential equations are adequate. When only the uncertainty is considered, the tools of probability theory illuminate the possible outcomes. When time and uncertainty are considered together we begin the study of advanced mathematical finance. Finance is the study of economic agents’ behavior in allocating financial resources and risks across alternative financial instruments and in time in an uncertain environment. Familiar examples of financial instruments are bank accounts, loans, stocks, government bonds and corporate bonds. Many less familiar examples abound. Economic agents are units who buy and sell financial resources in a market, from individuals to banks, businesses, mutual funds and hedge funds. Each agent has many choices of where to buy, sell, invest and consume assets, each with advantages and disadvantages. Each agent must distribute their resources among the many possible investments with a goal in mind. Advanced mathematical finance is often characterized as the study of the more sophisticated financial instruments called derivatives. A derivative is a financial agreement between two parties that depends on something that occurs in the future, such as the price or performance of an underlying asset. The underlying asset could be a stock, a bond, a currency, or a commodity. Derivatives have become one of the financial world’s most important risk-management tools. Finance is about shifting and distributing risk and derivatives are especially efficient for that purpose [38]. Two such instruments are futures and options. Futures trading, a key practice in modern finance, probably originated in seventeenth century Japan, but the idea can be traced as far back as ancient Greece. Options were a feature of the “tulip mania” in seventeenth century Holland. Both futures and options are called “derivatives”. (For the mathematical reader, these are called derivatives not because they involve a rate of change, but because their value is derived from some underlying asset.) Modern derivatives differ from their predecessors in that they are usually specifically designed to objectify and price financial risk. Derivatives come in many types. There are futures, agreements to tradesomethingatasetpriceatagivendates; options, therightbutnotthe obligation to buy or sell at a given price; forwards, like futures but traded directly between two parties instead of on exchanges; and swaps, exchanging flows of income from different investments to manage different risk exposure. 10 CHAPTER 1. BACKGROUND IDEAS For example, one party in a deal may want the potential of rising income from a loan with a floating interest rate, while the other might prefer the predictable payments ensured by a fixed interest rate. This elementary swap isknownasa“plainvanillaswap”. Morecomplexswapsmixtheperformance of multiple income streams with varieties of risk [38]. Another more complex swap is a credit-default swap in which a seller receives a regular fee from the buyer in exchange for agreeing to cover losses arising from defaults on the underlying loans. These swaps are somewhat like insurance [38]. These more complex swaps are the source of controversy since many people believe that they are responsible for the collapse or near-collapse of several large financial firms in late 2008. Derivatives can be based on pretty much anything as long as two parties are willing to trade risks and can agree on a price. Businesses use derivatives to shift risks to other firms, chiefly banks. About 95% of the world’s 500 biggest companies use derivatives. Derivatives with standardized termsaretradedinmarketscalledexchanges. Derivativestailoredforspecific purposesorrisksareboughtandsold“overthecounter”frombigbanks. The “over the counter” market dwarfs the exchange trading. In November 2009, the Bank for International Settlements put the face value of over the counter derivatives at $604.6 trillion. Using face value is misleading, after off-setting claims are stripped out the residual value is $3.7 trillion, still a large figure [48]. Mathematical models in modern finance contain deep and beautiful ap- plications of differential equations and probability theory. In spite of their complexity, mathematical models of modern financial instruments have had a direct and significant influence on finance practice. Early History The origins of much of the mathematics in financial models traces to Louis Bachelier’s 1900 dissertation on the theory of speculation in the Paris markets. Completed at the Sorbonne in 1900, this work marks the twin births of both the continuous time mathematics of stochastic processes and the continuous time economics of option pricing. While analyzing option pricing, Bachelier provided two different derivations of the partial differential equation for the probability density for the Wiener process or Brown- ian motion. In one of the derivations, he works out what is now called the Chapman-Kolmogorov convolution probability integral. Along the way, Bachelier derived the method of reflection to solve for the probability func-