Discrete Stochastic Models for Finance
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Discrete Stochastic Models for Finance Marc et Francine DIENER October 31, 2006
Voir
- binomial trees
- arbitrage-free markets
- short after
- lee model
- option pricing
- rates derivatives
- short-term interest-rates
- discrete stochastic
- term structure
- option
Discrete
Stochastic
Models
Marc et Francine
October
31,
for
DIENER
2006
Finance
2
Contents
0 Conditional expectation 0.1 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.2A . . . . . . . . . . . . . . . .. . . . . . . . . . . .-measurable random variable 0.3 Properties of the conditional expectation . . . . . . . . . . . . . . . . . . . . . . . 1 Vanilla options 1.1 Basic approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 What are derivatives ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Option pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 En route towards the stochastic approach . . . . . . . . . . . . . . . . . . . . . . 1.3 Model-free properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Arbitrage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 The butterfly spread . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 The Call-Put relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Exotic options 2.1 What is an exotic option ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Barrier options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Pricing an exotic option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Profit’n Loss 3.0 Actualization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Martingales with respect toF(S . . . . . . . . . . . . . . . . . . . . . . . . . .) . 3.2 Profit and loss of a predictable strategy . . . . . . . . . . . . . . . . . . . . . . . 3.3 Martingales representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Appendix 1 : self-financing strategies . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Appendix 2 : hedging away the risk of options : the Black and Scholes Story . . 4 Arbitrage probabilities 4.1 Uncomplete markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Profit-and-Loss and martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Arbitrage-free markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 The fundamental theorem of arbitrage-free probabilities . . . . . . . . . . 4.3.2 Existence ofP∗. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Pricing with an arbitrage-free probability . . . . . . . . . . . . . . . . . . 5 A stochastic interest-rates model 5.1 Some general facts on the present value of future money . . . . . . . . . . . . . . 5.1.1 Where are the risks ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Zero coupons and the term structure . . . . . . . . . . . . . . . . . . . . . 5.1.3 Short-term interest-rates and actualization . . . . . . . . . . . . . . . . . 5.1.4 Stochastic rollover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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7 7 9 10 15 15 15 16 21 21 21 22 23 25 25 25 26 29 29 30 31 32 34 34 35 35 36 37 37 38 38 41 41 41 42 43 43
4
A
CONTENTS
5.2 The Ho and Lee model for the term structure . . . . . . . . . . . . . . . . . . . . 5.3 The model as a three-parameters model :π,δ, andN. . . . . . . . . . . . . . . 5.3.1 No arbitrage condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Binomial condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exercises with Maple A.1 The netϕ(t S . . . . .) and the delta-hedge in a binomial tree model A.2 American options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2.1 The dynamic programming approach . . . . . . . . . . . . . . . A.2.2 Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3 Convergence of the CRR price towards the Black-Scholes price . . . . A.4 The Ho and Lee model for interest-rates, and interest-rates derivatives
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Foreword
These notes have been prepared on the occasion of the authors visit to the Mathematics De-partment of the University of the Philippines, in Manila-Diliman, in the framework of the Asia Link program of the European Community. Their purpose is to help all our colleagues here to offer to Filipino students attractive teachings onn recent subjets in Applied Mathematics. Here we have chosen, just as we did in our home university in Nice and Sophia-Antipolis, to introduce the main ideas of a scientific break-though that occurd during the 1970’ on the New York Stock Exchange (NYSE) and the Chicago Board on Trades (CBOT), when Black and Scholes introduced their famous formula giving the way of pricing and hedging options on stocks. As it was acknoledged in 1997 by the Nobel jury, at the same time as they published the basic ideas behind it, Merton derived it using the tools of stochastic calculus, a wonderfull mathematical object to which very little mathematician where trained at this time. Things have changed a lot in the interval ! (Almost) all young mathematician wants to know about maths applied to finance, and this is not always easy to do, at least when there is a necessity of full rigor, as stochastic calculus requires usually some confidence with Borel-Lebesgue-Kolmogorov theory ofσ course, one may-algebras and filtrations by such beasts ! Of wishjusttouseaverybrilliantspin-offofit:thefamousItˆoCalculus1and that could be tought independentlyfromtheconstructionoftheItˆointegrale.Intheapproachthatweadoptedhere, we made use of an other Nobel-Prize winner : Sharp, who encouraged Cox, Ross, and Rubinstein to think of a model of stock prices based on binomial trees. So we used this clever finite model to introduce first the elementary ideas that allow, in this model, to hedge the risks, using basic linear algebra and easy-to-understand reasonning. Then, we gradually introduced the major probabilistic (or “stochastic”) concepts, but again in a finite case, so to avoid the technicalities of measure theory. Even avoiding that, this is by no means trivial. Here we have collected the less immediat ideas in chapter 0, but we adopted to introduce the definitions and results gradually, when they could be used to rephrase tricks that could be understood on a financial problem. At the first glance, the stochastic version may seem a bit pedantic in the context where it is introduced (and it is !) but short after we could show how, using the formalism, one can attack problems much less obvious ! In this spirit, these lectures might be considered asAn Introduction to Mathematical Finance with Stochastic Calculus in View, as we consider that a mathematician should both have the (geometric, information-managment, risk-hedging oriented) intuition of stochasticcalculusandsomeideasonthetoolsusuallyusedinarigorousintroductionoftheItoˆ integrale, that could be introduced in later work-group seminars. Actually, this way of mixing chapter 0 to the next ones is not reflected in these notes, and this will have to be done in a later version. Again, in an attempt to help students to grasp as concretely as possible the new ideas, we used an other trick : to program small examples of financial products, usingMaple, a mathematician-friendly langage with nice graphical output. Usually, the method adopted was to give a first version of the program, that would introduce the commands needed, and then to 1hsewdluoereHacAhymedfocnerFehtotrepapaedttmiuboswhinblDfo¨rooyihtsgacihetriontmentalso Sciences as Germany was to invade France, requiring the paper would not be opened for at least fifty years, in order that his ideas would not fall into the Enemy’s hands, and then committed suicide. 5
