Introduction to Stochastic Calculus Applied to Finance Second Edition Damien Lamberton and Bernard Lapeyre Numerical Methods for Finance, John A. D. Introduction to stochastic calculus applied to finance / Damien Lamberton and Bernard Lapeyre ; translated by Nicolas Rabeau and François Mantion Lamberton. Lamberton D., Lapeyre P. – Introduction to Stochastic Calculus Applied to Finance – Download as PDF File .pdf), Text File .txt) or view presentation slides online.

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Skip to main content. Log In Sign Up. Introduction to Stochastic Calculus Applied to Finance. The field of financial mathematics forms an ever-expanding slice of the financial sector. This series aims to capture new developments and summarize what is known over the whole spectrum of this field. It will include a broad range of textbooks, reference works and handbooks that are meant to appeal to both academics and practitioners. The inclusion of numerical code and concrete real-world examples is highly encouraged.

Edelman, and John J.

## International Journal of Stochastic Analysis

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For this new edition, we have not tried to be exhaus- tive on all new developments but to select some techniques or concepts that could be incorporated at reasonable cost in terms of length and mathemati- cal sophistication. This was partly done by adding new exercises. The main addition concern: We are indebted, in addition to those cited in the introduction, to a number of colleagues whose suggestions have been helpful for this new edition.

Contents Introduction 9 1 Discrete-time models 15 1. Cox, Ross and Rubinstein model. Since then, as the option markets have evolved, Black- Scholes and Merton results have developed to become clearer, more general and mathematically more rigorous.

The theory seems to be advanced enough to attempt to make it accessible to students. The writer of the option needs to specify: The price of the option is the premium. When the option is traded on an organ- ised market, the premium is quoted by the market. Lapsyre, the problem is to price the option. Also, even if the option is traded on an organized market, it can be finanxe to detect some possible abnormalities in the market.

Let us examine the case of a European call option on a stock, whose price at time t yo denoted by St. Let us call T the expiration date and K the exercise price. Obviously, if K is greater than STstochsatic holder of the option has no interest whatsoever in exercising the option.

If the option is exercised, the writer must be able to deliver a stock at price K. At the time of laeyre the option, which will be considered as the origin of time, ST is unknown and therefore two questions have to be asked: How much should the buyer pay for the option?

That is the problem of pricing the option. That is the problem of hedging the option. At this point, we will only show how we can derive formulae relating European put and call prices from the no arbitrage assumption.

Consider a put and a call with the same maturity T and exercise price Kon the same underlying asset which is worth St at time t. We shall assume that it is possible to borrow or invest money at a constant rate r. Let us denote by Ct and Pt respectively the prices of the call and the put at time t. If this amount is positive, we invest it at rate r until time Twhereas if it is negative we borrow it at the same rate. At time Ttwo outcomes are possible: There are many similar examples in the book by Cox and Rubinstein To achieve this, we need to model stock prices more precisely.

Moreover, the formula depends on only one non-directly observable parameter, the so-called volatility. In the last few years, many extensions of the Black-Scholes approach has been considered.

From a thorough study of the Black-Scholes model, we will attempt to give to the reader the means to understand those extensions. The link between the mathematical concept of martingale and the economic notion of arbitrage is brought to light. The second chapter deals with American options.

### Introduction to stochastic calculus applied to finance, by Damien Lamberton and Bernard Lapeyre

Thanks to the theory of optimal stopping in a discrete time set-up, which uses quite elementary methods, we introduce the reader to all the ideas that can be developed in continuous time. Chapter 3 is an introduction to the main results in stochastic calculus that we will use in Chapter 4 to study the Black-Scholes model.

As far as European options are concerned, this model leads to explicit formulae. These questions are addressed in Chapter 5. Chapter 6 is a relatively quick introduction to the main interest rate models and Chapter 7 looks at the problems of option pricing and hedging when the price of the underlying asset follows a simple jump process.

These models are rather less optimistic than the Black-Scholes model and seem to be closer to reality. However, their mathematical treatment is still a matter of research, in the framework of so- called incomplete markets.

Also, a few exercises and longer questions are listed at the end of each chapter. A good level in probability theory is assumed to read this book. The reader is referred to Dudley and Williams for prerequisites. However, some basic results are also proved in the Appendix. Acknowledgments This finnace is paplied on the lecture notes of a course taught at l’Ecole des Ponts since The organisation of this lecture series would not have been possible without the encouragement of N.

A few people kindly read the earlier draft of our book and helped us with their remarks. Amongst them are S. Finally, we thank our colleagues at the university and at INRIA for their advice and their motivating comment: Le Gall and D. Chapter 1 Discrete-time models The objective of this chapter is to present the main ideas related to option theory within the very simple mathematical framework of discrete-time mod- els.

Cox, Ross and Rubinstein’s model is detailed at the end of the chapter in the form of a problem with its solution. The horizon N will often correspond to the maturity of the options. The interpretation is the following: The following proposition makes this clear in terms of discounted prices. The following are equivalent: The equivalence between i and ii results from Remark 1.

More precisely, we can prove the following propo- sition. In this model, short-selling and borrowing are allowed, but, by the following admisibility condition, the value of the portfolio must remain non-negative at all times. The investor must be able to pay back his debts in the riskless or the risky assets at any time.

Most models exclude any arbitrage opportunity, and the objective of the next section is to characterize these models with the notion of martingale. The sum of two martingales is a martingale. Obviously, similar properties can be shown for supermartingales and submartingales. Xn is sometimes called the martingale transform of Mn by Hn.

A conse- quence of this proposition and Proposition 1. Clearly, Xn is an adapted sequence. A market is viable if there is no arbitrage opportunity. The following result is sometimes referred to as the Fundamental Theorem of Asset Pricing. According to Proposition 1. That contradicts the assumption of market viability and completes the proof of the lemma.

According to Lemma 1. There are some options dependent on the whole path of the underlying asset, i. That is the case of the so-called Asian options, 2 Or, more generally, a contingent claim. The market is complete if every contingent claim is at- tainable. The interest of complete markets is that it allows us to derive a simple theory of contingent claim pricing and hedging. The Cox- Ross-Rubinstein model, which we shall study in the next section, is a very simple example of a complete market model.

It follows from Proposition 1. SN At any time, the value of an admissible strategy replicating h is completely determined by h. In other words, the investor is perfectly hedged.