## The Wiener Process and Rare Events in Financial Markets

At every instant of an ordinary trading day, there are three states of the world prices may go up by one tick, decrease by one tick, or show no change. In fact, the price of a liquid instrument rarely changes by more than a minimum tick. Hence, pricing financial assets in continuous time may proceed quite realistically with just three states of the world, as long as one ignores me events. Unfortunately, most markets for financial assets and derivative products may from time to time exhibit...

## Calculus in Deterministic and Stochastic Environments

The mathematics of derivative assets assumes that time passes continuously. As a result, new information is revealed continuously, and decision-makers may face instantaneous changes in random news. Hence, technical tools for pricing derivative products require ways of handling random variables over infinitesimal time intervals. The mathematics of such random variables is known as stochastic calculus. Stochastic calculus is an internally consistent set of operational rules that are different...

## 1

Clearly, X, is not a martingale with respect to the distribution in Eq. 41 and with respect to the information on current and past Xt. 8It is not clear why the variance of A A', should be proportional to A. For example, is it Possible that question is more complicated to answer than it seems. It will be at the core of the nest We have not yet defined integrals of random incremental changes. But, this last result gives a clue on how to generate a martingale X, . Consider the new process It is...

## Info

They both represent the same thing at time T. So, in the case of gold futures, we can indeed say that the equality in 1 holds at expiration. At t lt T, F t may not equal S,. Yet we can determine a function that ties 5, to F i . Futures and forwards are linear instruments. This section will discuss forwards their differences from futures will be briefly indicated at the end. DEFINITION A forward contract is an obligation to buy sell an underlying asset at a specified forward price on a...

## I

Where f' x is the 'th order derivative of f x with respect to x We are not going to elaborate on why the expansion in 64 is valid if f x is continuous and smooth enough. Taylor series expansion is taken for granted. We will, however, discuss some of its implications. First, note that at this point the expression in 64 is not an approximation. The right-hand side involves an infinite series. Each element involves simple powers of x only, but there are an infinite number of such ele- s ments....

## Primer on the Arbitrage Theorem

All current methods of pricing derivative assets utilize the notion of arbitrage. In arbitrage pricing methods this utilization is direct. Asset prices are obtained from conditions that preclude arbitrage opportunities. In equilibrium pricing methods, lack of arbitrage opportunities is part of general equilibrium conditions. In its simplest form, arbitrage means taking simultaneous positions in different assets so that one guarantees a riskless profit higher than the risklcss return given by...

## S2 Ik

The ift0 is the discount in riskless borrowing. 1. You are given the price of a nondividend paying stock St and a European call option C, in a world where there are only two possible states The true probabilities of the two states are given by P .5, Pd .5 . The current price is St 280. The annual interest rate is constant at r 5 . The time is discrete, with A 3 months. The option has a strike price of X 280 and expires at time t A. a Find the risk-neutral martingale measure P using the...

## St

1.1 The Ito Integral and SDEs 206 1.2 The Practical Relevance of the Ito Integral 207 2.1 The Riemann-Stieltjes Integral 209 2.2 Stochastic Integration and Riemann Sums 211 2.3 Definition The Ito Integral 213 3 Properties of the Ito Integral 220 3.1 The Ito Integral Is a Martingale 220 4 Other Properties of the Ito Integral 226 5 Integrals with Respect to Jump Processes 227 2 Types of Derivatives 231 3.1 The Notion of Size in Stochastic Calculus 235 3.4 Terms involving Cross Products 239 5.1...

## J

Here St and r are the security price and risk-free return, respectively. P is the risk-adjusted probability. According to this, utilization of risk-adjusted probabilities will convert all discounted asset prices into martingales. It is important to realize that, in finance, the notion of martingale is always associated with two concepts. First, a martingale is always defined with respect to a certain probability. Hence, in Section 3,4 the discounted stock price,

## Brief Introduction

This book is an introduction to quantitative tools used in pricing financial derivatives. Hence, it is mainly about mathematics. It is a simple and heuristic introduction to mathematical concepts that have practical use in financial markets. Such an introduction requires a discussion of the logic behind asset pricing. In addition, at various points we provide examples that also require an understanding of formal asset pricing methods. All these necessitate a brief discussion of the securities...

## Bibliography 509 Index 513

This edition is divided into two parts. The first part is essentially the revised and expanded version of the first edition and consists of 15 chapters. The sccond part is entirely new and is made of 7 chapters on more recent and more complex material. Overall, the additions amount to nearly doubling the content of the first edition. The first 15 chapters are revised for typos and other errors and are supplemented by several new sections. The major novelty, however, is in the 7 chapters...