Profesor

CHAPTER

2

Fundamentals of Stochastic Calculus

T

his chapter provides a summary of the concepts of stochastic calculus needed in financial engineering calculations. This chapter is an attempt to condense the fundamentals of a complex subject in a manner that is accessible to readers with a modest mathematical background. Readers who already have a background in stochastic calculus cango directly to the next chapter. The exposition in this chapter is nonrigorous and intuitive. For a more comprehensive treatment of stochastic calculus, the reader may consult the excellent works of Karatzas and Shreve (1988) and Protter (1995). The book by Oksendal (1995) is applications-oriented and highly recommended.

BASIC DEFINITIONS
Unlike regular calculus, which deals with deterministicfunctions, integrals, and differential equations, stochastic calculus deals with stochastic processes, functions of stochastic processes, integrals involving processes, and differential equations involving processes. A stochastic process is defined in a probability space. Before discussing stochastic processes in detail, we elaborate on the elements of the probability space.

PROBABILITY SPACEA stochastic process is defined in a probability space, which we denote by ( , F, P). In the probability space we have the following elements. is the space of all possible outcomes of an observation or experiment, also known as the sample space. I F is known as the filtration. The filtration is a set of so-called “ fields,” or “ -algebras.” The filtration determines, or encodes, the I

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10QUANTITATIVE METHODS IN DERIVATIVES PRICING

information that is revealed by observing the time evolution of the stochastic process. I P is the probability measure. It assigns probabilities to subsets of . We will now describe these items in greater detail.

Sample Space
The outcomes contained in the sample space depend on what we are interested in observing. For example, if we areconsidering the number of times a stock price has moved upward within a period of time, would be a set of integers, where each integer represents a possible number of = {0, 1, …, n}. upward moves by the stock, namely A more relevant example, which will serve as a basis for discussion in the next few sections, is when we are interested in the trajectories of up and down moves of a stock price in a givenperiod of time. would then be the set of up and down sequences that can be observed in that period of time. For example, if the period of time contains three observations, will consist of 23 = 8 sequences, each one indicating the succession of up and down moves, namely, = {uuu, uud, udu, udd, duu, dud, ddu, ddd}. The observations we are interested in are realizations of stochastic processes. (Theprice of a stock, as we will see later, can be characterized by a stochastic process.) Therefore, the sample space of interest to us is the set of possible trajectories of a stochastic process in a given time interval.

Filtration and the Revelation of Information
Information about the true outcome is represented by subsets of . Considering our three-observation example, before any observationis made, we can say the following about the true outcome of the stock trajectory: The true trajectory will not be part of the empty set, ∅, and will be part of the sample space, . Therefore, before any observation is made, information about the true outcome of the price trajectory is represented by the following set of subsets of : F 0 = { ∅, } (2.1)

At the first observation of our threeobservation example, we can say the following about the eventual trajectory that will turn out to be true: a) The true trajectory will not be contained in the empty set, ∅, and will be contained in the sample space, ; and b) The true trajectory will be part of either the set U = {uuu, uud, udu, udd} or the set D = {duu, dud, ddu, duu}. At the first observation, the information about the true outcome...