Stochastic Processes with Applications to Finance

Masaaki Kijima

• Publisher: Chapman and Hall/CRC
• Year: 2016
• ISBN:
  9781482211535
• Edition: 2
• Topic: Business
• Subtopic: Finance
• Index: Business

Financial engineering has been proven to be a useful tool for risk management, but using the theory in practice requires a thorough understanding of the risks and ethical standards involved. Stochastic Processes with Applications to Finance, Second Edition presents the mathematical theory of financial engineering using only basic mathematical tools

CHAPTER 1

Elementary Calculus: Toward Ito’s Formula

Undoubtedly, one of the most useful formulas in financial engineering is Ito’s formula. A goal of this chapter is to derive Ito’s formula from Taylor’s expansion. The derivation is not mathematically rigorous, but the idea is helpful in many practical situations of financial engineering. Important results from elementary calculus are also presented for the reader’s convenience. See, for example, Bartle (1976) for more details.

1.1 Exponential and Logarithmic Functions

Exponential and logarithmic functions naturally arise in the theory of finance when we consider a continuous-time model. This section summarizes important properties of these functions.

Consider the limit of sequence {a_n} defined by

an=(1+1n)n,n=1,2,....(1.1)

Note that the sequence {a_n} is strictly increasing in n (Exercise 1.1). Associated with the sequence {a_n} is the sequence {b_n} defined by

bn=(1+1n)n+1,n=1,2,....(1.2)

It can be readily shown that the sequence {b_n} is strictly decreasing in n (Exercise 1.1) and a_n < b_n for all n. Since

limn→∞bnan=limn→∞(1+1n)=1,

we conclude that the two sequences {a_n} and {b_n} converge to the same limit. The limit is usually called the base of natural logarithm and denoted by e (the reason for this will become apparent later). That is,

e=limn→∞(1+1n)n.(1.3)

The value is an irrational number (e = 2.718281828459 · · ·), ...

Table of contents

• Cover
• Half Title
• Series Page
• Title Page
• Copyright Page
• Contents
• Preface to the Second Edition
• Preface to the First Edition
• 1 Elementary Calculus: Toward Ito’s Formula
• 2 Elements in Probability
• 3 Useful Distributions in Finance
• 4 Derivative Securities
• 5 Change of Measures and the Pricing of Insurance Products
• 6 A Discrete-Time Model for the Securities Market
• 7 Random Walks
• 8 The Binomial Model
• 9 A Discrete-Time Model for Defaultable Securities
• 10 Markov Chains
• 11 Monte Carlo Simulation
• 12 From Discrete to Continuous: Toward the Black–Scholes
• 13 Basic Stochastic Processes in Continuous Time
• 14 A Continuous-Time Model for the Securities Market
• 15 Term-Structure Models and Interest-Rate Derivatives
• 16 A Continuous-Time Model for Defaultable Securities
• References
• Index