Quantitative Finance for Physicists

Chapter 9

Option Pricing

This chapter begins with an introduction of the notion of financial

derivative in Section 9.1. The general properties of the stock options

are described in Section 9.2. Furthermore, the option pricing theory is

presented using two approaches: the method of the binomial trees

(Section 9.3) and the classical Black-Scholes theory (Section 9.4).

A paradox related to the arbitrage free portfolio paradigm on which

the Black-Scholes theory is based is described in the Appendix section.

9.1 FINANCIAL DERIVATIVES

In finance, derivatives

1

are the instruments whose price depends

on the value of another (underlying) asset [1]. In particular, the

stock option is a derivative whose price depends on the underlying

stock price. Derivatives have also been used for many other assets,

including but not limited to commodities (e.g., cattle, lumber,

copper), Treasury bonds, and currencies.

An example of a simple derivative is a forward contract that obliges

its owner to buy or sell a certain amount of the underlying asset at a

specified price (so-called forward price or delivery price) on a specified

date (delivery date or maturity). The party involved in a contract as a

buyer is said to have a long position, while a seller is said to have a short

position. A forward contract is settled at maturity when the seller

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