The Black-Scholes Model for Option Pricing -Meeting-2, 09-05-2007

Introduction

Reference: Computational Methods for Option Pricing Yves Achdou and Olivier Pironneau SIAM, 2005

Option Pricing: Recap Call (long) Put (short) Buyers (holders) Right to exercise (Buy) (speculate) Right to exercise (Sell) (hedge) Sellers (writers) Obligated to Sell Obligated to Buy

Types European American Asian Vanilla & Exotic

Vanilla European Model Contract that gives the owner a right to buy a fixed number of shares of a specific common stock at a fixed price at a certain date. S or St : Spot price (price of the asset) K: Strike or exercise price T: expiry or maturity date Ct: Price of the Call option Pt: Price of the Put option

Problem Statement An Option has a value. Is it possible to evaluate the market price Ct of the call option at time t, 0  t  T ? Assumptions: No cost for transactions, Transactions are instantaneous, No arbitrage, and Cannot make instantaneous benefits without taking any risks.

Pricing at Maturity ST : Spot price at maturity Value of the call at maturity:

The Black-Scholes Model

Probability: Basics  : a set A : a –algebra of subsets of  P : a nonnegative measure on  such that: P()=1 The triple (,A,P) is called a probability space.

…. Probability: Basics X : a real-valued random variable on (,A,P) is an A–measurable real-valued function on ; For each Borel subset B on R: Filtration : Ft represents a certain past history available at time t.

The Black-Scholes Model A continuous-time model involving a risky asset (St) and a risk-free asset (St0) Evolution of risk-free asset is given by an ODE: r(t) is instantaneous rate If r is contant

… The Black-Scholes Model Evolution of risky asset is a solution to the following stochastic DE Deterministic term (drift): dt , where  is an average rate of growth of the asset price, and Random term that models variations in response to external effects.

… The Black-Scholes Model Bt is a standard Brownian motion on a probability space (,A,P) A real-valued continuous stochastic process whose increments are independent and stationary. t : the volatility (assumed constant)

Pricing the Option The Black-Scholes Formula: