1. The Black-Scholes Model for Option Pricing
-Meeting-2,

2. Introduction

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

4. 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

5. Types
European
American
Asian
Vanilla & Exotic

6. 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

7. 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.

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

9. The Black-Scholes Model

10. 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.

11. …. 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.

12. 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

13. … 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.

14. … 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)

15. Pricing the Option
The Black-Scholes Formula: