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The Black-Scholes Model for Option Pricing

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. Types. European American Asian Vanilla & Exotic. Vanilla European Model.

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The Black-Scholes Model for Option Pricing

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  1. The Black-Scholes Modelfor Option Pricing -Meeting-2, 09-05-2007

  2. Introduction

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

  4. Option Pricing: Recap

  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. • SorSt : 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:

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