derivatives09-4 - Option Valuation

# derivatives09-4 - Option Valuation - Session 4 Option...

This preview shows pages 1–15. Sign up to view the full content.

Session 4 Option Valuation

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Section Outline Valuing Stock Options by The Black- Scholes Model Options on Stock Indices and Currencies
The Black-Scholes Random Walk Assumption Consider a stock whose price is S In a short period of time of length t the return on the stock ( S / S ) is assumed to be normal with mean μ∆ t and standard deviation μ is expected return and σ is volatility t σ

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
The Lognormal Property These assumptions imply ln S T is normally distributed with mean: and standard deviation : Because the logarithm of S T is normal, S T is lognormally distributed T σ T S ) 2 / ( ln 2 0 σ - μ +
The Lognormal Property continued where φ [ m , v ] is a normal distribution with mean m and variance v [ ] [ ] T T S S T T S S T T 2 2 0 2 2 0 , ) 2 ( ln , ) 2 ( ln ln σ σ - μ φ σ σ - μ + φ or

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
The Lognormal Distribution E S S e S S e e T T T T T ( ) ( ) ( ) = = - 0 0 2 2 2 1 var μ μ σ
The Expected Return The expected value of the stock price is S 0 e μ T The expected return on the stock with continuous compounding is μ σ 2 /2 The arithmetic mean of the returns over short periods of length t is μ The geometric mean of these returns is μ σ 2 /2

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
The Volatility The volatility is the standard deviation of the continuously compounded rate of return in 1 year The standard deviation of the return in time t is If a stock price is \$50 and its volatility is 25% per year what is the standard deviation of the price change in one day? t σ
Estimating Volatility from Historical Data 1. Take observations S 0 , S 1 , . . . , S n at intervals of τ years 2. Define the continuously compounded return as: 3. Calculate the standard deviation, s , of the u i ´s 4. The historical volatility estimate is: u S S i i i = - ln 1 τ = σ s ˆ

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Nature of Volatility Volatility is usually much greater when the market is open (i.e. the asset is trading) than when it is closed For this reason time is usually measured in “trading days” not calendar days when options are valued
The Concepts Underlying Black- Scholes The option price and the stock price depend on the same underlying source of uncertainty We can form a portfolio consisting of the stock and the option which eliminates this source of uncertainty The portfolio is instantaneously riskless and must instantaneously earn the risk-free rate

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
The Black-Scholes Formulas T d T T r K S d T T r K S d d N S d N e K p d N e K d N S c rT rT σ - = σ σ - + = σ σ + + = - - - = - = - - 1 0 2 0 1 1 0 2 2 1 0 ) 2 / 2 ( ) / ln( ) 2 / 2 ( ) / ln( ) ( ) ( ) ( ) ( where
Rearrangement of d1, d2 T T Ke S d T T Ke S d rT rT σ 2 1 ) ln( 2 1 ) ln( 2 1 - = + = - -

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Properties of B-S formula When S/Ke -rT increases, the chances of exercising the call option increase, from the formula, d1 and d2 increase and N(d1)
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 02/21/2010 for the course FINA 221 taught by Professor Na during the Spring '09 term at HKUST.

### Page1 / 51

derivatives09-4 - Option Valuation - Session 4 Option...

This preview shows document pages 1 - 15. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online