Binomial Option Pricing Model Finance (Derivative Securities) 312 Tuesday, 3 October 2006 Readings: Chapter 11 & 16

Simple Binomial Model  Suppose that: Stock price is currently $20 In three months it will be either $22 or $18 3-month call option has strike price of 21 Stock Price = $22 Option Price = $1 Stock Price = $18 Option Price = $0 Stock price = $20 Option Price = ?

Option Pricing  Consider a portfolio: long  shares, short 1 call option  Portfolio is riskless when 22  – 1 = 18    =  – 1 18 

Option Pricing  Riskless portfolio: long 0.25 shares, short 1 call option  Value of the portfolio in three months: 22 x  0.25 – 1 = 4.50  Value of portfolio today ( r = 12%): 4.5 e –0.12  0.25) =  Value of shares: 0.25  20 = 5  Value of option: 5 – = 0.633

Generalisation  A derivative lasts for time T and is dependent on a stock Su ƒ u Sd ƒ d SƒSƒ

Generalisation  Consider the portfolio that is long  shares and short 1 derivative  The portfolio is riskless when Su  – ƒ u = Sd  – ƒ d or Su  – ƒ u Sd  – ƒ d

Generalisation  Value of portfolio at time T: Su  – ƒ u  Value of portfolio today: (Su  – ƒ u )e –rT  Cost of portfolio today: S  – f  Hence ƒ = S  – ( Su  – ƒ u )e –rT

Generalisation  Substituting for  we obtain ƒ = [ pƒ u + (1 – p)ƒ d ]e –rT where

Risk-Neutral Valuation  Variables p and ( 1 – p ) can be interpreted as the risk-neutral probabilities of up and down movements  Value of a derivative is its expected payoff in a risk-neutral world discounted at the risk-free rate  Expected stock price: pS 0 u + (1 – p)S 0 d Substitute for p, gives S 0 e rT

Risk-Neutral Valuation  Since p is a risk-neutral probability 20 e 0.12(0.25) = 22 p + 18 (1 – p)  p =  Alternatively, using the formula: Su = 22 ƒ u = 1 Sd = 18 ƒ d = 0 S ƒS ƒ p (1  – p )

Risk-Neutral Valuation Su = 22 ƒ u = 1 Sd = 18 ƒ d = 0 SƒSƒ  Value of option: e –0.12(0.25) ( x x 0) = 0.633

Two-Step Tree  Value at node B e –0.12(0.25) ( x x  0) =  Value at node A e –0.12(0.25) ( x  x  0) = A B C D E F

Valuing a Put Option A B C D E F

Valuing American Options A B C D E F

Delta  Delta (  ) is the ratio of the change in the price of a stock option to the change in the price of the underlying stock  The value of  varies from node to node

Determining u and d  Determined from stock volatility

Tree Parameters  Conditions: e r  t = pu + (1 – p)d  2  t = pu 2 + (1 – p)d 2 – [pu + (1 – p)d ] 2 u = 1/ d  Where  t is small:

Complete Tree S0S0 S0uS0u S0dS0d S0S0 S0S0 S0u2S0u2 S 0 d 2 S0u3S0u3 S0uS0u S0dS0d S 0 d 3

Example: Put Option  Parameters S 0 = 50; K = 50; r = 10%;  = 40%; T = 5 months = ;  t = 1 month =  Implying that: u = ; d = ; a = ; p =

Example: Put Option

Effect of Dividends  For known dividend yield: All nodes ex-dividend for stocks multiplied by (1 – δ), where δ is dividend yield  For known dollar dividend: Deduct PV of dividend from initial node Construct tree Add PV of dividend to each node before ex- dividend date