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

2. 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 = ?

3. Option Pricing  Consider a portfolio: long  shares, short 1 call option  Portfolio is riskless when 22  – 1 = 18    = 0.25 22  – 1 18 

4. 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) = 4.3670  Value of shares: 0.25  20 = 5  Value of option: 5 – 4.367 = 0.633

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

6. 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

7. 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

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

9. 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

10. Risk-Neutral Valuation  Since p is a risk-neutral probability 20 e 0.12(0.25) = 22 p + 18 (1 – p)  p = 0.6523  Alternatively, using the formula: Su = 22 ƒ u = 1 Sd = 18 ƒ d = 0 S ƒS ƒ p (1  – p )

11. Risk-Neutral Valuation Su = 22 ƒ u = 1 Sd = 18 ƒ d = 0 SƒSƒ 0.6523 0.3477  Value of option: e –0.12(0.25) (0.6523 x 1 + 0.3477 x 0) = 0.633

12. Two-Step Tree  Value at node B e –0.12(0.25) (0.6523 x 3.2 + 0.3477 x  0) = 2.0257  Value at node A e –0.12(0.25) (0.6523 x  2.0257 + 0.3477 x  0) = 1.2823 20 1.2823 22 18 24.2 3.2 19.8 0.0 16.2 0.0 2.0257 0.0 A B C D E F

13. Valuing a Put Option 50 4.1923 60 40 72 0 48 4 32 20 1.4147 9.4636 A B C D E F

14. Valuing American Options 50 5.0894 60 40 72 0 48 4 32 20 1.4147 12.0 A B C D E F

15. 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

16. Determining u and d  Determined from stock volatility

17. 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:

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

19. Example: Put Option  Parameters S 0 = 50; K = 50; r = 10%;  = 40%; T = 5 months = 0.4167;  t = 1 month = 0.0833  Implying that: u = 1.1224 ; d = 0.8909 ; a = 1.0084 ; p = 0.5076

20. Example: Put Option

21. 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