Chapter 3

1. Chapter 3 Tree Models for Stocks and Options

2. §1. A Stock Model Assume the stock can move to only two prices over one unit of time Then (q=1 – p)

3. We call pd+qd the drift of the stock price • pu + qd > 1, drift up, • pu + qd < 1, drift down, • pu + qd = 1, no drift. • Over several units of time

5. We have • In general

6. Recombining Trees

7. §2. Pricing a Call Option with the Tree model • Assume • The option expires at t=3, find its value.

8. Stock price tree 133.1 121 108.9 110 99 100 89.1 90 81 72.9 u = 1.1, d = 0.9

9. Sd = dS0 Su = uS0 • The risk-neutral probability is the same for all nodes q = 0.7564

10. For a typical node • We have q 1 - q

11. This is the European call option price Option price tree

12. Risk neutral prob. q = 0.7564 §3. Pricing an American Option • Assume • The put option expires at t = 3, find its value. Same stock data as before

13. Stock price tree 133.1 121 110 108.9 99 100 89.1 90 81 72.9

14. Option price tree 0 0 10.9 = 100 – 89.1 27.1 = 100 – 72.9

15. 10.9 q 1 - q 27.1 Maximal value Recall q = 0.7564, 1 – q = 0.2436 a q x b 1 - q x = e– 0.05 (0.7564∙10.9 + 0.2436∙27.1) = 14.12 But we can get more by exercising the option Profit = 100 – 81 = 19

16. Option price tree 0 0 10.9 27.1

17. 0 0 q q 1 - q 1 - q 0 10.9 x = e– 0.05 (0.7564∙0 + 0.2436∙10.9) = 2.53 Early Exercise = 100 – 99 = 1 x = e– 0.05 (0.7564∙0 + 0.2436∙0) = 0 Early Exercise = 0

18. Option price tree 0 0 10.9 27.1

19. 2.53 0 q q 1 - q 1 - q 19 2.53 x = e– 0.05 (0.7564∙0 + 0.2436∙2.53) = 0.59 Early Exercise = 0 x = e– 0.05 (0.7564∙2.53 + 0.2436∙19) = 6.22 Early Exercise = 10

20. Option price tree 0 0 10.9 27.1

21. What do you think about pricing the American call options? We discussed before that the American call option will never be exercised early (if there is non dividend).

22. §4. Pricing Exotic Option—Knockout Options • Down-Out barrier: Same as the European option except that whenever the price goes below the barrier, the option is cancelled. • Up-Out barrier: Same as the European option except that whenever the price goes above the barrier, the option is cancelled.

23. Assume • The option expires at t = 3 with down-out barrier at $95.

24. Stock price tree 133.1 121 110 108.9 99 100 89.1 90 81 72.9

25. 0 0 0 0 Option price tree 28.1 21.12 15.85 3.9 2.81 11.40

26. Barrier option value < Standard European option value. • The difference in values will be greater when the current stock price is closer to the barrier. Why? • What will different participants do when the stock price is close to the barrier? • Down and out? • Up and out?

27. Another way of calculating the value of the knockout option: Look at paths! Stock price tree uuu 133.1 121 110 108.9 uud or udu 99 100 89.1 90 81 72.9

28. We have P[uuu] = q3 = 0.75643 = 0.4328 P[uud or udu] =2q2(1 – q) = 0.2787 Thus the expectation of the knockout option at t = 3 is E[V3] = 0.4328×28.1 + 0.2787×3.9 = 13.2486. Hence the option value is V0 = e–0.05×3×13.2486 = 11.40.

29. §5. Pricing Exotic Option—Lookback Options • Lookback option: at the expiration date the holder is paid the maximum value of the stock over the whole period. Example: For 3 months with time step of 1 month S0 = 100 u = 1.2 r = 0.05 d = 0.9

30. Stock price tree uuu 172.8 172.8 144 120 129.6 108 100 97.2 90 81 72.9

31. Stock price tree uuu 172.8 172.8 144 uud 144 120 129.6 108 100 97.2 90 81 72.9

32. Stock price tree uuu 172.8 172.8 144 uud 144 120 129.6 udu 129.6 108 100 97.2 90 81 72.9

33. Stock price tree uuu 172.8 172.8 144 uud 144 120 129.6 udu 129.6 duu 129.6 108 100 97.2 90 81 72.9

34. Stock price tree uuu 172.8 172.8 144 uud 144 120 129.6 udu 129.6 duu 129.6 108 100 udd 120 97.2 90 81 72.9

35. Stock price tree uuu 172.8 172.8 144 uud 144 120 129.6 udu 129.6 duu 129.6 108 100 udd 120 97.2 dud 108 90 81 72.9

36. Stock price tree uuu 172.8 172.8 144 uud 144 120 129.6 udu 129.6 duu 129.6 108 100 udd 120 97.2 dud 108 90 ddu 100 81 72.9

37. Stock price tree uuu 172.8 172.8 144 uud 144 120 129.6 udu 129.6 duu 129.6 108 100 udd 120 97.2 dud 108 90 ddu 100 81 72.9 ddd 100

39. What is the risk-neutral probability q? q = (e0.05/12 – 0.9)/(1.2 – 0.9) = 0.34725

40. Thus the expectation of the Lookback option at t = 3 is E[V3] = 115.314. Hence the option value is V0 = e–0.05×3/12×115.314 = 113.88. What is the drawback of this computation? If the time steps n is large, we need to compute 2n nodes.

41. §6. Adjusting the Binomial Tree Model to Real-World Data • Stock price S is a random variable. In a time period △t, the relative return is (S – S0) / S0 • The average relative return during the period is the expectation E[(S – S0) / S0] = E[S / S0 – 1] = E[△S / S0] • The drift parameter μis defined by μ△t = E[S / S0 – 1]

42. We therefore have 1 + μ △t = E[S / S0] This could be estimated from real stock price data. The volatility parameter σis defined by σ2△t = E[(S – S0)/ S0 – μ △t]2 = D[(S – S0)/ S0] This could also be estimated from real stock price data. Note that if there is no volatility, then μ△t = △S / S0 So S = S0 eμ△t

43. How to get information for binomial tree? Recall E[X] = pa + (1 – p)b D[X] = E[X – E[X]]2 = (1 – p)2(a – b)2p + p2(a – b)2(1 – p) = p(1 – p)(a – b)2

44. Thus E[S/S0] = pu + (1 – p)d D[S/S0] = p(1 – p)(u – d)2 We have 1 + μ△t= pu + (1 – p)d σ2△t= p(1 – p)(u – d)2

45. Hull-White Algorithm 1. Derive u, d, p from real stock data for μ, σ: Set p = ½, then we have therefore

46. 2. Determine μ, σfrom real-world data: Assume we have stock prices S0, S1, …, Sn for each period △t . Let Sk = XkSk – 1, k = 1, …, n ThenXk = Sk/Sk – 1is a Bernoulli random variable P(Xk = u) = P(Xk = d) = 1/2 X1, … , Xn are independent samples for S/S0 during the time period △t .

47. Mean Standard deviation

48. Example: (p.64) Netscape Stock in May 1998

49. We thus have If the time period △t = 1 day, then From S0 = 27.25, we can build the stock binomial tree

50. If the time period △t = 7 day, then From S0 = 27.25, we can build the stock binomial tree