1. Basic Numerical Procedures Chapter 19 1 資管所 柯婷瑱 2009/07/17

2. Approaches to Derivatives Valuation Trees Monte Carlo simulation Finite difference methods 2

3. Binomial Trees Binomial trees are frequently used to approximate the movements in the price of a stock or other asset In each small interval of time the stock price is assumed to move up by a proportional amount u or to move down by a proportional amount d 3

4. Movements in Time  t (Figure 19.1, page 408) 4 Su Sd S p 1 – p

5. Generalization (Figure 11.2, page 239) A derivative lasts for time T and is dependent on a stock 5 S 0 u ƒ u S 0 d ƒ d S0ƒS0ƒ

6. Generalization (continued) Consider the portfolio that is long  shares and short 1 derivative The portfolio is riskless when S 0 u  – ƒ u = S 0 d  – ƒ d or 6 S 0 u  – ƒ u S 0 d  – ƒ d

7. Generalization (continued) Value of the portfolio at time T is S 0 u  – ƒ u Value of the portfolio today is (S 0 u  – ƒ u )e –rT Another expression for the portfolio value today is S 0  – f Hence ƒ = S 0  – ( S 0 u  – ƒ u )e –rT 7

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

9. Tree Parameters for asset paying a dividend yield of q Parameters p, u, and d are chosen so that the tree gives correct values for the mean & variance of the stock price changes in a risk- neutral world Mean: e (r-q)  t = pu + (1– p )d Variance:  2  t = pu 2 + (1– p )d 2 – e 2(r-q)  t A further condition often imposed is u = 1/ d 9

10. Tree Parameters for asset paying a dividend yield of q (continued) When  t is small a solution to the equations is 10 Cox-Ross-Rubinstein binomial tree.

11. The Complete Tree (Figure 19.2, page 410) 11 S0u 2S0u 2 S0u 4S0u 4 S 0 d 2 S0d 4S0d 4 S0S0 S0uS0u S0dS0d S0S0 S0S0 S0u 2S0u 2 S0d 2S0d 2 S0u 3S0u 3 S0uS0u S0dS0d S0d 3S0d 3

12. Backwards Induction We know the value of the option at the final nodes We work back through the tree using risk-neutral valuation to calculate the value of the option at each node, testing for early exercise when appropriate 12

13. Example: Put Option (Example 19.1, page 410) S 0 = 50; K = 50; r =10%;  = 40%; T = 5 months = 0.4167;  t = 1 month = 0.0833 The parameters imply u = 1.1224; d = 0.8909; a = 1.0084; p = 0.5073 13

14. Example (continued) Figure 19.3, page 411 14

15. Calculation of Delta Delta is calculated from the nodes at time  t 15

16. Calculation of Gamma Gamma is calculated from the nodes at time 2  t 16

17. Calculation of Theta Theta is calculated from the central nodes at times 0 and 2  t 17

18. Calculation of Vega We can proceed as follows Construct a new tree with a volatility of 41% instead of 40%. Value of option is 4.62 Vega is 18

19. Trees for Options on Indices, Currencies and Futures Contracts As with Black-Scholes: – For options on stock indices, q equals the dividend yield on the index – For options on a foreign currency, q equals the foreign risk-free rate – For options on futures contracts q = r 19

20. A know dividend yield 20 ex-dividend date

21. 21

22. A known dollar dividend D 22 70.69895 62.98909 56.1256.11698 5049.99731 44.54544.5426 39.68514 35.35549 59.47495 52.98909 56.1247.20798 5039.9973144.89298 44.54535.6336 29.6851433.3186 26.44649 s50 p0.5073 u1.1224 d0.8909

23. Binomial Tree for Stock Paying Known Dividends Procedure: – Draw the tree for the stock price less the present value of the dividends – Create a new tree by adding the present value of the dividends at each node This ensures that the tree recombines and makes assumptions similar to those when the Black-Scholes model is used 23

24. Alternative Binomial Tree time steps are so large that negative probabilities

25. Alternative Binomial Tree Instead of setting u = 1/d we can set each of the 2 probabilities to 0.5 and 25 it’s not s straightforward to calculate delta, gamma, and rho because the tree is no longer centered at the initial stock price

26. Trinomial Tree (Page 424) 26 SS Sd Su pupu pmpm pdpd

27. Time Dependent Parameters in a Binomial Tree (page 425) We have assumed that r, q, r f,and are constants. In practice, they are usually assumed to be time dependent. – r is forward interest rate 27

28. Monte Carlo Simulation and Options 28

29. Sampling Stock Price Movements (Equations 19.13 and 19.14, page 427) In a risk neutral world the process for a stock price is We can simulate a path by choosing time steps of length  t and using the discrete version of this where  is a random sample from  (0,1) 29

30. A More Accurate Approach (Equation 19.15, page 428) 30

31. Monte Carlo Simulation and Options When used to value European stock options, Monte Carlo simulation involves the following steps: 1.Simulate 1 path for the stock price in a risk neutral world 2.Calculate the payoff from the stock option 3.Repeat steps 1 and 2 many times to get many sample payoff 4.Calculate mean payoff 5.Discount mean payoff at risk free rate to get an estimate of the value of the option 31

32. Table 19.2 Monte Carlo Simulation to Check Black-Scholes

33. Extensions When a derivative depends on several underlying variables we can simulate paths for each of them in a risk-neutral world to calculate the values for the derivative 33

34. Number of Trials Denote the mean by μ and the standard deviation by ω. The variable μ is the simulation’s estimate of the value of the derivative. The standard error of the estimate is where M is the number of trails. A 95% confidence interval for the price of the derivative is therefore given by

35. Standard Errors in Monte Carlo Simulation The standard error of the estimate of the option price is the standard deviation of the discounted payoffs given by the simulation trials divided by the square root of the number of observations. To double the accuracy of a simulation, we must quadruple the number of trials. 35

36. Example 19.8 In Table 19.2, the value of the option is calculated as the average of 1000 numbers. The standard deviation of the numbers is 7.68 In this case, ω=7.68 and M=1000. The standard error of the estimate is The spreadsheet therefore gives a 95% confidence interval for the option value as (4.98-1.96x0.24) to (4.98+1.96x0.24), or 4.51 to 5.45.

37. Application of Monte Carlo Simulation Monte Carlo simulation can deal with path dependent options, options dependent on several underlying state variables, and options with complex payoffs It cannot easily deal with American-style options 37

38. Determining Greek Letters For  1. Make a small change to asset price 2. Carry out the simulation again using the same random number streams 3. Estimate  as the change in the option price divided by the change in the asset price Proceed in a similar manner for other Greek letters 38

39. Sampling Through the Tree Instead of sampling from the stochastic process we can sample paths randomly through a binomial or trinomial tree to value a derivative 39 Suppose we have a binomial tree where the probability of an up movement is 0.6. At each node, we sample a random number between 0 and 1.

40. Example 40

41. Example Asian option TrialPathAverage stock priceOption payoff 1UUUUD64.9814.98 2UUUDD59.829.82 ：：：： ：：：： ：：：： 9UUUDU62.2512.25 10DDUUD45.560 Average7.08 discount6.79 41 Using N-step binomial tree and sampling from the 2 N paths that are possible.