Basic Numerical Procedure

1. Basic Numerical Procedure

2. Content 1 Binomial Trees 2 Using the binomial tree for options on indices, currencies, and futures contracts 3 Binomial model for a dividend-paying stock 4 Alternative procedures for constructing trees 5 Time-dependent parameters 6 Monte Carlo simulation 7 Variance reduction procedures 8 Finite difference methods

3. Su p S 1 – p Sd Binomial Trees • In each small interval of time （Δt）the stock price is assumed to move up by a proportional amount u or to move down by a proportional amount d

4. Risk-Neutral Valuation 1. Assume that the expected return from all traded assets is the risk-free interest rate. 2. Value payoffs from the derivative by calculating their expected values and discounting at the risk-free interest rate.

5. Determination of p, u, and d Mean: e(r-q)Dt= pu + (1– p )d Variance:s2Dt = pu2 + (1– p )d2 – e2(r-q)Dt A third condition often imposed is u = 1/ d

6. A solution to the equations, when terms of higher order than Dt are ignored, is

7. S0u4 S0u3 S0u2 S0u2 S0u S0u S0 S0 S0 S0d S0d S0d2 S0d 2 S0d3 S0d4 Tree of Asset Prices • At time iΔt：

8. Working Backward through the Tree Example : American put option S0= 50; K = 50; r =10%; s = 40%; T = 5 months = 0.4167; Dt = 1 month = 0.0833 The parameters imply： u = 1.1224; d = 0.8909; a = 1.0084; p = 0.5073

9. F D C E B A G Example (continued)

10. In practice, a smaller value of Δt, and many more nodes, would be used. DerivaGem shows： Example (continued)

11. Expressing the Approach Algebraically

12. S0u4 S0u3 S0u2 S0u2 S0u S0u S0 S0 S0 S0d S0d S0d2 S0d 2 S0d3 S0d4 Estimating Delta and Other Greek Letters • delta（Δ）：at time Δt

13. S0u4 S0u3 S0u2 S0u2 S0u S0u S0 S0 S0 S0d S0d S0d2 S0d 2 S0d3 S0d4 • gamma（Γ）： at time 2Δt

14. S0u4 S0u3 S0u2 S0u2 S0u S0u S0 S0 S0 S0d S0d S0d2 S0d 2 S0d3 S0d4 • theta（Θ）：

15. Vega（ν）： • Rho（ρ）：

16. F D C E B A G Example

17. Using the binomial tree for options on indices, currencies, and futures contracts As with Black-Scholes： • For options on stock indices,qequals the dividend yield on the index • For options on a foreign currency, qequals the foreign risk-free rate • For options on futures contracts： q = r

18. Example

19. Example

20. Binomial model for a dividend-paying stock • Known Dividend Yield： before： after： Several known dividend yields：

21. Known Dollar Dividend： • i≦k： i=k+1： i=k+2：

22. Simplify the problem • The stock price has two components：a part that is uncertain and a part that is the present value of all future dividends during the life of the option. Step 1：A tree can be structured in the usual way to model . Step 2：By adding to the stock price at each nodes, the present value of future dividends, the tree can be converted into model S.

23. Example

24. Control Variate Technique 1. Using the same tree to calculate both the value of the American option（ ）and the value of the European option（ ）. 2. Calculating the Black-Scholes price of the European option（ ）. 3. This gives the estimate of the value of the American option as

25. Example • B-S model： • ∴

26. Alternative procedures for constructing trees • Instead of setting u = 1/d we can set each of the 2 probabilities to 0.5 and

27. Example

28. Su pu pm S S pd Sd Trinomial Trees

29. Adaptive mesh model（Figlewski and Gao,1999）

30. Time-dependent parameters

31. Monte Carlo simulation • When used to value an option, Monte Carlo simulation uses the risk-neutral valuation result. It involves the following steps: 1. Simulate a random path for S in a risk neutral world. 2. Calculate the payoff from the derivative. 3. Repeat steps 1 and 2 to get many sample values of the payoff from the derivative in a risk neutral world. 4. Calculate the mean of the sample payoffs to get an estimate of the expected payoff. 5. Discount this expected payoff at risk-free rate to get an estimate of the value of the derivative.

32. Monte Carlo simulation (continued) • In a risk neutral world the process for a stock price is • We can simulate a path by choosing time steps of length Δt and using the discrete version of this where ε is a random sample from f(0,1)

33. Monte Carlo simulation (continued)

34. Derivatives Dependent on More than One Market Variable • 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：

35. Generating the Random Samples from Normal Distributions • How to get two correlated samples ε1 and ε2 from univariate standard normal distributions x1 and x2？

36. Cholesky decomposition

37. Number of Trials • Denote the mean by μ and the standard deviation by ω. • The standard error of the estimate is where M is the number of trials. • A 95% confidence interval for the price f of the derivative is • To double the accuracy of a simulation, we must quadruple the number of trials.

38. Applications • Advantage： 1. It tends to be numerically more efficient （increases linearly）than other procedures（ increases exponentially）when there are more stochastic variables. 2. It can provide a standard error for the estimates. 3. It is an approach that can accommodate complex payoffs and complex stocastic processes.

39. Applications (continued) • An estimate for the hedge parameter is • Sampling through a Tree：

40. Variance reduction procedures • Antithetic Variable Techniques： standard error of the estimate is • Control Variate Technique：

41. Variance reduction procedures(continued) • Importance Sampling: • Stratified Sampling: • Moment Matching: • Using Quasi-Random Sequences:

42. Finite difference methods • Define ƒi,jas the value of ƒ at time iDt when the stock price is jDS • ΔT=T/N; ΔS=Smax /M

43. Implicit Finite Difference Method • Forward difference approximation • backward difference approximation

44. Implicit Finite Difference Methods(continued)

45. Implicit Finite Difference Methods(continued)

46. Implicit Finite Difference Methods(continued)

47. Explicit Finite Difference Methods

48. Explicit Finite Difference Methods(continued)

49. Explicit Finite Difference Methods(continued)

50. ƒi +1, j +1 ƒi , j +1 ƒi , j ƒi +1, j ƒi , j ƒi +1, j ƒi , j –1 ƒi +1, j –1 Implicit Method Explicit Method Difference between implicit and explicit finite difference methods