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
Monte Carlo simulation：與標的物歷史資料有關，或多個變因影響商品價格
Binomial Trees、 Finite difference methods ：擁有提前執行權的商品
這些數值方法亦可計算delta值、gamma值、vega值

3. 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 Su Sd S p
1 – p

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 )d 2 – 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. Tree of Asset Prices S0u 4 S0u 3 S0u 2 S0u S0 S0d S0d 2 S0d 3 S0d 4
At time iΔt：
S0u 2
S0u 4
S0d 2
S0d 4 S0
S0u
S0d
S0u 3
S0d 3

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

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

10. Example (continued)
In practice, a smaller value of Δt, and many more nodes, would be used. DerivaGem shows：
steps 5 30 50
100
500 f0
4.49
4.263
4.272
4.278
4.283

11. Expressing the Approach Algebraically

12. Estimating Delta and Other Greek Letters
delta（Δ）：at time Δt
S0u 2
S0u 4
S0d 2
S0d 4 S0
S0u
S0d
S0u 3
S0d 3

13. gamma（Γ）： at time 2Δt
S0u 2
S0u 4
S0d 2
S0d 4 S0
S0u
S0d
S0u 3
S0d 3

14. theta（Θ）：
S0u 2
S0u 4
S0d 2
S0d 4 S0
S0u
S0d
S0u 3
S0d 3

15. Vega（ν）：
Rho（ρ）：

16. Example G F E D C B A

17. Using the binomial tree 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

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. Trinomial Trees S Sd Su pu pm pd

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

51. Change of Variable

52. Change of Variable (continued)

53. Relation to Trinomial Tree Approaches
The three probabilities sum to unity.

54. Relation to Trinomial Tree Approaches (continued)

55. Other Finite Difference Methods
Hopscotch method
Crank-Nicolson scheme
Quadratic approximation

56. Summary
We have three different numerical procedures for valuing derivatives when no analytic solution: trees, Monte Carlo simulation, and finite difference methods.
Trees: derivative price are calculated by starting at the end of the tree and working backwards.
Monte Carlo simulation: works forward from the beginning, and becomes relatively more efficient as the number of underlying variables increases.
Finite difference method: similar to tree approaches. The implicit finite difference method is more complicated but has the advantage that does not have to take any special precautions to ensure convergence.