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Basic Numerical Procedures

Basic Numerical Procedures. Chapter 19. 資管所 柯婷瑱 2009/07/17. Approaches to Derivatives Valuation. Trees Monte Carlo simulation Finite difference methods. Binomial Trees. Binomial trees are frequently used to approximate the movements in the price of a stock or other asset

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Basic Numerical Procedures

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

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

  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

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

  5. S0u ƒu S0 ƒ S0d ƒd Generalization (Figure 11.2, page 239) A derivative lasts for time T and is dependent on a stock

  6. Generalization(continued) • Consider the portfolio that is long D shares and short 1 derivative • The portfolio is riskless when S0uD – ƒu = S0dD – ƒd or S0uD – ƒu S0dD – ƒd

  7. Generalization (continued) • Value of the portfolio at time Tis S0uD – ƒu • Value of the portfolio today is (S0uD – ƒu)e–rT • Another expression for the portfolio value today is S0D – f • Hence ƒ = S0D – (S0uD – ƒu)e–rT

  8. Generalization(continued) Substituting for D we obtain ƒ = [ pƒu + (1 – p)ƒd ]e–rT where

  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)Dt= pu + (1– p )d Variance: s2Dt = pu2 + (1– p )d2 – e2(r-q)Dt A further condition often imposed is u = 1/ d

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

  11. S0u3 S0u2 S0u S0u S0 S0 S0 S0d S0d S0d2 S0d3 The Complete Tree(Figure 19.2, page 410) S0u4 S0u2 S0d 2 S0d4

  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

  13. Example: Put Option(Example 19.1, page 410) 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

  14. Example (continued)Figure 19.3, page 411

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

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

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

  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

  19. Trees 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

  20. A know dividend yield ex-dividend date

  21. A known dollar dividend D

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

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

  25. Su pu pm S S pd Sd Trinomial Tree (Page 424)

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

  27. Monte Carlo Simulation and Options

  28. 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 Dt and using the discrete version of this where e is a random sample from f(0,1)

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

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

  32. 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. 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

  34. 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. 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.

  36. 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. Determining Greek Letters For D: 1. Make a small change to asset price 2. Carry out the simulation again using the same random number streams 3. Estimate D 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. 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 • 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.

  39. Example

  40. Example Asian option • Using N-step binomial tree and sampling from the 2Npaths that are possible.

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