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Basic Numerical Procedures Chapter 19 資管所 柯婷瑱 2009/07/17 1 Approaches to Derivatives Valuation • Trees • Monte Carlo simulation • Finite difference methods 2 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 Movements in Time Dt (Figure 19.1, page 408) Su S Sd 4 Generalization (Figure 11.2, page 239) A derivative lasts for time T and is dependent on a stock Su 0 S0 ƒ ƒu S0d ƒd 5 Generalization (continued) • Consider the portfolio that is long D shares and short 1 derivative S0uD – ƒu S0dD – ƒd • The portfolio is riskless when S0uD – ƒu = S0dD – ƒd or ƒu f d D S 0u S 0 d 6 Generalization (continued) • Value of the portfolio at time T is 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 7 Generalization (continued) Substituting for D we obtain ƒ = [ pƒu + (1 – p)ƒd ]e–rT e d where p ud rT 8 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 riskneutral world Mean: Variance: e(r-q)Dt = pu + (1– p )d s2Dt = pu2 + (1– p )d 2 – e2(r-q)Dt A further condition often imposed is u = 1/ d 9 Tree Parameters for asset paying a dividend yield of q (continued) When Dt is small a solution to the equations is ue s Dt d e s Dt ad p ud a e ( r q ) Dt Cox-Ross-Rubinstein binomial tree. 10 The Complete Tree (Figure 19.2, page 410) S0u 3 S0u 4 S0u 2 S0u S0 S0u S0 S0d S 0d S0u 2 S0 S0d 2 S0d 3 S0d 2 S0d 4 11 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 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 13 Example (continued) Figure 19.3, page 411 89.07 0.00 79.35 0.00 70.70 0.00 70.70 0.00 62.99 0.00 62.99 0.64 50.00 2.66 50.00 3.77 50.00 4.49 56.12 0.00 56.12 1.30 56.12 2.16 44.55 5.45 44.55 6.38 44.55 6.96 39.69 10.31 39.69 10.36 35.36 14.64 35.36 14.64 31.50 18.50 28.07 21.93 14 Calculation of Delta Delta is calculated from the nodes at time Dt 2.16 6.96 Delta 0.41 5612 . 44.55 15 Calculation of Gamma Gamma is calculated from the nodes at time 2Dt 0.64 3.77 3.77 10.36 D1 0.24; D 2 0.64 62.99 50 50 39.69 D1 D 2 Gamma = 0.03 1165 . 16 Calculation of Theta Theta is calculated from the central nodes at times 0 and 2Dt 3.77 4.49 Theta = 4.3 per year 01667 . or - 0.012 per calendar day 17 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 4.62 4.49 013 . per 1% change in volatility 18 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 A know dividend yield s0u s0 s0d ex-dividend date 20 21 A known dollar dividend D 59.47495 70.69895 52.98909 62.98909 56.12 50 56.12 56.11698 49.99731 44.545 50 44.5426 39.68514 35.35549 47.20798 39.99731 44.545 44.89298 35.6336 29.68514 33.3186 26.44649 s 50 p 0.5073 u 1.1224 d 0.8909 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 s (r q Dt ) a -d p , a d p is negative u -d negative probabilities Alternative Binomial Tree Instead of setting u = 1/d we can set each of the 2 probabilities to 0.5 and ue ( r q s 2 / 2 ) Dt s Dt d e ( r q s 2 / 2 ) Dt s Dt it’s not s straightforward to calculate delta, gamma, and rho because the tree is no longer centered at the initial stock price 25 Trinomial Tree (Page 424) Su u es 3 Dt d 1/ u Dt s r pu 2 12s 2 2 1 6 2 pm 3 Dt s 2 1 r pd 2 12s 2 6 pu S pm S pd Sd 26 Time Dependent Parameters in a Binomial Tree (page 425) • We have assumed that r, q, rf ,and are s 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 S dt sS dz dS • We can simulate a path by choosing time steps of length Dt and using the discrete version of this DS ˆS Dt sS e Dt where e is a random sample from f(0,1) 29 A More Accurate Approach (Equation 19.15, page 428) Use d ln S ˆ s 2 / 2 dt s dz The discrete version of this is ln S (t Dt ) ln S (t ) ˆ s 2 / 2 Dt se Dt or S (t Dt ) S (t ) e ˆ s / 2 Dt se 2 Dt 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 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 M where M is the number of trails. • A 95% confidence interval for the price of the derivative is therefore given by 1.96 M f 1.96 M 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 7.68 1000 0.24 • 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. 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 89.07 0.00 79.35 0.00 70.70 0.00 62.99 0.64 56.12 2.16 50.00 4.49 70.70 0.00 62.99 0.00 56.12 1.30 50.00 3.77 44.55 6.96 56.12 0.00 50.00 2.66 44.55 6.38 39.69 10.36 44.55 5.45 39.69 10.31 35.36 14.64 35.36 14.64 31.50 18.50 28.07 21.93 40 Example Asian option Using N-step binomial tree and sampling from the 2Npaths that are possible. Trial Path Average stock price Option payoff 1 UUUUD 64.98 14.98 2 UUUDD 59.82 9.82 : : : : : : : : : : : : 9 UUUDU 62.25 12.25 10 DDUUD 45.56 0 Average 7.08 discount 6.79 41