Ch21HullOFOD9thEdition

Chapter 21Basic Numerical Procedures Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014 1 Approaches to Derivatives Valuation Trees Monte Carlo simulation Finite difference methods Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014 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 Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014 3 Movements in Time t (Figure 21.1, page 451) Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014 4 Su Sd S 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)At = pu + (1– p )d Variance: o 2 At = pu 2 + (1– p )d 2 – e 2(r−q)At A further condition often imposed is u = 1/ d Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014 5 Tree Parameters for asset paying a dividend yield of q (continued) When At is small a solution to the equations is Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014 6 t q r t t e a d u d a p e d e u A ÷ A o ÷ A o = ÷ ÷ = = = ) ( The Complete Tree (Figure 21.2, page 453) Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014 7 S 0 u 4 S 0 u 2 S 0 d 2 S 0 d 4 S 0 S 0 u S 0 d S 0 S 0 S 0 u 2 S 0 d 2 S 0 u 3 S 0 u S 0 d S 0 d 3 Backwards I nduction 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 Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014 8 Example: Put Option (Example 21.1, page 453-455) S 0 = 50; K = 50; r =10%; o = 40%; T = 5 months = 0.4167; At = 1 month = 0.0833 In this case Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014 9 5073 0 8909 0 1224 1 8909 0 0084 1 8909 0 1 1224 1 0084 1 12 1 4 0 12 1 1 0 . . . . . . . . / . / . = ÷ ÷ = = = = = = = × p u d e u e a Example (continued; Figure 21.3, page 454) Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014 10 At each node: Upper value = Underlying Asset Price 89.07 Lower value = Option Price 0.00 Values in red are a result of exercise. 79.35 0.00 70.70 70.70 0.00 0.00 62.99 62.99 0.64 0.00 56.12 56.12 56.12 2.16 1.30 0.00 50.00 50.00 50.00 4.49 3.77 2.66 44.55 44.55 44.55 6.96 6.38 5.45 39.69 39.69 10.36 10.31 35.36 35.36 14.64 14.64 31.50 18.50 28.07 21.93 Node Time: 0.0000 0.0833 0.1667 0.2500 0.3333 0.4167 Calculation of Delta Delta is calculated from the nodes at time At Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014 11 Delta = ÷ ÷ = ÷ 216 696 5612 4455 041 . . . . . Calculation of Gamma Gamma is calculated from the nodes at time 2At Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014 12 A A A A 1 2 2 064 377 62 99 50 0 24 377 10 36 50 39 69 064 1165 003 = ÷ ÷ = ÷ = ÷ ÷ = ÷ ÷ = . . . . ; . . . . . . Gamma= 1 =0.5(62.99-50)+0.5(50-39.69) Calculation of Theta Theta is calculated from the central nodes at times 0 and 2At Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014 13 Theta= per year or - . per calendar day 377 4 49 01667 4 3 0012 . . . . ÷ = ÷ 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 Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014 14 4 62 4 49 013 . . . ÷ = per 1% change in volatility Trees for Options on I ndices, Currencies and Futures Contracts As with Black-Scholes-Merton: 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 Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014 15 Binomial Tree for Stock Paying Known Dividends Procedure: Construct a 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-Merton model is used for European options Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014 16 Control Variate Technique Value American option, f A Value European option using same tree, f E Value European option using Black-Scholes – Merton, f BS Option price =f A +(f BS – f E ) Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014 17 Alternative Binomial Tree (page 465) Instead of setting u = 1/d we can set each of the 2 probabilities to 0.5 and Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014 18 t t q r t t q r e d e u A o ÷ A o ÷ ÷ A o + A o ÷ ÷ = = ) 2 / ( ) 2 / ( 2 2 Trinomial Tree (Page 467) Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014 19 6 1 2 12 3 2 6 1 2 12 / 1 2 2 2 2 3 + | | . | \ | o ÷ o A ÷ = = + | | . | \ | o ÷ o A = = = A o r t p p r t p u d e u d m u t S S Sd Su p u p m p d Time Dependent Parameters in a Binomial Tree (page 468) Making r or q a function of time does not affect the geometry of the tree. The probabilities on the tree become functions of time. We can make o a function of time by making the lengths of the time steps inversely proportional to the variance rate. Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014 20 Monte Carlo Simulation and How could you calculate t by randomly sampling points in the square? Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014 21 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 payoffs 4. Calculate mean payoff 5. Discount mean payoff at risk free rate to get an estimate of the value of the option Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014 22 Sampling Stock Price Movements In a risk neutral world the process for a stock price is where is the risk-neutral return We can simulate a path by choosing time steps of length At and using the discrete version of this where c is a random sample from |(0,1) Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014 23 t S t S S A c o + A µ = A ˆ dS S dt S dz = +  µ o µ ˆ A More Accurate Approach (Equation 21.15, page 471) Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014 24 ( ) ( ) ( ) t t e t S t t S t t t S t t S dz dt S d A c o + A o ÷ µ = A + A oc + A o ÷ µ = ÷ A + o + o ÷ µ = or is this of version discrete The Use 2 / ˆ 2 2 2 ) ( ) ( 2 / ˆ ) ( ln ) ( ln 2 / ˆ ln 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 Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014 25 Sampling from Normal Distribution (Page 473) In Excel =NORMSINV(RAND()) gives a random sample from |(0,1) Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014 26 To Obtain 2 Correlated Normal Samples Obtain independent normal samples x 1 and x 2 and set Use a procedure known as Cholesky’s decomposition when samples are required from more than two normal variables (see page 473) 2 2 1 2 1 1 1 µ ÷ + µ = c = c x x x Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014 27 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. Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014 28 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 Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014 29 Determining Greek Letters For A: 1. Make a small change to asset price 2. Carry out the simulation again using the same random number streams 3. Estimate A as the change in the option price divided by the change in the asset price Proceed in a similar manner for other Greek letters Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014 30 Variance Reduction Techniques Antithetic variable technique Control variate technique Importance sampling Stratified sampling Moment matching Using quasi-random sequences Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014 31 Sampling Through the Tree Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014 32 Instead of sampling from the stochastic process we can sample paths randomly through a binomial or trinomial tree to value a derivative At each node that is reached we sample a randon number between 0 and 1. If it is betweeb 0 and p, we take the up branch; if it is between p and 1, we take the down branch Finite Difference Methods Finite difference methods aim to represent the differential equation in the form of a difference equation We form a grid by considering equally spaced time values and stock price values Define ƒ i,j as the value of ƒ at time iAt when the stock price is jAS Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014 33 The Grid Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014 34 Finite Difference Methods (continued) Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014 35 2 2 Set 2 1 or 2 1 1 2 2 1 1 2 2 1 1 2 2 2 2 S Δ ƒ ƒ ƒ S ƒ S Δ S Δ ƒ ƒ S Δ ƒ ƒ S ƒ ΔS ƒ ƒ S ƒ rƒ S ƒ S σ S ƒ S q r t ƒ i,j i,j i,j i,j i,j i,j i,j i,j i,j ÷ + = c c | | . | \ | ÷ ÷ ÷ = c c ÷ = c c = c c + c c ÷ + c c ÷ + ÷ + ÷ + ) ( I mplicit Finite Difference Method Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014 36 : form the of form the of equation an node each for obtain to Set 1 1 1 1 ƒ ƒ c ƒ b ƒ a t Δ ƒ ƒ t ƒ ,j i i,j j i,j j i,j j i,j ,j i + + ÷ + = + + ÷ = c c Explicit Finite Difference Method Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014 37 ƒ c ƒ b ƒ a ƒ j i, j , i S f S f ,j i * j ,j i * j ,j i * j i,j 1 1 1 1 1 2 1 + + + ÷ + + + = + c c c c : form the of equations solving involves This method difference finite explicit the obtain we point ) ( the at are they as point ) ( the at same the be to assumed are and If 2 I mplicit vs Explicit Finite Difference Method The explicit finite difference method is equivalent to the trinomial tree approach The implicit finite difference method is equivalent to a multinomial tree approach Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014 38 I mplicit vs Explicit Finite Difference Methods (Figure 21.16, page 484) Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014 39 ƒ i , j ƒ i +1, j ƒ i +1, j –1 ƒ i +1, j +1 ƒ i +1, j ƒ i , j ƒ i , j –1 ƒ i , j +1 Implicit Method Explicit Method Other Points on Finite Difference Methods It is better to have ln S rather than S as the underlying variable Improvements over the basic implicit and explicit methods: Hopscotch method Crank-Nicolson method Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014 40