Basic Numerical Procedures

Chapter 21Basic Numerical ProceduresOptions, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 201411Approaches to Derivatives Valuation

TreesMonte Carlo simulationFinite difference methodsOptions, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 201422Binomial TreesBinomial trees are frequently used to approximate the movements in the price of a stock or other assetIn 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 dOptions, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 201433Movements in Time Dt(Figure 21.1, page 451) Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 20144Su SdS p1 p4Tree Parameters for asset paying a dividend yield of qParameters 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(rq)Dt = pu + (1 p )d Variance:s2Dt = pu2 + (1 p )d 2 e2(rq)Dt

A further condition often imposed is u = 1/ d Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 201455

Tree Parameters for asset paying a dividend yield of q(continued)When Dt is small a solution to the equations isOptions, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 20146

6The Complete Tree(Figure 21.2, page 453)Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 20147S0u4 S0u2 S0d2 S0d4 S0 S0u S0d S0 S0 S0u2 S0d2 S0u3 S0u S0d S0d3 7Backwards InductionWe know the value of the option at the final nodesWe work back through the tree using risk-neutral valuation to calculate the value of the option at each node, testing for early exercise when appropriateOptions, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 201488Example: Put Option(Example 21.1, page 453-455)S0 = 50; K = 50; r =10%; s = 40%; T = 5 months = 0.4167; Dt = 1 month = 0.0833In this case

Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 20149

9Example (continued; Figure 21.3, page 454)

Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 201410

10Calculation of DeltaDelta is calculated from the nodes at time Dt

Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 201411

11Calculation of GammaGamma is calculated from the nodes at time 2Dt

Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 201412

=0.5(62.99-50)+0.5(50-39.69)12Calculation of ThetaTheta is calculated from the central nodes at times 0 and 2Dt

Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 201413

13Calculation of VegaWe can proceed as followsConstruct a new tree with a volatility of 41% instead of 40%. Value of option is 4.62Vega is

Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 201414

14Trees for Options on Indices, Currencies and Futures ContractsAs with Black-Scholes-Merton:For options on stock indices, q equals the dividend yield on the indexFor options on a foreign currency, q equals the foreign risk-free rateFor options on futures contracts q = rOptions, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 20141515Binomial Tree for Stock Paying Known DividendsProcedure:Construct a tree for the stock price less the present value of the dividendsCreate a new tree by adding the present value of the dividends at each nodeThis ensures that the tree recombines and makes assumptions similar to those when the Black-Scholes-Merton model is used for European optionsOptions, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 20141616Control Variate TechniqueValue American option, fAValue European option using same tree, fEValue European option using Black-Scholes Merton, fBSOption price =fA+(fBS fE) Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 20141717Alternative Binomial Tree(page 465)Instead of setting u = 1/d we can set each of the 2 probabilities to 0.5 andOptions, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 201418

18Trinomial Tree (Page 467)Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 201419

SSSdSupupmpd19Time 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 s 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 20142020Monte Carlo Simulation and pHow could you calculate p by randomly sampling points in the square?Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 201421

21Monte 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 world2.Calculate the payoff from the stock option3.Repeat steps 1 and 2 many times to get many sample payoffs4.Calculate mean payoff5.Discount mean payoff at risk free rate to get an estimate of the value of the optionOptions, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 20142222Sampling Stock Price Movements

In a risk neutral world the process for a stock price is

where is the risk-neutral returnWe 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)Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 201423

23A More Accurate Approach(Equation 21.15, page 471)Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 201424

24ExtensionsWhen 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 20142525Sampling from Normal Distribution (Page 473)In Excel =NORMSINV(RAND()) gives a random sample from f(0,1) Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 20142626To Obtain 2 Correlated Normal SamplesObtain independent normal samples x1 and x2 and set

Use a procedure known as Choleskys decomposition when samples are required from more than two normal variables (see page 473)

Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 20142727Standard Errors in Monte Carlo SimulationThe 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 20142828Application of Monte Carlo SimulationMonte Carlo simulation can deal with path dependent options, options dependent on several underlying state variables, and options with complex payoffsIt cannot easily deal with American-style optionsOptions, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 20142929Determining Greek LettersFor D:1. Make a small change to asset price2. Carry out the simulation again using the same random number streams3. 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 lettersOptions, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 20143030Variance Reduction TechniquesAntithetic variable techniqueControl variate techniqueImportance samplingStratified samplingMoment matchingUsing quasi-random sequencesOptions, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 20143131Sampling Through the TreeOptions, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 201432Instead of sampling from the stochastic process we can sample paths randomly through a binomial or trinomial tree to value a derivativeAt 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 branch32Finite Difference MethodsFinite difference methods aim to represent the differential equation in the form of a difference equationWe form a grid by considering equally spaced time values and stock price valuesDefine i,j as the value of at time iDt when the stock price is jDSOptions, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 20143333The Grid

Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 20143434Finite Difference Methods(continued)Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 201435

35Implicit Finite Difference MethodOptions, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 201436

36Explicit Finite Difference MethodOptions, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 201437

37Implicit vs Explicit Finite Difference MethodThe explicit finite difference method is equivalent to the trinomial tree approachThe implicit finite difference method is equivalent to a multinomial tree approachOptions, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 20143838 Implicit vs Explicit Finite Difference Methods (Figure 21.16, page 484) Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 201439i , ji +1, j i +1, j 1i +1, j +1i +1, ji , ji , j 1i , j +1Implicit MethodExplicit Method39Other Points on Finite Difference MethodsIt is better to have ln S rather than S as the underlying variableImprovements over the basic implicit and explicit methods:Hopscotch methodCrank-Nicolson methodOptions, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 20144040