1、Numerical ProceduresChapter 18,Binomial Trees,Binomial 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 d,Movements in
2、Time dt(Figure 18.1),Su,Sd,S,p,1 p,1. Tree Parameters for aNondividend Paying Stock,We choose the tree parameters p, u, and d so that the tree gives correct values for the mean & standard deviation of the stock price changes in a risk-neutral world er dt = pu + (1 p )d s2dt = pu 2 + (1 p )d 2 pu + (
3、1 p )d 2A further condition often imposed is u = 1/ d,2. Tree Parameters for aNondividend Paying Stock(Equations 18.4 to 18.7),When dt is small, a solution to the equations is,The Complete Tree(Figure 18.2),S0,S0u,S0d,S0,S0,S0u2,S0d2,S0u2,S0u3,S0u4,S0d2,S0u,S0d,S0d4,S0d3,Backwards Induction,We know
4、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 appropriate,Example: Put Option,S0 = 50; X = 50; r =10%; s = 40%; T = 5 months = 0.4167; dt = 1 month = 0.0833The pa
5、rameters imply u = 1.1224; d = 0.8909; a = 1.0084; p = 0.5076,Example (continued)Figure 18.3,Calculation of Delta,Delta is calculated from the nodes at time dt,Calculation of Gamma,Gamma is calculated from the nodes at time 2dt,Calculation of Theta,Theta is calculated from the central nodes at times
6、 0 and 2dt,Calculation of Vega,We can proceed as followsConstruct a new tree with a volatility of 41% instead of 40%. Value of option is 4.62Vega is,Trees and Dividend Yields,When a stock price pays continuous dividends at rate q we construct the tree in the same way but set a = e(r q )dt As with Bl
7、ack-Scholes: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 = r,Binomial Tree for Dividend Paying Stock,Procedure:Draw the tree for the stock price less the present value of
8、 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 model is used,Extensions of Tree Approach,Time dependent interest ratesThe control variate technique,Alternativ
9、e Binomial Tree,Instead of setting u = 1/d we can set each of the 2 probabilities to 0.5 and,Trinomial Tree (Page 409),Adaptive Mesh Model,This is a way of grafting a high resolution tree on to a low resolution treeWe need high resolution in the region of the tree close to the strike price and optio
10、n maturity,Monte Carlo Simulation,When used to value European stock options, this 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 payoff4.Calculate mean payoff5.
11、Discount mean payoff at risk free rate to get an estimate of the value of the option,Sampling Stock Price Movements (Equations 18.13 and 18.14, page 411),In a risk neutral world the process for a stock price isWe can simulate a path by choosing time steps of length dt and using the discrete version
12、of thiswhere e is a random sample from f(0,1),A More Accurate Approach(Equation 18.15, page 411),Extensions,When a derivative depends on several underlying variables we can simulate paths for each of them in a risk-neutral world tocalculate the values for the derivative,Sampling from Normal Distribu
13、tion (Page 412),One simple way to obtain a sample from f(0,1) is to generate 12 random numbers between 0.0 & 1.0, take the sum, and subtract 6.0,To Obtain 2 Correlated Normal Samples,Standard Errors in Monte Carlo Simulation,The standard error of the estimate of the option price is the standard devi
14、ation of the discounted payoffs given by the simulation trials divided by the square root of the number of observations.,Application of Monte Carlo Simulation,Monte Carlo simulation can deal with path dependent options, options dependent on several underlying state variables, and options with comple
15、x payoffsIt cannot easily deal with American-style options,Determining Greek Letters,For 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 priceProceed in a simi
16、lar manner for other Greek letters,Variance Reduction Techniques,Antithetic variable techniqueControl variate techniqueImportance samplingStratified samplingMoment matchingUsing quasi-random sequences,Representative Sampling Through the Tree,We can sample paths randomly through a binomial or trinomi
17、al tree to value an optionAn alternative is to choose representative paths Paths are representative if the proportion of paths through each node is approximately equal to the probability of the node being reached,Finite Difference Methods,Finite difference methods aim to represent the differential e
18、quation in the form of a difference equationDefine i,j as the value of at time idt when the stock price is jdS,Finite Difference Methods(continued),Implicit Finite Difference Method (Equation 18.25, page 420),Explicit Finite Difference Method (Equation 18.32, page 422),Implicit vs Explicit Finite Di
19、fference Method,The explicit finite difference method is equivalent to the trinomial tree approachThe implicit finite difference method is equivalent to a multinomial tree approach,Implicit vs Explicit Finite Difference Methods(Figure 18.16, page 422),i +1, j +1,i +1, j,i , j,i , j 1,i , j +1,Implic
20、it Method,Explicit Method,Other Points on Finite Difference Methods,It is better to have ln S rather than S as the underlying variableImprovements over the basic implicit and explicit methods:Hopscotch methodCrank-Nicolson method,The Barone Adesi & Whaley Analytic Approximation for American Call OptionsAppendix 18A, page 433),The Barone Adesi & Whaley Analytic Approximation for American Put Options,
