Hull经典衍生品教科书第9版官方PPT-第21章.ppt

Click to edit Master title style,Click to edit Master text styles,Second level,Third level,Fourth level,Fifth level,*,Options,Futures,and Other Derivatives,9th Edition,Copyright John C.Hull 2014,*,Chapter 21Basic Numerical Procedures,1,Options,Futures,and Other Derivatives,9th Edition,Copyright John C.Hull 2014,Approaches to Derivatives Valuation,Trees,Monte Carlo simulation,Finite difference methods,2,Options,Futures,and Other Derivatives,9th Edition,Copyright John C.Hull 2014,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,Options,Futures,and Other Derivatives,9th Edition,Copyright John C.Hull 2014,Movements in Time,D,t,(Figure 21.1,page 451),Su,Sd,S,p,1,p,4,Options,Futures,and Other Derivatives,9th Edition,Copyright John C.Hull 2014,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,(,rq,),D,t,=,pu,+(1,p,),d,Variance:,s,2,D,t,=,pu,2,+(1,p,),d,2,e,2(,rq,),D,t,A further condition often imposed is,u,=1/,d,5,Options,Futures,and Other Derivatives,9th Edition,Copyright John C.Hull 2014,Tree Parameters for asset paying a dividend yield of q,(continued),When,D,t,is small a solution to the equations is,6,Options,Futures,and Other Derivatives,9th Edition,Copyright John C.Hull 2014,The Complete Tree,(Figure 21.2,page 453),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,3,7,Options,Futures,and Other Derivatives,9th Edition,Copyright John C.Hull 2014,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,8,Options,Futures,and Other Derivatives,9th Edition,Copyright John C.Hull 2014,Example:Put Option,(Example 21.1,page 453-455),S,0,=50;,K,=50;,r,=10%;,s,=40%;,T,=5 months=0.4167;,D,t,=1 month=0.0833,In this case,9,Options,Futures,and Other Derivatives,9th Edition,Copyright John C.Hull 2014,Example,(continued;Figure 21.3,page 454),10,Options,Futures,and Other Derivatives,9th Edition,Copyright John C.Hull 2014,Calculation of Delta,Delta is calculated from the nodes at time,D,t,11,Options,Futures,and Other Derivatives,9th Edition,Copyright John C.Hull 2014,Calculation of Gamma,Gamma is calculated from the nodes at time 2,D,t,=0.5(62.99-50)+0.5(50-39.69),12,Options,Futures,and Other Derivatives,9th Edition,Copyright John C.Hull 2014,Calculation of Theta,Theta is calculated from the central nodes at times,0,and 2,D,t,13,Options,Futures,and Other Derivatives,9th Edition,Copyright John C.Hull 2014,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,14,Options,Futures,and Other Derivatives,9th Edition,Copyright John C.Hull 2014,Trees for Options on Indices,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,15,Options,Futures,and Other Derivatives,9th Edition,Copyright John C.Hull 2014,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,16,Options,Futures,and Other Derivatives,9th Edition,Copyright John C.Hull 2014,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,),17,Options,Futures,and Other Derivatives,9th Edition,Copyright John C.Hull 2014,Alternative Binomial Tree,(page 465),Instead of setting,u,=1/,d,we can set each of the 2 probabilities to 0.5 and,18,Options,Futures,and Other Derivatives,9th Edition,Copyright John C.Hull 2014,Trinomial Tree,(Page 467),S,S,Sd,Su,p,u,p,m,p,d,19,Options,Futures,and Other Derivatives,9th Edition,Copyright John C.Hull 2014,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,s,a function of time by making the lengths of the time steps inversely proportional to the variance rate.,20,Options,Futures,and Other Derivatives,9th Edition,Copyright John C.Hull 2014,Monte Carlo Simulation and,p,How could you calculate,p,by randomly sampling points in the square?,21,Options,Futures,and Other Derivatives,9th Edition,Copyright John C.Hull 2014,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,22,Options,Futures,and Other Derivatives,9th Edition,Copyright John C.Hull 2014,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,D,t,and using the discrete version of this,where,e,is a random sample from,f,(0,1),23,Options,Futures,and Other Derivatives,9th Edition,Copyright John C.Hull 2014,A More Accurate Approach,(Equation 21.15,page 471),24,Options,Futures,and Other Derivatives,9th Edition,Copyright John C.Hull 2014,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,25,Options,Futures,and Other Derivatives,9th Edition,Copyright John C.Hull 2014,Sampling from Normal Distribution,(Page 473),In Excel =NORMSINV(RAND()gives a random sample from,f,(0,1),26,Options,Futures,and Other Derivatives,9th Edition,Copyright John C.Hull 2014,To Obtain 2 Correlated Normal Samples,Obtain independent normal samples,x,1,and,x,2,and set,Use a procedure known as Choleskys decomposition when samples are required from more than two normal variables(see page 473),27,Options,Futures,and Other Derivatives,9th Edition,Copyright John C.Hull 2014,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.,28,Options,Futures,and Other Derivatives,9th Edition,Copyright John C.Hull 2014,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,29,Options,Futures,and Other Derivatives,9th Edition,Copyright John C.Hull 2014,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,30,Options,Futures,and Other Derivatives,9th Edition,Copyright John C.Hull 2014,Variance Reduction Techniques,Antithetic variable technique,Control variate technique,Importance sampling,Stratified sampling,Moment matching,Using quasi-random sequences,31,Options,Futures,and Other Derivatives,9th Edition,Copyright John C.Hull 2014,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,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,32,Options,Futures,and Other Derivatives,9th Edition,Copyright John C.Hull 2014,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,i,j,as the value of,at time,i,D,t,when the stock price is,j,D,S,33,Options,Futures,and Other Derivatives,9th Edition,Copyright John C.Hull 2014,The Grid,34,Options,Futures,and Other Derivatives,9th Edition,Copyright John C.Hull 2014,Finite Difference Methods,(continued),35,Options,Futures,and Other Derivatives,9th Edition,Copyright John C.Hull 2014,Implicit Finite Difference Method,36,Options,Futures,and Other Derivatives,9th Edition,Copyright John C.Hull 2014,Explicit Finite Difference Method,37,Options,Futures,and Other Derivatives,9th Edition,Copyright John C.Hull 2014,Implicit 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,38,Options,Futures,and Other Derivatives,9th Edition,Copyright John C.Hull 2014,Implicit vs Explicit Finite Difference Methods,(Figure 21.16,page 484),i,j,i,+1,j,i,+1,j,1,i,+1,j,+1,i,+1,j,i,j,i,j,1,i,j,+1,Implicit Method,Explicit Method,39,Options,Futures,and Other Derivatives,9th Edition,Copyright John C.Hull 2014,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,40,Options,Futures,and Other Derivatives,9th Edition,Copyright John C.Hull 2014,