1、Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,20.1,More on Models and Numerical ProceduresChapter 20,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,20.2,Models to be Considered,Constant elasticity of variance (CEV)Jump diffusionStochastic volati
2、lityImplied volatility function (IVF),Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,20.3,CEV Model (p456),When a = 1 we have the Black-Scholes caseWhen a 1 volatility rises as stock price risesWhen a 1 volatility falls as stock price rises,Options, Futures, and Other Deri
3、vatives, 5th edition 2002 by John C. Hull,20.4,CEV Models Implied Volatilities,K,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,20.5,Jump Diffusion Model (page 457),Merton produced a pricing formula when the stock price follows a diffusion process overlaid with random jump
4、s,dp is the random jump k is the expected size of the jump l dt is the probability that a jump occurs in the next interval of length dt,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,20.6,Jumps and the Smile,Jumps have a big effect on the implied volatility of short term o
5、ptionsThey have a much smaller effect on the implied volatility of long term options,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,20.7,Time Varying Volatility,Suppose the volatility is s1 for the first year and s2 for the second and thirdTotal accumulated variance at the
6、 end of three years is s12 + 2s22The 3-year average volatility is,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,20.8,Stochastic Volatility Models (page 458),When V and S are uncorrelated a European option price is the Black-Scholes price integrated over the distribution o
7、f the average variance,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,20.9,The IVF Model (page 460),Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,20.10,The Volatility Function,The volatility function that leads to the model matching all European
8、 option prices is,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,20.11,Strengths and Weaknesses of the IVF Model,The model matches the probability distribution of stock prices assumed by the market at each future timeThe models does not necessarily get the joint probabilit
9、y distribution of stock prices at two or more times correct,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,20.12,Numerical Procedures,Topics:Path dependent options using treesLookback optionsBarrier optionsOptions where there are two stochastic variablesAmerican options us
10、ing Monte Carlo,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,20.13,Path Dependence: The Traditional View,Backwards induction works well for American options. It cannot be used for path-dependent optionsMonte Carlo simulation works well for path-dependent options; it cann
11、ot be used for American options,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,20.14,Extension of Backwards Induction,Backwards induction can be used for some path-dependent optionsWe will first illustrate the methodology using lookback options and then show how it can be
12、used for Asian options,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,20.15,Lookback Example (Page 462),Consider an American lookback put on a stock whereS = 50, s = 40%, r = 10%, dt = 1 month & the life of the option is 3 monthsPayoff is Smax-ST We can value the deal by c
13、onsidering all possible values of the maximum stock price at each node (This example is presented to illustrate the methodology. A more efficient ways of handling American lookbacks is in Section 20.6.),Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,20.16,Example: An Ameri
14、can Lookback Put Option (Figure 20.2, page 463),S0 = 50, s = 40%, r = 10%, dt = 1 month,A,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,20.17,Why the Approach Works,This approach works for lookback options becauseThe payoff depends on just 1 function of the path followed
15、by the stock price. (We will refer to this as a “path function”)The value of the path function at a node can be calculated from the stock price at the node & from the value of the function at the immediately preceding nodeThe number of different values of the path function at a node does not grow to
16、o fast as we increase the number of time steps on the tree,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,20.18,Extensions of the Approach,The approach can be extended so that there are no limits on the number of alternative values of the path function at a nodeThe basic i
17、dea is that it is not necessary to consider every possible value of the path function It is sufficient to consider a relatively small number of representative values of the function at each node,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,20.19,Working Forward,First wor
18、k forwards through the tree calculating the max and min values of the “path function” at each nodeNext choose representative values of the path function that span the range between the min and the maxSimplest approach: choose the min, the max, and N equally spaced values between the min and max,Opti
19、ons, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,20.20,Backwards Induction,We work backwards through the tree in the usual way carrying out calculations for each of the alternative values of the path function that are considered at a nodeWhen we require the value of the derivati
20、ve at a node for a value of the path function that is not explicitly considered at that node, we use linear or quadratic interpolation,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,20.21,Part of Tree to Calculate Value of an Option on the Arithmetic Average(Figure 20.2, p
21、age 464),0.5056,0.4944,S=50, X=50, s=40%, r=10%, T=1yr, dt=0.05yr. We are at time 4dt,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,20.22,Part of Tree to Calculate Value of an Option on the Arithmetic Average (continued),Consider Node X when the average of 5 observations
22、is 51.44Node Y: If this is reached, the average becomes 51.98. The option price is interpolated as 8.247 Node Z: If this is reached, the average becomes 50.49. The option price is interpolated as 4.182Node X: value is (0.50568.247 + 0.49444.182)e0.10.05 = 6.206,Options, Futures, and Other Derivative
23、s, 5th edition 2002 by John C. Hull,20.23,A More Efficient Approach for Lookbacks (Section 20.6, page 465),Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,20.24,Using Trees with Barriers(Section 20.7, page 467),When trees are used to value options with barriers, convergence
24、 tends to be slowThe slow convergence arises from the fact that the barrier is inaccurately specified by the tree,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,20.25,True Barrier vs Tree Barrier for a Knockout Option: The Binomial Tree Case,Barrier assumed by treeTrue bar
25、rier,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,20.26,True Barrier vs Tree Barrier for a Knockout Option: The Trinomial Tree Case,Barrier assumed by treeTrue barrier,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,20.27,Alternative Solutions t
26、o the Problem,Ensure that nodes always lie on the barriersAdjust for the fact that nodes do not lie on the barriersUse adaptive meshIn all cases a trinomial tree is preferable to a binomial tree,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,20.28,Modeling Two Correlated V
27、ariables (Section 20.8, page 472),APPROACHES:1.Transform variables so that they are not correlated & build the tree in the transformed variables2.Take the correlation into account by adjusting the position of the nodes3.Take the correlation into account by adjusting the probabilities,Options, Future
28、s, and Other Derivatives, 5th edition 2002 by John C. Hull,20.29,Monte Carlo Simulation and American Options,Two approaches:The least squares approachThe exercise boundary parameterization approachConsider a 3-year put option where the initial asset price is 1.00, the strike price is 1.10, the risk-
29、free rate is 6%, and there is no income,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,20.30,Sampled Paths,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,20.31,The Least Squares Approach (page 474),We work back from the end using a least squares
30、approach to calculate the continuation value at each timeConsider year 2. The option is in the money for five paths. These give observations on S of 1.08, 1.07, 0.97, 0.77, and 0.84. The continuation values are 0.00, 0.07e-0.06, 0.18e-0.06, 0.20e-0.06, and 0.09e-0.06,Options, Futures, and Other Deri
31、vatives, 5th edition 2002 by John C. Hull,20.32,The Least Squares Approach continued,Fitting a model of the form V=a+bS+cS2 we get a best fit relationV=-1.070+2.983S-1.813S2for the continuation value VThis defines the early exercise decision at t=2. We carry out a similar analysis at t=1,Options, Fu
32、tures, and Other Derivatives, 5th edition 2002 by John C. Hull,20.33,The Least Squares Approach continued,In practice more complex functional forms can be used for the continuation value and many more paths are sampled,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,20.34,T
33、he Early Exercise Boundary Parametrization Approach (page 477),We assume that the early exercise boundary can be parameterized in some wayWe carry out a first Monte Carlo simulation and work back from the end calculating the optimal parameter valuesWe then discard the paths from the first Monte Carl
34、o simulation and carry out a new Monte Carlo simulation using the early exercise boundary defined by the parameter values.,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,20.35,Application to Example,We parameterize the early exercise boundary by specifying a critical asset price, S*, below which the option is exercised.At t=1 the optimal S* for the eight paths is 0.88. At t=2 the optimal S* is 0.84In practice we would use many more paths to calculate the S*,
