Lecture 18Stochastic Models & Numerical MethodsOptions, Futures, Derivatives / April 7, 2008 1Alternatives to Black-ScholesWe can generalize the underlying stochastic process of the asset to try to better capture thebehavior of b oth the asset and any derivatives based on the stock. There are three important typesof such Levy processes:• A diffusion process is a model with only continuous changes. Our Brownian motion is adiffusion process.• A pure jump model is a m odel in which only jumps occur at se parated times.• A mixed jump-diffusion model is a continuous diffusion model with jumps occurring over thecontinuous process.Options, Futures, Derivatives / April 7, 2008 2The Constant Elasticity of Variance ModelAn constant elasticity of variance (CEV) model is one alternative to Black-Scholes.This is a diffusion model where the risk-neutral process for a stock price S isdS = (r − q)Sdt + σSαdzwhere r is the risk-free rate, q is the dividend yield, dz is a Wiener proce ss, σ is the volatilityparameter, and α is a positive constant.• When α = 1 the CEV model is geometric Brownian motion.• When α < 1 the CEV model the volatility increases as the stock price decreases. This is in linewith the volatility smile.• When α > 1 the CEV model the volatility increases as the stock price increases. This is in linewith the volatility smile.The valuation formulas for European call and put options under the CE model arec = S0e−qTh1 − χ2(a, b + 2, c)i− Ke−rTχ2(c, b, a)p = Ke−rTh1 − χ2(c, b, a)i− S0e−qTχ2(a, b + 2, c)Options, Futures, Derivatives / April 7, 2008 3when 0 < α < 1, andc = S0e−qTh1 − χ2(c, −b, a)i− Ke−rTχ2(a, 2 − b, c)p = Ke−rTh1 − χ2(a, 2 − b, c)i− S0e−qTχ2(c, −b, a)when 1 > α, witha =hKe−(r−q) Ti2(1−α)(1 − α)2vb =11 − αc =S2(1−α)(1 − α)2vwherev =σ22(r − q)(α − 1)he2(r−q )( α−1)T− 1iand χ2(z, k, v) is the cumulative probability that a variable with a non-central χ2distributionwith noncentrality parameter v and k degrees of freedom is less than z.The CEV m odel is useful for valuing e xotic equity options. The parameters of the model can bechosen to fit the prices of plain v anilla options options as closely as possibly by minimizing the sumof the squared differences between model prices and market prices.Options, Futures, Derivatives / April 7, 2008 4Merton’s Mixed Jump-Diffusion ModelAnother method mixes continuous changes with jumps. Setλ ≡ Average number of jumps per yeark ≡ Average jump size measured as a percentage of the asset priceThe percentage jump size is assumed to be drawn from a probability distribution in the model.• The prob. of a jump in time ∆t is λ∆t.• The average growth rate in the asset price from the jumps is λk.• The risk-neutral process for the asset price isdSS= (r − q − λk)dt + σdz + dpwhere dz is the Wiener process, dp is the Poisson process generating the jumps, and σ is thevolatility of the Brownian motion. The processes dz and dp are assumed to be independent ofeach other.Options, Futures, Derivatives / April 7, 2008 5An important e xample of the Merton model is where the logarithm of the size of the percentagejump is normal. Assume that the standard deviation of the normal distribution is s. Then the priceof a European option price is∞Xn=0e−λ0T`λ0T´nn!fnwhere λ0= λ(1 + k). The variable fnis the Black-Scholes option pric e when the dividend yieldq, the variance rate isσ2+ns2Tand the risk-free interest rate isr − λk +nγTwhere γ = ln(1 + k).• The Merton m odel yields heavier left and heavier right tails than Black-Scholes.• Useful for pricing currency options, since these options have heavy right and left tails in thevolatility smile.• As in CEV we choose the model parameters by minimizing the sum of the squared differencebetween model prices and market prices.Options, Futures, Derivatives / April 7, 2008 6Variance-Gamma ModelThe variance-gamma model is a pure jump model. A gamma process is is a pure jump processwhere small jumps occur very frequently and large jumps occur infrequently.To define the model we first define g as a the change over time T in a variable that follows agamma process with mean rate 1 and variance rate of v. The probability density for g isφ(g) =gTv−1e−gvvTvΓ(Tv)where Γ(·) denotes the gamma function.Define STas the asset price as time T , S0denote the asset price today, r the risk-free interestrate, q the dividend yield. Under the variance-gamma model and in a risk-neutral world, ln SThasa normal probability distribution, conditional on g.Options, Futures, Derivatives / April 7, 2008 7Let θ denote a skewness parameter such that• θ = 0 then ln STis symmetric• θ < 0 then ln STis negatively skewed (as for equities)• θ > 0 then ln STis positively skewedThen the c onditional mean isln S0+ (r − q)T + ω + θgand the conditional standard deviation isσ√gwhereω =Tvln 1 − θv −σ2v2!The function g in the variance-gamma distribution can be thought of as a way to controlOptions, Futures, Derivatives / April 7, 2008 8Stochastic Volatility ModelsBlack-Scholes model assumes that volatility is constant, but we know that volatility should changein time. The variance-gamma model reflects this with its parameter g.• Low information and a low volatility• High information and a high volatilityAn alternative to the variance-gamma m odel is a model where the process followed by the volatilityvariable is specified explicitly.Suppose first that we make the volatility parameter in the geom etric Brownian motion is a knownfunction of time. The first neutral process followed by the asset price is thendS = (r − q)Sdt + σ(t)SdzWe can use the EWMA or GARCH(1,1) models at this point to explicitly write σ(t).Options, Futures, Derivatives / April 7, 2008 9However, in practice the variance σ has some stochastic component. This leads t o the followingmodeldSS= (r − q)dt +√V dzdV = a(VL− V )dt + ξVαdzVwhere a, VL, ξ, and α are c onstants and dz and dzVare Wiener processes.• Stochastic volatility models have little impact on short-lived options, but they manifest largedifferences on long-lived options.• Impact of a stochastic volatility on the performance of delta hedging can be large.Options, Futures, Derivatives / April 7, 2008 10Implied Volatility Function (IVF) ModelThe parameters of the m odels we have discussed so far can be chosen so that they provide