《金融期货与期权》（英文版） Chapter 20 More on models and Numerical procedures

20.1 More on models and Numerical Procedures Chapter 20 Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 20.1 More on Models and Numerical Procedures Chapter 20

20.2 Models to be considered Constant elasticity of variance (CEV) Jump diffusion Stochastic volatility Implied volatility function(VF) Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 20.2 Models to be Considered • Constant elasticity of variance (CEV) • Jump diffusion • Stochastic volatility • Implied volatility function (IVF)

20.3 CEV Model (p456 ds=(r-gSsat +os dz When a =1 we have the black- Scholes case When a> 1 volatility rises as stock price rises hen a< 1 volatility falls as stock price rises Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

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 case – When a > 1 volatility rises as stock price rises – When a < 1 volatility falls as stock price rises dS r q Sdt S dz a = ( − ) +

0.4 CEV Models Implied volatilities K Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 20.4 CEV Models Implied Volatilities imp K a 1

20.5 Jump diffusion model (page 457) Merton produced a pricing formula when the stock price follows a diffusion process overlaid with random jumps ds s=(u-nk)dt+odz +dp Ap is the random jump k is the expected size of the jump n 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

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 jumps • 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 dS / S = ( − lk)dt + dz + dp

20.6 Jumps and the smile Jumps have a big effect on the implied volatility of short term options They 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

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 options • They have a much smaller effect on the implied volatility of long term options

20.7 Time varying volatility Suppose the volatility is o for the first year and o, for the second and third Total accumulated variance at the end of three years isσ2+22 The 3-year average volatility is +20 +20 3 Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 20.7 Time Varying Volatility • Suppose the volatility is 1 for the first year and 2 for the second and third • Total accumulated variance at the end of three years is 1 2 + 22 2 • The 3-year average volatility is 2 2 2 2 2 1 2 1 2 2 3 2 ; 3  +   =  +   =

20.8 Stochastic Volatility Models (page 458) S (r-q)dt+vdrs dv=a(vi-v)dt+er dzy When v and s are uncorrelated a European option price is the Black Scholes price integrated over the distribution of the average variance Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

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 of the average variance L V S dV a V V dt V dz r q dt V dz S dS a = − +  = − + ( ) ( )

20.9 The vf Model (page 460) The impied volati lity function model is designed to create a process for the asset price that exactly matches observed option prices. The usual model ds=(r-gSat +osdz is replaced b ds=[r(t)-g(t]dt +o(s, t)Saz Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 20.9 The IVF Model (page 460) dS r t q t dt S t Sdz dS r q Sdt Sdz [ ( ) ( )] ( , ) ( ) = − +  = − +  i s replaced by prices.The usual model price that exactly matches observed option designed to create a process for the asset The impied volatility function model i s

20.10 The volatility function The volatility function that leads to the model matching all European option prices Is l(K,m)2cn/O+q(lme+Kr()-)moK K(0'Cmk aK2) Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

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 option prices is ( ) ( ) [ ( ) ( )] [ ( , )] 2 2 2 2 2 K c K c t q t c K r t q t c K K t mkt mkt mkt mkt     + + −    =