文档介绍：Chapter 26 More on Models and Numerical Procedures
Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012
1
Three Alternatives to Geometric Brownian Motion
Constant elasticity of variance (CEV)
Mixed Jump diffusion
Variance Gamma
Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012
2
CEV Model (pages 600-601)
When a = 1 the model is Black-Scholes
When a > 1 volatility rises as stock price rises
When a < 1 volatility falls as stock price rises
European options can be value analytically in terms of the cumulative non-central chi square distribution
Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012
3
CEV Models Implied Volatilities
Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012
4
simp
K
a < 1
a > 1
Mixed Jump Diffusion Model (page 601-603)
Merton produced a pricing formula when the asset 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
lk is the expected return from jumps
Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012
5
Simulating a Jump Process
In each time step
Sample from a binomial distribution to determine the number of jumps
Sample to determine the size of each jump
Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012
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, 8th Edition, Copyright © John C. Hull 2012
7
The Variance-Gamma Model (page 603-605)
Define g as change over time T in a variable that follows a gamma process. This is a process where small jumps occur frequently and there are occasional large jumps
Conditional on