期权、期货及其他衍生产品第9版-赫尔】Ch(5)

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Time Varying Volatility
The variance rate substituted into BSM
should be the average variance rate
Suppose the volatility is 1 for the first year and 2 for the second and third
(equations 27.2 and 27.3, page 631)
dS (r q)dt S
V dzSdBiblioteka a(VL V )dt V adzV
When V and S are uncorrelated a European
option price is the Black-Scholes-Merton price
lk is the expected return from jumps
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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
integrated over the distribution of the average
variance rate
Total accumulated variance at the end of three years is 12 + 222
The 3-year average volatility is given by
32 12 222;
12 222 3
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Stochastic Volatility Models
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The Variance-Gamma Model (page
628-630)
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
q, a parameter defining skewness
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Understanding the VarianceGamma Model
g defines the rate at which information arrives during time T (g is sometimes referred to as measuring economic time)
Conditional on g, ln ST is normal. Its variance proportional to g
There are 3 parameters
v, the variance rate of the gamma process 2, the average variance rate of ln S per unit time
dS / S (r q lk)dt dz dp
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
Chapter 27 More on Models and Numerical Procedures
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Three Alternatives to Geometric Brownian Motion
Constant elasticity of variance (CEV) Mixed Jump diffusion Variance Gamma
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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
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CEV Model (pages 625-626)
dS (r q)Sdt S adz
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
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CEV Models Implied Volatilities
imp
a<1
a> 1
K
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Mixed Jump Diffusion Model
(page 626-628)
Merton produced a pricing formula when the asset price follows a diffusion process overlaid with random jumps
If g is large the change in ln S has a relatively large mean and variance
If g is small relatively little information arrives and the change in ln S has a relatively small mean and variance