CH20 More on Models and Numerical Procedures.ppt

More on Models and Numerical ProceduresChapter 20
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Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
Models to be Considered
Constant elasticity of variance (CEV)
Jump diffusion
Stochastic volatility
Implied volatility function (IVF)
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Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
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
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Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
CEV Models Implied Volatilities
simp
K
a < 1
a > 1
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Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
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
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Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
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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Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
Time Varying Volatility
Suppose the volatility is s1 for the first year and s2 for the second and third
Total accumulated variance at the end of three years is s12 + 2s22
The 3-year average volatility is
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Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
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
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Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
The IVF Model (page 460)
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Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
The Volatil