Unformatted text preview: Hint: One way is to use Lemma 2.3.4. c) Let ψ : (0 , ∞ ) → [0 , ∞ ) be a function such that R ψ dμ = 1. Show that P ∗ [ A ] := Z A ψ ( σ ) d e P is also a risk-neutral measure. d) Show that E ∗ [ ( S T − K ) + ] = Z v ( T,S ,s ) ψ ( s ) μ ( ds ) , where v ( T,S ,s ) is the price of the plain vanilla European call option ( S T − K ) + in a Black-Scholes model with zero interest rate, spot S , and volatility s > 0. e) Conclude that inf P ∗ ∈P E ∗ [ ( S T − K ) + ] = ( S − K ) + and sup P ∗ ∈P E ∗ [ ( S T − K ) + ] = S , where P denotes the set of all risk-neutral measures, and give a verbal interpretation of this fact. Hint: We already know from class and from Exercise 2 in homework set 4 that v ( T,S ,s ) & ( S − K ) + for s ↓ 0 and v ( T,S ,s ) % S as s ↑ ∞ . 1...
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- Spring '08
- Brownian Motion, Mathematical finance, Stochastic volatility, Upson, simple stochastic volatility