# ex_5 - Hint One way is to use Lemma 2.3.4 c Let ψ(0 ∞...

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ORIE 569 Problem set 5 March 6, 2008 Homework must be turned in by Thursday, April 10, at 10 a.m. Solutions turned in at a later time will not be graded. Place homework in the 569 homework box on the bridge between Upson and Rhodes. Exercise 1. We are considering a very simple stochastic volatility model in which volatil- ity is sampled randomly at time t = 0 and stock prices afterwards follow a corresponding risk-neutral geometric Brownian motion. More precisely, we set interest rates to zero and let, under e P , S t = S 0 e σ f W t 1 2 σ 2 t for a constant S 0 , a Brownian motion f W , and a random variable σ that is independent of f W and has distribution μ ( A ) := Z 0 φ ( x ) I A ( x ) dx. Here, φ : (0 , ) (0 , ) denotes a strictly positive density function with R 0 φ ( x ) dx = 1. a) Let ( F t ) be the Fltration generated by S and show that the random variable σ is F ε -measurable for all ε > 0. Hint: How would you determine the volatility σ if you know (a piece of) the path of S ? b) Show that S is a e P -martingale, i.e., e P is a risk-neutral measure.
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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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