smile-lecture11

smile-lecture11 - E4718 Spring 2008: Derman: Lecture...

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E4718 Spring 2008: Derman: Lecture 11:Stochastic Volatility Models Cont. Page 1 of 28 5/1/08 smile-lecture11.fm Lecture 11: Stochastic Volatility Models Cont.
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E4718 Spring 2008: Derman: Lecture 11:Stochastic Volatility Models Cont. Page 2 of 28 5/1/08 smile-lecture11.fm 11.1 Valuing Options With Stochastic Volatility Having understood the qualitative features of stochastic volatility models, we return to examining a full stochastic volatility model. Let’s derive a partial dif- ferential equation for valuing an option in the presence of stochastic volatility by extending the Black-Scholes riskless-hedging argument. Assume a general stochastic evolution process for the stock and its volatility are as follows: Eq.11.1 The coefficients and are general functions that can accommodate geometric Brownian motion, mean reversion, or more general behaviors. Now consider an option that has the value and another option , both dependent on the same stochastic vol and stock price, but with different strikes and/or expirations too. We can create a portfolio , short shares of S and short options U to hedge V. From Ito’s lemma, we have Collecting the dt , dS and d σ terms together we get dS μ Sdt σ SdW + = d σ pS σ t ,, () dt q S σ t dZ + = dWdZ ρ = p S σ t qS σ t VS σ t US σ t Π V Δ S δ U = Δδ d Π t V S V σ V d σ 1 2 -- S 2 2 V σ 2 S 2 1 2 σ 2 2 V q 2 S σ 2 V σ ρ ++ + + + = Δ δ t U S U σ U d σ 1 2 S 2 2 U σ 2 S 2 1 2 σ 2 2 U q 2 S σ 2 U σ ρ +++ + + ⎝⎠ ⎜⎟ ⎛⎞ d Π t V 1 2 S 2 2 V σ 2 S 2 1 2 σ 2 2 V q 2 S σ 2 V σ ρ + δ t U 1 2 S 2 2 U σ 2 S 2 1 2 σ 2 2 U q 2 S σ 2 U σ ρ + = S V δ S U Δ d σ σ V δ σ U
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E4718 Spring 2008: Derman: Lecture 11:Stochastic Volatility Models Cont. Page 3 of 28 5/1/08 smile-lecture11.fm We want to create a riskless hedge, and all the risk lies in the and terms. We can eliminate all the randomness by continuous hedging, choosing Δ and δ to satisfy which gives the hedge ratios Eq.11.2 Then with these hedges in place, the change in value of the hedged portfolio is given by Eq.11.3 Since the increase in the value of is deterministic, if there is to be no riskless arbitrage, it must yield the riskless return per unit time, and so Eq.11.4 Comparing the last two equations, we have Eq.11.5 dS d σ S V δ S U Δ –0 = σ V δ σ U = Δ S V δ S U = δ σ V σ U = d Π dt t V 1 2 -- S 2 2 V σ 2 S 2 1 2 σ 2 2 V q 2 S σ 2 V σ qS ρ ++ + δ t U 1 2 S 2 2 U σ 2 S 2 1 2 σ 2 2 U q 2 S σ 2 U σ ρ + ⎝⎠ ⎜⎟ ⎛⎞ = Π d Π r Π r V Δ S δ U [] == t V 1 2 S 2 2 V σ 2 S 2 1 2 σ 2 2 V q 2 S σ 2 V σ ρ rV + δ t U 1 2 S 2 2 U σ 2 S 2 1 2 σ 2 2 U q 2 S σ 2 U σ ρ
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This note was uploaded on 01/31/2011 for the course PSYCH 121 taught by Professor John during the Summer '10 term at UC Davis.

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smile-lecture11 - E4718 Spring 2008: Derman: Lecture...

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