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A Comparison of Option Prices Under Different Pricing Measures in a Stochastic Volatility Model with Correlation

  • Vicky Henderson

    ()

  • David Hobson
  • Sam Howison
  • Tino Kluge
Registered author(s):

    This paper investigates option prices in an incomplete stochastic volatility model with correlation. In a general setting, we prove an ordering result which says that prices for European options with convex payoffs are decreasing in the market price of volatility risk. As an example, and as our main motivation, we investigate option pricing under the class of q-optimal pricing measures. The q-optimal pricing measure is related to the marginal utility indifference price of an agent with constant relative risk aversion. Using the ordering result, we prove comparison theorems between option prices under the minimal martingale, minimal entropy and variance-optimal pricing measures. If the Sharpe ratio is deterministic, the comparison collapses to the well known result that option prices computed under these three pricing measures are the same. As a concrete example, we specialize to a variant of the Hull-White or Heston model for which the Sharpe ratio is increasing in volatility. For this example we are able to deduce option prices are decreasing in the parameter q. Numerical solution of the pricing pde corroborates the theory and shows the magnitude of the differences in option price due to varying q. Copyright Springer Science + Business Media, Inc. 2005

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    File URL: http://hdl.handle.net/10.1007/s11147-005-1005-x
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    Article provided by Springer in its journal Review of Derivatives Research.

    Volume (Year): 8 (2005)
    Issue (Month): 1 (June)
    Pages: 5-25

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    Handle: RePEc:kap:revdev:v:8:y:2005:i:1:p:5-25
    Contact details of provider: Web page: http://www.springerlink.com/link.asp?id=102989

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    1. David Heath & Eckhard Platen & Martin Schweizer, 2001. "A Comparison of Two Quadratic Approaches to Hedging in Incomplete Markets," Mathematical Finance, Wiley Blackwell, vol. 11(4), pages 385-413.
    2. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-43.
    3. Hull, John C & White, Alan D, 1987. " The Pricing of Options on Assets with Stochastic Volatilities," Journal of Finance, American Finance Association, vol. 42(2), pages 281-300, June.
    4. Schweizer, Martin, 1999. "A minimality property of the minimal martingale measure," Statistics & Probability Letters, Elsevier, vol. 42(1), pages 27-31, March.
    5. Frey, RĂ¼diger, 1997. "Derivative Asset Analysis in Models with Level-Dependent and Stochastic Volatility," Discussion Paper Serie B 401, University of Bonn, Germany.
    6. Bergman, Yaacov Z & Grundy, Bruce D & Wiener, Zvi, 1996. " General Properties of Option Prices," Journal of Finance, American Finance Association, vol. 51(5), pages 1573-1610, December.
    7. Francesca Biagini & Paolo Guasoni & Maurizio Pratelli, 2000. "Mean-Variance Hedging for Stochastic Volatility Models," Mathematical Finance, Wiley Blackwell, vol. 10(2), pages 109-123.
    8. Alan L. Lewis, 2000. "Option Valuation under Stochastic Volatility," Option Valuation under Stochastic Volatility, Finance Press, number ovsv, December.
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