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Partial Hedging In A Stochastic Volatility Environment

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  • Mattias Jonsson
  • K. Ronnie Sircar

Abstract

We consider the problem of partial hedging of derivative risk in a stochastic volatility environment. It is related to state‐dependent utility maximization problems in classical economics. We derive the dual problem from the Legendre transform of the associated Bellman equation and interpret the optimal strategy as the perfect hedging strategy for a modified claim. Under the assumption that volatility is fast mean‐reverting and using a singular perturbation analysis, we derive approximate value functions and strategies that are easy to implement and study. The analysis identifies the usual mean historical volatility and the harmonically averaged long‐run volatility as important statistics for such optimization problems without further specification of a stochastic volatility model. The approximation can be improved by specifying a model and can be calibrated for the leverage effect from the implied volatility skew. We study the effectiveness of these strategies using simulated stock paths.

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  • Mattias Jonsson & K. Ronnie Sircar, 2002. "Partial Hedging In A Stochastic Volatility Environment," Mathematical Finance, Wiley Blackwell, vol. 12(4), pages 375-409, October.
    • Handle: RePEc:bla:mathfi:v:12:y:2002:i:4:p:375-409
    DOI: 10.1111/j.1467-9965.2002.tb00130.x
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    Cited by:

    1. Andrew E. B. Lim, 2004. "Quadratic Hedging and Mean-Variance Portfolio Selection with Random Parameters in an Incomplete Market," Mathematics of Operations Research, INFORMS, vol. 29(1), pages 132-161, February.
    2. Bayraktar, Erhan & Hu, Xueying & Young, Virginia R., 2011. "Minimizing the probability of lifetime ruin under stochastic volatility," Insurance: Mathematics and Economics, Elsevier, vol. 49(2), pages 194-206, September.
    3. Ilhan, Aytaç & Jonsson, Mattias & Sircar, Ronnie, 2009. "Optimal static-dynamic hedges for exotic options under convex risk measures," Stochastic Processes and their Applications, Elsevier, vol. 119(10), pages 3608-3632, October.
    4. Zhou, Qing & Wu, Weixing & Wang, Zengwu, 2008. "Cooperative hedging with a higher interest rate for borrowing," Insurance: Mathematics and Economics, Elsevier, vol. 42(2), pages 609-616, April.
    5. Maxim Bichuch & Ronnie Sircar, 2014. "Optimal Investment with Transaction Costs and Stochastic Volatility," Papers 1401.0562, arXiv.org, revised Aug 2014.
    6. Matthew Lorig & Ronnie Sircar, 2015. "Portfolio Optimization under Local-Stochastic Volatility: Coefficient Taylor Series Approximations & Implied Sharpe Ratio," Papers 1506.06180, arXiv.org.
    7. Glasserman, Paul & Kim, Kyoung-Kuk, 2009. "Saddlepoint approximations for affine jump-diffusion models," Journal of Economic Dynamics and Control, Elsevier, vol. 33(1), pages 15-36, January.
    8. Behzad Alimoradian & Karim Barigou & Anne Eyraud-Loisel, 2022. "Derivatives under market impact: Disentangling cost and information," Working Papers hal-03668432, HAL.

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