Partial hedging in a stochastic volatility environment

Mattias Jonsson, K. Ronnie Sircar

Research output: Contribution to journalArticle

26 Scopus citations

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.

Original languageEnglish (US)
Pages (from-to)375-409
Number of pages35
JournalMathematical Finance
Volume12
Issue number4
DOIs
StatePublished - Oct 2002

All Science Journal Classification (ASJC) codes

  • Accounting
  • Finance
  • Social Sciences (miscellaneous)
  • Economics and Econometrics
  • Applied Mathematics

Keywords

  • Asymptotic analysis
  • Dynamic programming
  • Hamilton-Jacobi-Bellman equations
  • Hedging of options
  • Stochastic volatility
  • Utility maximization

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