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PARTIAL HEDGING IN A STOCHASTIC VOLATILITY ENVIRONMENT
Author(s) -
Jonsson Mattias,
Sircar K. Ronnie
Publication year - 2002
Publication title -
mathematical finance
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.98
H-Index - 81
eISSN - 1467-9965
pISSN - 0960-1627
DOI - 10.1111/j.1467-9965.2002.tb00130.x
Subject(s) - stochastic volatility , implied volatility , volatility smile , sabr volatility model , forward volatility , volatility (finance) , local volatility , econometrics , variance swap , volatility swap , constant elasticity of variance model , heston model , economics , mathematical optimization , mathematics
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.