Superreplication Under Gamma Constraints
- 1 January 2000
- journal article
- Published by Society for Industrial & Applied Mathematics (SIAM) in SIAM Journal on Control and Optimization
- Vol. 39 (1) , 73-96
- https://doi.org/10.1137/s0363012998348991
Abstract
In a financial market consisting of a nonrisky asset and a risky one, we study the minimal initial capital needed in order to superreplicate a given contingent claim under a gamma constraint. This is a constraint on the unbounded variation part of the hedging portfolio. We first consider the case in which the prices are given as general Markov diffusion processes and prove a verification theorem which characterizes the superreplication cost as the unique solution of a quasi-variational inequality. In the context of the Black--Scholes model (i.e., when volatility is constant), this theorem allows us to derive an explicit solution of the problem. These results are based on a new dynamic programming principle for general "stochastic target" problems.Keywords
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