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Variance Optimal Hedging in the Black-Scholes Model for a given Number of Transactions

Claude Martini 1 Christophe Patry
1 MATHFI - Financial mathematics
Inria Paris-Rocquencourt, ENPC - École des Ponts ParisTech, UPEC UP12 - Université Paris-Est Créteil Val-de-Marne - Paris 12
Abstract : In the Black-Scholes option pricing paradigm it is assumed that the market-mak- er designs a continuous-time hedge. This is not realistic from a practical point of view. We introduce trading restrictions in the Black-Scholes model in the sense that hedging is only allowed a given number of times-only the number is fixed, the market-maker is free to choose the (stopping) times and hedge ratios. We identify the strategy which minimizes the variance of the tracking error for a given initial value of the portfolio. The minimal variance is shown to be the solution to a sequence of optimal stopping problems. Existence and uniqueness is proved. We design a lattice algorithm with complexity N3 (N being the number of lattice points) to solve the corresponding discrete problem in the Cox-Ross-Rubinstein setting. The convergence of the scheme relies on a viscosity solution argument. Numerical results and dynamic simulations are provided.
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https://hal.inria.fr/inria-00072895
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Submitted on : Wednesday, May 24, 2006 - 11:13:45 AM
Last modification on : Thursday, February 3, 2022 - 11:13:52 AM
Long-term archiving on: : Thursday, March 24, 2011 - 12:19:16 PM

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Claude Martini, Christophe Patry. Variance Optimal Hedging in the Black-Scholes Model for a given Number of Transactions. [Research Report] RR-3767, INRIA. 1999. ⟨inria-00072895⟩

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