Articlegreen2005

EVALUATING HEDGING ERRORS: AN ASYMPTOTIC APPROACH

85 citations0/0 files storedMathematical Finance

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Stochastic processes and financial applications Financial Markets and Investment Strategies Financial Risk and Volatility Modeling

Abstract

We propose a methodology for evaluating the hedging errors of derivative securities due to the discreteness of trading times or the observation times of market prices, or both. Utilizing a weak convergence approach, we derive the asymptotic distributions of the hedging errors as the discreteness disappears in several situations. First, we examine the hedging error due to discrete‐time trading when the true strategy is known, which generalizes the result of Bertsimas, Kogan, and Lo (2000) to continuous Itô processes. Then we consider a data‐driven strategy, when the true strategy is unknown. This strategy is free of parametric model assumptions, therefore it is expected to serve as a benchmark for the evaluation of parametric strategies. Finally, we consider a case study of the Black‐Scholes delta‐hedging strategy when the volatility is unknown in the proposed framework. The results obtained give us a prospect for further developments of the framework under which various parametric strategies could be compared in a unified manner.

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References

2007Brownian Motion and Stochastic Calculus 1998Brownian Motion and Stochastic Calculus 1998Methods of Mathematical Finance 1987Limit Theorems for Stochastic Processes 1998Nonparametric Estimation of State‐Price Densities Implicit in Financial Asset Prices 1985Option Pricing and Replication with Transactions Costs 2004Econometric Analysis of Realized Covariation: High Frequency Based Covariance, Regression, and Correlation in Financial Economics 1990Hedging of contingent claims under incomplete information

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2008Effectiveness of CPPI strategies under discrete-time trading 2008Asymptotic analysis of hedging errors in models with jumps 2009Weak convergence of error processes in discretizations of stochastic integrals and Besov spaces 2015Asymptotics for fixed transaction costs 2011Discretization error of stochastic integrals 2014Asymptotically optimal discretization of hedging strategies with jumps

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