Evaluating the hedging error in price processes with jumps present

In this draft, we consider a hedging strategy concerning only the continuous parts of two asset price processes which have jumps. Two consistent estimators of the hedging strategy, $\hat{\rho}$ and $\tilde{\rho}$, are presented in terms of realized bipower variation and threshold quadratic variation, respectively. Based on $\hat{\rho}$, estimators for operational risk, market risk (risk due to jumps) and total risk are investigated. It turns out that the variance of $\hat{\rho}$ enters into the bias of the operational risk estimator, whereas the variance is mainly due to jump influenced bipower estimation error. The convergence rate of the operational risk estimator (properly centralized) is $O_P((\overline{\Delta t})^{1/2})$. The convergence rate of the market risk is however $O_P((\overline{\Delta t})^{1/4})$. Based on $\tilde{\rho}$, the total risk is also studied, and it has the same convergence rate as that based on $\hat{\rho}$. Besides the interest in financial econometrics, it is also of significance in a statistical sense when we are interested in estimating the quadratic variation of the corresponding unhedgeable residual process.

Keywords

hedging strategy, threshold variation, realized bipower variation, quadratic variation, volatility, jump diffusion, variation of time

2010 Mathematics Subject Classification

Primary 60G44, 62M09, 62M10, 91Gxx. Secondary 60G42, 62G20, 62Pxx, 91B84.

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Published 10 January 2014