Improved method for static replication under the CEV model

- I improve the DEK's static hedging method for continuous barrier options under the CEV model.
- The idea is to match both the value-matching condition and the theta-matching condition on the barrier.
- I reform a portfolio consisting of European and binary options regarding to the two conditions.
- This improved method improves not only the replication but also the convergence speed.

This paper studies the hedging performance of static replication approach proposed by Derman, Ergener, and Kani (DEK, 1995) for continuous barrier options under the constant elasticity of variance (CEV) model of Cox (1975) and Cox and Ross (1976), and then focuses on how to improve the DEK method. Given the time-varying volatility feature of the CEV model, I show that the DEK static hedging portfolio exhibits serious mismatches of the theta values on the barrier, particularly when one of the component options of the portfolio is around the neighborhood of expiration, which primarily explains why static portfolio values are greater than zero on the barrier except at the matching points. The DEK method (hereafter, the improved DEK method) is improved by re-forming a static replication portfolio consisting of plain vanilla options and cash-or-nothing binary options with different maturities to match both the value-matching condition and the theta-matching condition on the barrier. The numerical analyses indicate that under the CEV model, the improved DEK method significantly reduces replication errors for an up-and-out call option.