Dynamic hedging with a deterministic local volatility function model

Abstract

We compare the dynamic hedging performance of the deterministic local volatility function approach with the implied/constant volatility method. Using an example in which the underlying price follows an absolute diffusion process, we illustrate that hedge parameters computed from the implied/constant volatility method can have significant error even though the implied volatility method is able to calibrate the current option prices of different strikes and maturities. In particular the delta hedge parameter produced by the implied/constant volatility method is consistently significantly larger than that of the exact delta when the underlying price follows an absolute diffusion.In order to compute a better hedge parameter, accurate estimation of the local volatility function in a region surrounding the current asset price is crucial. We illustrate that a suitably implemented volatility function method can estimate this local volatility function sufficiently accurately to generate more accurate hedge parameters. Hedging using this volatility function for the absolute diffusion example leads to a smaller average absolute hedging error when compared with using the implied/constant volatility rate.When comparing the hedging performance in the S&P 500 index option market as well as the S&P 500 futures option market, we similarly observe that the delta hedge parameter from the implied/constant volatility method is typically greater than that using the volatility function approach. Examination of the hedging error reveals that using a larger delta factor greater than that of the true volatility yields more positive 1

Model risk adjusted hedge ratios

“…They find that delta hedging alone does not lead to satisfying results and that sticky strike models perform best. However, these findings are contradicted by Coleman, Kim, Li, and Verma (2001), who show that BSM deltas are too large for S&P 500 index options and that the average hedging error using a parameterization of the "true" local volatility is smaller than it is using the BSM model. More recently Crépey (2004), Vähämaa (2004), and Alexander and Nogueira (2007a) support this argument and conclude that, on average, "true" local volatility deltas are indeed more effective than BSM deltas.…”

“…Following Coleman et al (2001) we replace u F with u K to obtain an ad hoc correction of the BSM model. As the smile in equity markets is usually downward sloping this will adjust the BSM delta downward and therefore this ad hoc approach is consistent with the negative correlation, which is often observed between implied volatilities and the underlying (see Vähämaa, 2004).…”

“…When the model's local volatility surface is a function of the current asset price S 0 as in Hagan et al (2002) this violates a key assumption of local volatility models, because the surface cannot be static. Secondly, following a result by Coleman et al (2001), both Crépey (2004, Vähämaa (2004) replace the implied volatility derivative to price by its derivative to strike, i.e. the slope of the smile curve.…”

Model-free hedge ratios and scale-invariant models

“…1a, the scale-invariant model-free delta is greater than the BS delta for all but the very high strikes. So if the BS model over-hedges in presence of the skew (as shown by Coleman et al, 2001) then scale-invariant models should perform worse than the BS model. A different picture emerges when MV hedge ratios are used.…”

Calibration and hedging under jump diffusion

“…It has been illustrated in Coleman et al (1999Coleman et al ( , 2001) that it is important to estimate the local volatility function sufficiently accurately for the purposes of pricing and hedging. Given a finite set of current option prices, the jump model calibration problem (2.5) is ill-posed even when the jump parameters (λ Q , µ Q , γ Q ) are fixed, and the effects of this can be significant since the option prices are more sensitive to volatilities.…”