Volatility Modelling in Option Pricing and its Impact on Payoff Replication Performance

Volatility modelling in option pricing has been shown to be of first-order importance in improving upon the Black-Scholes pricing biases. However, no consensus emerges on its impact on hedging performance, with more realistic volatility specifications sometimes decreasing the accuracy compared to BS. In light of this, we provide a thorough study of the hedging performance and behaviour of four models: constant volatility (Black-Scholes), local volatility (Practitioner Black-Scholes), GARCH-type (Heston-Nandi) and stochastic volatility (Heston). We apply a Delta-neutral payoff replication strategy on European call option quotes on the S&P500 and Apple. We wish to assess the relative hedging performance of our models and how moneyness, maturity and volatility impact the accuracy. Additionally, we further dive into the link between hedging and pricing accuracy, the effect of a frequent parameters re-calibration and the drivers of under/over-hedging. We use a pricing $MSE loss function on windows of two days and we calculate the hedging errors out-of-sample. Interesting results emerge: • The maturity, moneyness and volatility have a substantial impact on hedging accuracy. A larger maturity sometimes surprisingly leads to lower errors for far-from-the-money options. Then, near-the-money options are better hedged than far-from-the-money ones under a low volatility, while the reverse effect is interestingly observed for long-term options under a high volatility. Finally, we observe that HN and Heston are better suited to a more volatile asset: their errors are lower for Apple than the S&P500. • More advanced volatility modelling overall decreases hedging accuracy. Under a low volatility, the ranking is always the same: PBS, BS, Heston, HN. When volatility rises, Heston and HN sometimes outperform BS and PBS for sufficiently OTM options. Based on the end-moneyness, the ranking shows that HN and Heston beat BS and PBS when the Delta is close to 0 during a large proportion of the hedging period. • The worst models for hedging are overall the best ones for pricing, i.e. an over-fit of option prices leads to poor Delta estimates. Then, a frequent parameters re-calibration surprisingly increases the size of errors under a low volatility. When volatility rises, the ranking is modified for DOTM options, with HN and Heston beating BS and PBS. Finally, under/over-hedging is mainly driven by the re-balancing frequency: a lower frequency leads to more over-hedging. Three implications emerge: (1) there is not one ideal model: the choice depends on the applied exercise, the underlying specificities, the option model parameters and the procedure followed; (2) adding complexity to make the procedure and models more realistic may counter-intuitively lead to worse results; (3) given the negative link between hedging and pricing performance, a focus should be put on application-oriented calibration procedures.