A comparison of pricing and hedging performances of equity derivatives models

ABSTRACT

This article investigates the pricing/hedging conundrum, i.e. the observation of a mismatch between derivatives models’ pricing and hedging performances, that has so far been under-emphasized as the literature tends to focus on increasingly complicated option pricing models, without adequately addressing hedging performance. Hence, we analyse the ability of the Black–Scholes, Practitioner Black–Scholes, Heston–Nandi and Heston models to Delta-hedge a set of call options on the S&P500 index and Apple stock. We extend earlier studies in that we consider the impact of asset dynamics, apply a stringent payoff replication strategy, look at the impact of moneyness at maturity and test for the robustness to the parameters’ calibration frequency and Delta-Vega hedging. The study shows that adding risk factors to a model, as stochastic volatility, should only be considered in light of the data dynamics. Even then, however, more complicated models generally fare poorly for hedging purposes. Hence, a better fit of a model to option prices is not a good indicator of its hedging performance, and so of its ability to describe the underlying dynamics. This can be understood for reasons of over-fitting. Those findings hint to a potentially appealing hedging-based calibration of models’ parameters, rather than the standard pricing-based one.

KEYWORDS:

• Option pricing
• dynamic hedging
• stochastic volatility
• payoff replication
• calibration

JEL CLASSIFICATION:

• G12 (Asset Pricing)
• G13 (Contingent Pricing|Futures Pricing)

Acknowledgements

The authors would like to thank Prof. Mikael Petitjean for preliminary discussions on the paper content.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

¹ We restrict our analysis to Delta hedging in this article. Though other strategies exist (e.g. Delta-Gamma, Delta-Vega), Delta hedging remains the most common one, and Bakshi, Cao, and Chen (1997) explain that ‘factors as model misspecification and transaction costs may render this type of hedge more practical to adopt’. Delta-Vega hedging is considered as robustness test of our results in the section ‘Models’ out-of-sample Delta-Vega hedging performance’.

² The hedging horizon is the number of days that the Delta-neutral portfolio is re-balanced before being unwound.

³ This was already hinted by Bakshi, Cao, and Chen (1997): ‘Clearly, during a one-day or five-day interval, the chance for a significant price jump (or fall) to occur is very small. Thus, once stochastic volatility is modeled, hedging performance may not be improved any further by incorporating jumps into the option pricing framework […]’.

⁴ This brings some flexibility to hedging since the volatility input changes over the life of the option depending on the value of $S_t$ while it would stay constant if it depended on K only. The function can also feature the time to maturity but without significant improvement in terms of out-of-sample pricing and hedging.

⁵ The section ‘Sliding window technique’ will explain that the total period is split in 30 windows of 2 days and that half of them are used for calibration, and the other half to calculate the hedging errors out-of-sample. Longer maturities are used for estimation than for hedging.

⁶ AAPL options are of American exercise style, which, in contrast to the S&P500, raises the issue of call options that might be exercised early. However, the conditions of early exercise, regarding issuance of dividends (see Merton 1973) and negative interest rates, are not satisfied here.

⁷ We have also checked for the arbitrage condition $C_t<S_t-D_{pv}-K{e}^{-r(T-t)}$, but the inequality is not verified on any option.

⁸ $S_t$ in Equation (7) is the asset market price, i.e. not adjusted to future dividends.

⁹ The errors are calculated on the same windows than for the out-of-sample hedging errors. We re-include the options that were eliminated on the hedging windows because they had maturity dates extending beyond the date of analysis (i.e. those corresponding to the hedging criterion in Table 1). In the end, 1410 and 1349 options are used to compute the pricing errors for the S&P500 and Apple, respectively.

¹⁰ The $RMSE criterion is used in order to have consistent estimation and evaluation loss functions, since the parameters are estimated by minimizing the pricing $MSE in Equation (5). This allows to achieve an optimal out-of-sample loss as pointed out by Christoffersen and Jacobs (2003).

¹¹ This transpires through some parameters that do not represent their intended financial/economic meaning anymore, e.g. $\rho$ is very close to −1 under the S&P500, the vol of vol $\alpha$ takes very high values for both time series, even often higher than 100% for Apple, and $V_0$ and $h_0$ are sometimes absurdly close to 0 for Apple. Those observations drive us to challenge the standard pricing-based calibration of parameters in Section VI.

¹² For the S&P500, the moneyness categories are: $M\in [\leftarrow ,0.99]$, [0.99,1], [1,1.01], $[1.01,\to ]$, and for Apple: $M\in [\leftarrow ,0.97]$, [0.97,1], [1,1.03], $[1.03,\to ]$. The maturity categories become: DTM$\le$20, 20<DTM$\le$40 and DTM>40.

¹³ For BS and PBS, ${\nu }_t=S_t\sqrt{T-t}\varphi (d_1)$. For HN and Heston, we refer the reader to Rouah and Vainberg (2007).

¹⁴ We use two options because, in our dataset of SPX options, we do not have a single option that is traded without interruption on the whole period.

Additional information

Funding

This work was supported by the Fonds National de la Recherche Scientifique (F.R.S.-FNRS) under grant ASP [grant number FC 17775].