## Abstract

This study examines several alternative symmetric and asymmetric model specifications of regression-based deterministic volatility models to identify the one that best characterizes the implied volatility functions of S&P 500 Index options in the period 1996–2009. We find that estimating the models with nonlinear least squares, instead of ordinary least squares, always results in lower pricing errors in both in- and out-of-sample comparisons. In-sample, asymmetric models of the moneyness ratio estimated separately on calls and puts provide the overall best performance. However, separating calls from puts violates the put-call-parity and leads to severe model mis-specification problems. Out-of-sample, symmetric models that use the logarithmic transformation of the strike price are the overall best ones. The lowest out-of-sample pricing errors are observed when implied volatility models are estimated consistently to the put-call-parity using the joint data set of out-of-the-money options. The out-of-sample pricing performance of the overall best model is shown to be resilient to extreme market conditions and compares quite favorably with continuous-time option pricing models that admit stochastic volatility and random jump risk factors.

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## Notes

DVF is an interpolative (smile-consistent) regression-based approach in which implied volatilities are smoothed across moneyness and maturities, and as such, it is effective in relaxing the BS assumption of having a single volatility per day. This approach is also commonly known as the practitioner BS model (see Christoffersen and Jacobs 2004; Christoffersen et al. 2009) or as the ad hoc BS model (see Linaras and Skiadopoulos 2005; Kim 2009; Berkowitz 2010).

The moneyness ratio

*K*(i.e.,*K*=*S*/*X*, stock index value,*S*, over the option’s strike price,*X*) is used in the relative smile approach to model implied volatility as a fixed function of moneyness, while in the absolute smile approach, the strike price is used to model implied volatility as a fixed function of the strike price (see discussions in Kim 2009).For a review of smile/smirk consistent deterministic volatility models, see Skiadopoulos (2001); for applications that involve the use of DVF models, see Ncube (1996), Dumas et al. (1998), Brandt and Wu (2002), Christoffersen and Jacobs (2004), Linaras and Skiadopoulos (2005), Kim (2009), and Andreou et al. (2010), among others.

Relying on the overall best DVF model is important for other studies as well, such as that of Brandt and Wu (2002), where option parameters are estimated from liquid European options and are then applied to price illiquid, exotic or other derivatives (see also, Linaras and Skiadopoulos 2005; Berkowitz 2010; Chang et al. 2012). Moreover, several studies using nonparametric option pricing methods such as kernel regression or neural networks also have a need for a proper benchmark model (see Andreou et al. 2008).

Shimko (1993) was the first to show how to recover the risk-neutral probability density function by fitting a smooth curve to the implied volatility smirk. Moreover, Bakshi et al. (2003) show that the shape of the implied volatility function is directly linked to the risk-neutral skewness and kurtosis of the implied distribution. Panigirtzoglou and Skiadopoulos (2004) show how to model the dynamics of implied distributions for (smile-consistent) option pricing purposes.

Nonlinear optimization algorithms employed for estimating the nonlinear least squares versions of the DVF models might be affected by large differences in the levels of the observed variables. As suggested by Nocedal and Wright (1999, p. 27), the performance of an algorithm may depend critically on how the problem is formulated. In addition, the use of very high or low numbers, which can occur when the (untransformed) strike price is used, can cause underflow or overflow problems.

Dumas et al. (1998) use \( \sigma_{X}^{s} \) in most of their analysis, but in a few cases where option cross-sections have fewer than three expiration dates available, they use a reduced version that ignores some of the time-terms. In our case, there was no such problem, since there were no days with less than three expiration dates.

In a previous version of this paper, which used SPX data for the period 1998–2004, we also estimated all asymmetric DVF models with the threshold value for the dummy variable to be either

*K*= 1.05 or*K*= 1.10. These specifications do not work better than those where the threshold is set equal to 1 (*K*= 1).Data synchronicity is a minor issue for this highly active market. Among others, Constantinides et al. (2009), Christoffersen et al. (2006), Christoffersen and Jacobs (2004), and Chernov and Ghysels (2000) use daily closing prices of European options written on the S&P 500 Index. Other related studies that use daily closing prices include Ncube (1996), Peña et al. (1999), Engström (2002), Brandt and Wu (2002), Chen et al. (2009), Andreou et al. (2010) and Mozumder et al. (2012).

We use the following moneyness ratio classes: deep out-of-the-money (DOTM) for 0.75 ≤

*K*< 0.90, out-of-the-money (OTM) for 0.90 ≤*K*< 0.95, just out-of-the-money (JOTM) for 0.95 ≤*K*< 0.99, at-the-money (ATM) for 0.99 ≤*K*< 1.01, just in-the-money (JITM) for 1.01 ≤*K*< 1.05, in-the-money (ITM) for 1.05 ≤*K*< 1.10, and deep in-the-money (DITM) for 1.10 ≤*K*≤ 1.25. In terms of maturity, an option contract is classified as being of short-term maturity (where its maturity is ≤60 calendar days), as medium-term (where its maturity is between 61 and 180 calendar days) and as long-term (where its maturity is >180 calendar days).Patton (2011) reports that among many alternative loss functions Mean Squared Error is robust to the noise in the volatility proxy and yields correct rankings of models used to forecast volatility. Despite the overwhelming (theoretical and empirical) evidence in favor of the Mean Squared Error metric, we also compute the Mean Absolute Error, the Median Absolute Error, and the 5th and 95th Percentile of Absolute Error (results are available upon request—see also our robustness analysis in Sect. 5.1). Overall, we observe that all of our findings remain unchanged when using any of the alternative loss functions.

We use \( \sigma_{LnX}^{s} \) in BS because we find it to be superior in terms of out-of-sample pricing accuracy; yet, similar violations are obtained with all other DVF specifications.

Average bid–ask spreads for the seven moneyness classes vary between 1.57 and 2.86 for short-term options (category average is 1.64), between 1.83 and 2.73 for medium-term options (average is 2.04), and between 2.14 and 2.67 for long-term options (average is 2.29).

For brevity, in Table 3 we report

*t*-statistics for a subset of models that exhibit the overall best out-of-sample pricing accuracy. For all model comparisons, we have also computed Student’s*t*-statistics, as well as the Johnson (1978) modified*t*-statistics for non-normal distributions, which account for the presence of skewness in the model residuals. All results are qualitatively similar to those reported in Table 3 (the same holds true for the*t*-statistics in Table 6).Mozumder et al. (2012) document that more complex hedging strategies such those that rely on delta and delta-gamma approaches yield inaccurate approximations of option portfolio values, especially in the face of large swings in the price of the underlying asset. Yet, the more sophisticated delta-gamma approximation is not significantly more accurate than the easier to implement single-instrument delta-hedging approximation. These authors also report that even the most sophisticated option pricing models they employ in their empirical investigation deliver poor risk-management effectiveness during times of financial market turbulence.

As in the main analysis of this study, all additional (reduced-version) DVF specifications are estimated daily based on three different data sets: (1) all call options, (2) all put options, and (3) the joint data set of out-of-the-money call and put options. As discussed in Sect. 5.3, to validate the robustness of the estimation data set, we also estimated all DVF specifications using the joint data set of out-of-the-money call and put options, whereas moneyness was based on the options’ delta value. We estimate daily the additional 44 DVF models separately on each data set using both OLS and NLS. In this Appendix, we tabulate in- and out-of-sample pricing results (one-day, one-week, and two-weeks ahead) of all models considered in this study on all three evaluation data sets as discussed in the main analysis, using a variety of error metrics (i.e., Root Mean Squared Error, Mean Absolute Error, Median Absolute Error, and 5th and 95th Percentile of Absolute Error).

We find, though, a few asymmetric specification cases where DVF models that exclude the time-term

*T*^{2}perform slightly better (this is more pronounced when the estimation and evaluation data sets are different).The underperformance of the VIX is not surprising. In a financial econometrics context, Becker and Clements (2008) find that the VIX index produces volatility forecasts that are inferior to model-based forecasts of realized volatility.

SV and SVJ sometimes result in implausible implied parameters to rationalize the observed option prices. Bakshi et al. (1997) report that both are clearly mis-specified (see also Bates 1996 and discussions in Skiadopoulos 2001). Moreover, while it can take only a few seconds to fit a regression-based DVF model to daily option prices, it sometimes can take up to a few minutes to do the same using the SV, and even longer for the SVJ model. For these reasons, traders in practice usually prefer the regression-based DVF approach and rely less on mathematically advanced continuous-time option pricing models such as SV or SVJ (see also discussions in Dumas et al. 1998; Brandt and Wu 2002; Chen et al. 2009; Kim 2009).

Out-of-the-money calls are those with \( \Updelta_{c} < 0.50 \), where \( \Updelta_{c} \) is the delta value for a call option computed as: \( \partial c/\partial S = e^{{ - d_{y} T}} {\text{N}}\left( d \right) \), while out-of-the-money puts are those with \( \Updelta_{p} \ge - 0.50 \), where \( \Updelta_{p} \) is the delta value for a put option computed as: \( \partial p/\partial S = e^{{ - d_{y} T}} \left( {{\text{N}}\left( d \right) - 1} \right) \), whereas

*d*is given by Eq. (1.3). Bollen and Whaley (2004) explain that it is preferable to split the data in delta moneyness categories, since delta is sensitive to the volatility of the underlying asset, as well as to the option’s time-to-maturity. As in their case, the proxy for the volatility rate is the realized return volatility of the S&P 500 Index over the most recent 60 trading days.

## References

Ahoniemi K, Lanne M (2009) Joint modeling of call and put implied volatility. Int J Forecast 25(2):239–258

Andreou PC, Charalambous C, Martzoukos SH (2008) Pricing and trading European options by combining artificial neural networks and parametric models with implied parameters. Eur J Oper Res 185(3):1415–1433

Andreou PC, Charalambous C, Martzoukos SH (2010) Generalized parameter functions for option pricing. J Bank Financ 34(3):633–646

Bakshi G, Cao C, Chen Z (1997) Empirical performance of alternative options pricing models. J Financ 52(5):2003–2049

Bakshi G, Kapadia N, Madan D (2003) Stock return characteristics, skew laws, and the differential pricing of individual equity options. Rev Financ Stud 16(1):101–143

Bates DS (1991) The crash of ‘87: was it expected? The evidence from options markets. J Financ 46(3):1009–1044

Bates DS (1996) Jumps and stochastic volatility: exchange rate processes implicit in Deutsche Mark options. Rev Financ Stud 9(1):69–107

Bates DS (2000) Post-‘87 crash fears in the S&P 500 futures option market. J Econom 94(1–2):181–238

Becker R, Clements AE (2008) Are combination forecasts of S&P 500 volatility statistically superior? Int J Forecast 24(1):122–133

Berkowitz J (2010) On justifications for the ad hoc Black-Scholes method of option pricing. Stud Nonlinear Dyn Econom 14(1): Article 4

Black F, Scholes M (1973) The pricing of options and corporate liabilities. J Polit Econ 81(3):637–654

Bollen NPB, Whaley RE (2004) Does net buying pressure affect the shape of implied volatility functions? J Financ 59(2):711–753

Brandt MW, Wu T (2002) Cross-sectional tests of deterministic volatility functions. J Bank Financ 9(5):525–550

Chang EC, Ren J, Shi Q (2009) Effects of the volatility smile on exchange settlement practices: the Hong Kong case. J Bank Financ 33(1):98–112

Chang CC, Lin JB, Tsai WC, Wang YH (2012) Using Richardson extrapolation techniques to price American options with alternative stochastic processes. Rev Quant Financ Acc 39(3):383–406

Chen RR, Lee CF, Lee HH (2009) Empirical performance of the constant elasticity variance option pricing model. Rev Pac Basin Financ Mark Policies 12(2):177–217

Chernov M, Ghysels E (2000) Towards a unified approach to the joint estimation objective and risk neutral measures for the purpose of option valuation. J Financ Econ 56(3):407–458

Christoffersen P, Jacobs K (2004) The importance of the loss function in option valuation. J Financ Econ 72(2):291–318

Christoffersen P, Heston S, Jacobs K (2006) Option valuation with conditional skewness. J Econom 131(1–2):253–284

Christoffersen P, Heston SL, Jacobs K (2009) The shape and term structure of the index option smirk: why multifactor stochastic volatility models work so well. Manag Sci 55(12):1914–1932

Constantinides GM, Jackwerth JC, Perrakis S (2009) Mispricing of S&P 500 index options. Rev Financ Stud 22(3):1247–1277

Dumas B, Fleming J, Whaley RE (1998) Implied volatility functions: empirical tests. J Financ 53(6):2059–2106

Engström M (2002) Do Swedes smile? On implied volatility functions. J Multinatl Financ Manag 12(4–5):285–304

Heston SL (1993) A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev Financ Stud 6(2):327–343

Johnson NJ (1978) Modified

*t*test and confidence intervals for asymmetrical populations. J Am Stat Assoc 73(363):536–544Jones EP (1984) Option arbitrage and strategy with large price changes. J Financ Econ 13(1):91–113

Kim S (2009) The performance of traders’ rules in option market. J Futur Mark 29(11):999–1020

Kuo ID (2011) Pricing and hedging volatility smile under multifactor interest rate models. Rev Quant Financ Acc 36(1):83–104

Linaras CE, Skiadopoulos G (2005) Implied volatility trees and pricing performance: evidence from the S&P 100 options. Int J Theor Appl Financ 8(8):1085–1106

Merton RC (1976) Option pricing when underlying stock return are discontinuous. J Financ Econ 3(1–2):125–144

Mozumder S, Sorwar G, Dowd K (2012) Option pricing under non-normality: a comparative analysis. Rev Quant Financ Acc 40(2):273–292

Ncube M (1996) Modelling implied volatility with OLS and panel data models. J Bank Financ 20(1):71–84

Nocedal J, Wright JS (1999) Numerical optimization. Springer, New York

Panigirtzoglou N, Skiadopoulos G (2004) A new approach to modeling the dynamics of implied distributions: theory and evidence from the S&P 500 options. J Bank Financ 28(7):1499–1520

Patton AJ (2011) Volatility forecast comparison using imperfect volatility proxies. J Econom 160(1):246–256

Peña I, Rubio G, Serna G (1999) Why do we smile? On the determinants of the implied volatility function. J Bank Financ 23(8):1151–1179

Peña I, Rubio G, Serna G (2001) Smiles, bid–ask spreads and option pricing. Eur Financ Manag 7(3):351–374

Shimko DC (1993) Bounds of probability. RISK 6(4):33–37

Skiadopoulo G (2001) Volatility smile consistent option models: a survey. Int J Theor Appl Financ 4(3):403–437

Tompkins R (2001) Stock index futures markets: stochastic volatility models and smiles. J Futur Mark 21(1):43–78

Vähämaa S, Äijö J (2011) The Fed’s policy decisions and implied volatility. J Futur Mark 31(10):995–1010

White H (1980) A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity. Econometrica 48(4):817–838

## Acknowledgments

This article has benefited from the useful comments of Hans Bystrom (discussant) and the participants of the 16th Annual Conference of the Multinational Finance Society (Crete, June 2009). Comments by Christodoulos Louca (Cyprus University of Technology), George Skiadopoulos (University of Piraeus and University of Warwick), Panayiotis Theodossiou (Cyprus University of Technology), and Christos Savva (Cyprus University of Technology) are also acknowledged. The authors are also grateful to Isabella Karasamani (Cyprus University of Technology and Northcentral University) for proofreading the paper and providing insightful and helpful comments.

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Andreou, P.C., Charalambous, C. & Martzoukos, S.H. Assessing the performance of symmetric and asymmetric implied volatility functions.
*Rev Quant Finan Acc* **42**, 373–397 (2014). https://doi.org/10.1007/s11156-013-0346-z

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DOI: https://doi.org/10.1007/s11156-013-0346-z

### Keywords

- Option pricing
- Deterministic volatility functions
- Implied volatility forecasting
- Model selection
- Stochastic volatility