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“Capturing the Volatility Smile: Parametric Volatility Models Versus Stochastic Volatility Models” “Capturing the volatility smile: parametric volatility models versus stochastic volatility models” AUTHORS Belen Blanco Belen Blanco (2016). Capturing the volatility smile: parametric volatility models ARTICLE INFO versus stochastic volatility models. Public and Municipal Finance, 5(4), 15-22. doi:10.21511/pmf.05(4).2016.02 DOI http://dx.doi.org/10.21511/pmf.05(4).2016.02 RELEASED ON Monday, 26 December 2016 JOURNAL "Public and Municipal Finance" FOUNDER LLC “Consulting Publishing Company “Business Perspectives” NUMBER OF REFERENCES NUMBER OF FIGURES NUMBER OF TABLES 0 0 0 © The author(s) 2021. This publication is an open access article. businessperspectives.org Public and Municipal Finance, Volume 5, Issue 4, 2016 Belen Blanco (Australia) Capturing the volatility smile: parametric volatility models versus stochastic volatility models Abstract Black-Scholes option pricing model (1973) assumes that all option prices on the same underlying asset with the same expiration date, but different exercise prices should have the same implied volatility. However, instead of a flat implied volatility structure, implied volatility (inverting the Black-Scholes formula) shows a smile shape across strikes and time to maturity. This paper compares parametric volatility models with stochastic volatility models in capturing this volatility smile. Results show empirical evidence in favor of parametric volatility models. Keywords: smile volatility, parametric, stochastic, Black-Scholes. JEL Classification: C14, C68, G12, G13. Acknowledgement: I gratefully acknowledge financial support from the Spanish Ministry of Economy and Competi- tion (ECO2013-48328 and ECO2010-19314), and from the Ramón Areces Foundation. Introduction¤ inverse Fourier transform, what is potentially more precise than the approximation suggested by Hull In financial economics, there is a concern about and White (1987). More recently, Bakshi et al. how modelling the volatility, and more specifical- (1997), Ball and Roma (1994), Bates (1996), among ly, how modelling the volatility in option pricing. others, analyze if stochastic volatility or random There are two main alternatives available for this jumps resolve the anomalies in the Black-Scholes purpose: the first one is parametric volatility mod- model. They find that stochastic volatility models els (as the Dumas et al.’s model (1998)), and the seem to behave slightly better than jumps. second one is stochastic volatility models (as the Heston’s model (1993)). The main objective in this paper is to compare the performance of stochastic volatility models with that Under Black-Scholes option pricing model (1973) of parametric volatility models. To this aim, I use assumptions, all option prices on the same underly- data of transaction prices for future call1 options on ing asset with the same expiration date, but different the Spanish IBEX-35 stock exchange index. I em- exercise prices should have the same implied vola- th tility. Empirically, this is not what I observe using ploy a database transacted on 6 of November traded option prices. Instead of flat implied volatil- 2,015. To estimate stochastic volatility models, I use ity structure, prior literature finds implied volatility the Heston’s model (1993); on the other hand, to (inverting the Black-Scholes formula) showing a estimate parametric volatility models, I use the Du- smile shape across strikes and time to maturity (i.e., mas et al.’s model (1998), but with small variations Cont and Fontseca, 2001; Alerton, 2004). With the in the variables. aim to fit the implied volatilities, Derman & Kani The results obtained show empirical evidence in (1994), Dupire (1994) and Rubinstein (1994), favor of parametric volatility models, respect to the among others, develop a volatility function that fits stochastic volatility models. In particular, I find that the observed cross-section of option prices. A more the parametric volatility models fit the data better recent stream of literature uses parametric models to than the stochastic volatility models, because the fit the implied volatilities (Ncube, 1996; Dumas et last ones tend to overprice out of the money calls. al., 1998; Peña et al., 1999). This paper is organized as follows: the next section In contrast, other stream of literature uses stochastic contains a description of how obtaining the implied volatility models to fit the volatility surface. The volatility. In section 2, I present the parametric vola- paper of Hull and White (1987) was the first sys- tility models, which are based on some adaptations tematic approach in option pricing literature to rec- of the parametric models from Dumas et al. (1998), ognize nonconstant volatility. They show that the to estimate the surface across moneyness. In section price of a European option is the Black-Scholes 3, I describe the Heston’s model (1993) as an ap- price integrated over the probability distribution of proach to stochastic volatility models. The data are the average variance during the life of the option. Later, Heston (1993) shows that a closed-form solu- 1 tion for a European call can be derived as an integral In principle, call and put options should yield the same implied volatil- ity, based on the Call Put parity theorem. Some argue (Ncube, 1996) that of the future security price density, calculated by an since the put option is a natural hedging instrument, investors may be willing to pay more for it and, therefore, its implied volatility would be higher than the call counterpart. However, I will not take into account this ¤ Belen Blanco, 2016. possible bias in my model, since I am interested in obtaining a more Belen Blanco, Dr., The University of Adelaide, Australia. generic solution, from which both call and put options can be priced. 15 Public and Municipal Finance, Volume 5, Issue 4, 2016 described in section 4. The estimation results of the ı is the volatility rate, and N(d) is the cumulative volatility surfaces for both models are discussed in unit normal density function with upper integral section 5, I conclude in last section. limit d. The implied Black-Scholes volatility can be found individually from traded option prices: 1. Obtaining the implied volatility with the Newton-Raphson algorithm wBS ! .0 (3) The Black-Scholes European call option formula is wV the following one: The Newton-Raphson algorithm provides a numeri- ( tTq ) ( tTr ) C 0eS N(d1 ) Ke N(d2 ), (1) cal way to invert the Black-Scholes formula in order to recover ı from the market prices of the call op- where S0 is the underlying asset (in this case, the tion C (or Put option P) future contract of IBEX-35 index), q is the expected dividends paid over the option’s life, X is the op- f (V ) BS(V ) C .0 (4) tion’s strike price, (T-t) is the time to expiration, r is the risk-free interest rate, 2. The parametric volatility models 2 To estimate parametric volatility models, I use an ln(S0 / K) (r q V )(2/ T t) adaptation of the parametric models proposed by d1 , V T t (2) Dumas et al. (1998). In particular, I run the fol- lowing three models, as a function of moneyness d 2 d1 V T t , and time: Model 0: V(MN T), E0 H , 2 Model 1: V (MN T ), E0 E1 log(MN ) E2 log(MN ) H , (5) 2 Model 2: V (MN T ), E0 E1 log(MN ) E2 log(MN) E3T E 4T log(MN ) H. Model 0 is the volatility function representing the 2 SSE(E ) constant volatility as in the Black-Scholes model. R 1 m . (7) Model 1 captures the quadratic volatility smile 2 ¦ (yi y) across moneyness, and model 2 captures extends i 1 model 1 by capturing the variation across time, and From this formula, I can calculate the adjusted R2 a combined effect of time and moneyness. statistic: The moneyness in this work is defined as MN = K/F. If K/F > 1, the call option is out-the-money 2 § n 1 · 2 R 1 ¨ ¸ 1( R ), (8) (OTM), and when K/F < 1, the call option is in-the- © n p ¹ money (ITM). When K/F|1, I can say that the call option is at-the-money (ATM). where p is the number of variables. I will choose the model with higher R2 to compare it ȕ0 is the constant of the regression. ȕ1 coefficient captures the dislocation of the origin of the parabola to stochastic volatility model. with respect to the ATM options, and ȕ2 coefficient 3. The stochastic volatility models controls the size of the smile. ȕ3 and ȕ4 capture the term structure of the implied volatility. To estimate stochastic volatility models, I use the Heston’s (1993) model. Heston (1993) proposed the ȕ vector is estimated with nonlinear least-squares following model: function as follows: dS PS dt V S dW 1, m t t tt t 2 2 min SSE(E ) (yi prdi (w ,)) (6) ¦ dVt N(T Vt )dt V Vt dWt , (9) i 1 dW 1dW 2 Udt, where prdi(w) is the function that implements each t t specific model of the volatility surface equation. where {S }tt t0 and {V }tt t0 are the price and volatil- Later, I measure how successful the fit of the model 1 2 ity processes, respectively, and {W } , {W } is in explaining the variation of the data with the R2 tt t0 tt t0 statistic. For the nonlinear least squares estimation, are correlated Brownian motion processes (with is defined as the square of the correlation between correlation parameter ȡ). ȝ is the instantaneous ex- the observations (Greene, 2000): pected rate of return of the underlying asset. {V }tt t0 16 Public and Municipal Finance, Volume 5, Issue 4, 2016 is the instantaneous stochastic variance, ș is the long clustering. This is something that is observed in the term mean of the variance, and rate at which the vari- market, large price variations are more likely to be ance converges to this mean is ț.
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