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http://www.fedea.es/hojas/publicado.html

Gabriele Fiorentini (Universidad de Alicante)

Angel León (Universidad de Alicante)

and

Gonzalo Rubio (Universidad del País Vasco)

First draft: April 1998. This version: December 1998

JEL classification: G12, G13 Keywords: Stochastic; Volatility; Skewness; Kurtosis; Pricing

Abstract

This paper examines the stochastic volatility model suggested by Heston (1993). We employ a timeseries approach to estimate the model and we discuss the potential effects of time-varying skewness and kurtosis on the performance of the model. In particular, it is found that the model tends to overprice out-of-the-money calls and underprice in-the-money calls. It is also found that the daily volatility risk premium presents a quite volatile behavior over time; however, our evidence suggests that the volatility risk premium has a negligible impact on the pricing performance of Heston´s model.

Corresponding author: Prof. Gonzalo Rubio, Dpto. Fundamentos, Facultad de Ciencias Económicas, Universidad del País Vasco, Avda. L. Aguirre 83, 48015 Bilbao, Spain. Tel: 94- 4797470; Fax: 94-4797474; e-mail: jepruirg@bs.ehu.es

We thank Eduardo Schwartz, Enrique Sentana, Ignacio Peña, Gregorio Serna, Rafael Salinas, José Luis Fernández, Eliseo Navarro, Alejandro Balbás and Rafael Santamaría for their helpful comments and useful discussions. Earlier versions of this paper were presented at Universidad Carlos III, II Foro Finanzas-Segovia, and MEFF Institute for Derivative Assets. Gabriele Fiorentini and Gonzalo Rubio acknowledge the financial support provided by Dirección Interministerial Científica y Técnica (DGICYT) grants PB96-0339 and PB97-0621 respectively. The three authors have benefited from the research grant received from the Instituto Valenciano de Investigaciones Económicas (IVIE). The contents of this paper are the sole responsibility of the authors.

1. Introduction

A central point for the empirical testing of option pricing models is whether the actual distribution of the underlying asset implied by the option market data is consistent with the distribution assumed by the theoretical option pricing model. Thus, whenever we find biases in our popular option models, the issue becomes knowing who is the killer of the theoretical model analized.

Given the Black-Scholes (1973) (BS henceforth) assumptions, all option prices on the same underlying security with the same expiration date but with different exercise prices should have the same implied volatility. However, the well known volatility smile pattern suggests that the BS formula tends to misprice deep in-the-money and deep out-of-themoney options¹. There have been various attempts to deal with this apparent failure of the BS valuation model. In principle, as explained by Das and Sundaram (1998), the existence of the smile may be attributed to the well known presence of excess kurtosis in the conditional return distributions of the underlying assets. It is clear that excess kurtosis makes extreme observations more likely than in the BS case. This increases the value of out-of-the-money and in-the-money options relative to at-the-money options, creating the smile. However, at least in the U.S. market, the pattern shown by data contains a clear asymmetry in the shape of the smile. This may be due to the presence of skewness in the distribution which has the effect of accentuating just one side of the smile².

¹ After the October 1987 crash, the implied volatility computed from options on stock indexes in the US market inferred from the BS formula appears to be different across exercise prices. This is the so-called “volatility smile”. In fact, as pointed out by Rubinstein (1994), Aït-Sahalia and Lo (1997) and Dumas, Fleming and Whaley (1998), implied volatilities of the S&P 500 options decrease monotonically as the exercise price becomes higher relative to the current level of the underlying asset.

² In the case of the Spanish options market, contrary to the evidence of the US market, an almost perfect smile is generally found. Of course, we should be very cautious in making this type of comparisons. The Spanish evidence is based on short-term to expiration options. The U.S. evidence is generally based on longer-term options. See, in this regard, the paper by Bakshi and Chen (1997). Moreover, liquidity costs, proxied by the relative bid-ask spread, seem to cause the shape of the smile in the Spanish market to a significant degree. See Peña, Rubio, and Serna (1998).

Given this evidence, extensions to the BS model that exhibit excess kurtosis and skewness have been proposed in recent years along two lines of research: Jumpdiffusion models under a Poisson-driven jump process, and the stochastic volatility framework are the key developments in the theoretical option pricing literature³.

A recent and important attempt to summarize alternative option pricing models is carried out by Bakshi, Cao, and Chen (1997). They are able to derive a closed-form jumpdiffusion model that includes previously studied models. It allows not only stochastic volatility, but also stochastic interest rates and stochastic jumps. Moreover, following Bates (1996), they use a cross-sectional framework to implement their model, and analyze the performance and hedging behavior of the nested option pricing models. Das and Sundaram (1998) also examine the extent to which these models are able to capture the observed anomalies discussed in literature. Bakshi, Cao, and Chen find that both stochastic volatility and jumps are important for pricing. However, they suggest that recognizing stochastic volatility alone produces the best hedging performance. On the other hand, Das and Sundaram argue that, generally speaking, stochastic volatility models yield better pricing results than jumps, although none of them is able to explain all patterns of kurtosis, skewness and volatility smiles found in empirical pricing literature.

In this paper, given the previous evidence briefly discussed above, we concentrate on the stochastic volatility approach within a time-series context. The stochastic volatility framework of Hull and White (1987) was the first systematic approach in option pricing literature to recognize nonconstant volatility. When volatility is stochastic but uncorrelated with the underlying asset price, they show that the price of a European option is the BS price integrated over the probability distribution of the average variance during the life of the option⁴. Unfortunately, however, this framework generally requires a market price of volatility risk. In other words, with stochastic volatility, a second factor is introduced, requiring the option to satisfy a bivariate stochastic differential equation. Since the volatility -the second factor- is not spanned by existing securities, arbitrage pricing techniques are no longer valid. We must therefore introduce the explicit exogenous market price of volatility risk. We face similar problems if we introduce any other non-traded source of risks such as systematic jumps or transaction costs.

3 Alternatively, Corrado and Su (1996), and Backus, Foresi, Li and Wu (1997) adapt a Gram-Charlier series expansion of the normal density function to obtain skewness and kurtosis adjustment terms for the BS formula. Eberlein, Keller and Prause (1998) introduce the hyperbolic density to account for excess kurtosis and skewness, and they are even able to obtain a closed option pricing formula under this assumption. Rosenberg (1998) suggests the so called flexible density function methodology to estimate risk-neutral densities implied by option pricing data.

Another related (non-stochastic) approach allows the volatility to depend functionally on the underlying security price. Various alternative proposals for the functional volatility process have been

Recent theoretical advances in this literature include Stein and Stein (1991), Heston (1993), and Bates (1996). In particular, Heston (1993) shows that a closed-form solution for a European call can be derived as an integral of the future security price density, which itself may be calculated by an inverse Fourier transform. This method may also be applied when correlation between the increments of the driving Brownian motions of the underlying asset and the volatility is non-zero. Thus, while Hull and White (1987) is an approximation, Fourier inversion methods are potentially more precise. Of course, estimation methods remain quite challenging⁵.

The key objective of the paper is to analyze the empirical performance of the stochastic volatility model proposed by Heston (1993) relative to the BS framework. At the same time, the empirical and theoretical behavior of the parameters characterizing the diffusion process assumed for the instantaneous variance is studied. Their behavior is discussed relative to the appropriate skewness and kurtosis in Heston´s model. This provides us with insights clarifying the reasons behind the poor performance found for the stochastic volatility option pricing model.

suggested. The well known constant elasticity of variance model due to Cox and Ross (1976) is the most prominent one.

An alternative approach for dealing with nonconstant volatility has been suggested by Rubinstein (1994), Jackwerth and Rubinstein (1996) and Jackwerth (1996), and a related series of papers by Derman and Kani (1994), Dupire (1994), Chriss (1995), Derman, Kani, and Chriss (1996). Instead of imposing a parametric functional form for volatility, they construct a binomial or trinomial numerical procedure so that a perfect fit with observed option prices is achieved. This procedure captures (by construction) the most salient characteristics of the data. In particular, the implied tree employed in the numerical estimation must correctly reproduce the volatility smile. The most popular models within this family use recombining binomial trees implied by the smile from given prices of European options. Once the appropriate prices and transition probabilities corresponding to the nodes and links of the tree are calculated, any American or path-dependent option may be priced consistently with the market

Our main interest is to investigate option pricing in markets where there are a very limited number of exercise prices traded simultaneously, and where liquidity is also less generalized than in the US market. This has serious consequences for the empirical implementation of the cross-sectional estimation of implied parameters proposed by Bates (1996) and Bakshi, Cao and Chen (1997). There are just not enough prices available in order to estimate jointly all parameters of the theoretical option models. This forces us to turn to a time-series framework which may be relevant for option markets narrower than the U.S. market. Besides this practical argument, it should also be pointed out that the cross-sectional approach may easily ignore relevant information in the original series that may not be embedded in the option prices⁶. We employ the indirect inference estimation technique of Gorurieroux, Monfort and Renault (1993) to avoid discretization biases when estimating diffusion processes.

We employ a database of intraday transaction prices for options on the Spanish IBEX-35 stock exchange index. This is one of the most popular option contracts traded in Europe. We are particularly concerned with exploring alternative options markets which are probably narrower than the fully investigated S&P 100 index options traded at the Chicago Board Options Exchange (CBOE).

This paper is organized as follows: the next section contains a brief summary of the Spanish options market. The data are described in Section 3. The theoretical model employed in the paper appears in Section 4. Section 5 presents the empirical results regarding the time-series estimation of the stochastic volatility parameters, and the theoretical discussion on the relationship between these parameters and the appropriate skewness and kurtosis in Heston´s framework. The empirical assessment to the extent of each model´s misspecification is contained in Section 6. Finally, in Section 7, we conclude with a summary and discussion.

2. The Spanish IBEX-35 Index Options

⁶ This issue is extensively discussed by Chernov and Ghysels (1998). They suggest a univariate timeseries methodology based on the Efficient Method of Moments using only option prices. However, their

The Spanish IBEX-35 index is a value-weighted index comprising the 35 most liquid Spanish stocks traded in the continuous auction market system. The official derivative market for risky assets, which is known as MEFF, trades a futures contract on the IBEX-35, the equivalent option contract for calls and puts, and individual option contracts for blue-chip stocks. Trading in the derivative market started in 1992. The market has experienced tremendous growth from the very beginning. Relative to the volume traded in the Spanish continuous market, trading in MEFF represented 40% of the regular continuous market in 1992, 156% in 1994, and 170% in 1995. The number of all traded contracts in MEFF relative to the contracts traded in the CBOE reached 20% in1995.

The IBEX-35 option contract is a cash settled European option with trading during the three nearest consecutive months and the other three months of the March-June-September-December cycle. The expiration day is the third Friday of the contract month. Trading occurs from 10:30 to 17:15. During the sample period covered by this research, the contract size is 100 Spanish pesetas times the IBEX-35 index, and prices are quoted in full points, with a minimum price change of one index point or 100 pesetas⁷. The exercise prices are given in 50 index point intervals.

It is important to point out that liquidity is concentrated in the nearest expiration contract. Thus, during 1995 and 1996 almost 90% of crossing transactions occurred in this type of contracts. Finally, it should be noticed that options and futures contracts are directly associated. The futures contract has exactly the same contract specifications as the IBEX-35 options. This will allow us to employ the futures price rather than the spot price in our empirical exercise. In fact, this is what is usually done by practitioners.

3. The data

For this paper, our database is comprised of all call options on the IBEX-35 index traded daily on MEFF during the period January 1996 through April 1996. Given the concentration in liquidity, our daily set of observations includes only calls with the nearest expiration day. Moreover, we eliminate all transactions taking place during the last week before expiration. In other words, for each monthly expiration date cycle, we only take into account prices for the first three weeks of the cycle.

approach obliges them to employ only highly liquid at-the-money options to estimate the diffusion process for the volatility.

⁷ This has recently been changed to 1,000 pesetas.

As usual in this type of research, our primary concern is the use of simultaneous prices for the options and the underlying security. The data, which are based on all reported transactions during each day throughout the sample period, do not allow us to observe simultaneously enough options with the same time-to-expiration on exactly the same underlying security price but with different exercise prices. In order to avoid large variations in the underlying security price, we restrict our attention to the 45-minute window from 16:00 to 16:45. It turns out that, on average and during our sample period, almost 25% of crossing transactions occur during this interval. Moreover, care was also taken to eliminate the potential problems with artificial trading that are most likely to occur at the end of the day. Thus, all trades after 16:45 were eliminated so that we avoid data which may reflect trades to influence market maker margin requirements. At the same time, using data from the same period each day avoids the possibility of intraday effects in the IBEX-35 index options market.

These exclusionary criteria yield a final daily sample of 768 observations. The implied volatility for each of our 768 options is estimated next. Note that we take as the underlying asset the average of the bid and ask price quotation given for each futures contract associated with each option during the 45-minute interval⁸. Recall that we are allowed to use futures prices given that the expiration day of the futures and options contracts systematically coincides during the expiration date cycle. Moreover, note that dividends are already taken into account by the futures price. To proxy for riskless interest rates, we use the daily series of annualized repo T-bill rates with either one week, two weeks or three weeks to maturity. One of these three interest rates will be employed depending upon how close the option is to the expiration day. Finally, as discussed by French (1984), volatility appears to be a phenomenon that is basically related to trading days. However, interest rates are paid by the calendar day. Thus, in order to estimate the implied volatility of each option in our sample, we employ Black´s (1976) option pricing formula adjusted by two time measures to reflect both trading days and calendar days until expiration. These implied volatilities will be used later as the basis for comparison with Heston´s implied instantaneous volatilities.

It might be that lack of liquidity in the futures market is responsible for the lack of variation in the price of the underlying asset during the 45-minute window. However, this is not the case. In fact, the

4. The theoretical framework of stochastic volatility

We briefly describe the stochastic volatility model that we want to investigate in this paper. As already mentioned, we concentrate our attention on the model proposed by Heston (1993). He obtains a closed-form solution for the price of a European call option on an asset with stochastic volatility. Heston works with Fourier transforms of conditional probabilities that the option expires in-the-money. The characterization of these probabilities is achieved through their characteristic function.

The stochastic volatility model proposed by Heston generalizes Geometric Brownian Motion by allowing the volatility of the return process itself to evolve stochastically over time in a square root mean-reverting fashion:

\[\mathrm{d} S _ {\mathrm{t}} = \mu S _ {\mathrm{t}} \mathrm{dt} + \sqrt {V _ {\mathrm{t}}} S _ {\mathrm{t}} \mathrm{dW} _ {\mathrm{lt}}\]

\[\mathrm{dV} _ {\mathrm{t}} = \kappa (\theta - \mathrm{V} _ {\mathrm{t}}) \mathrm{dt} + \sigma \sqrt {\mathrm{V} _ {\mathrm{t}}} \mathrm{dW} _ {2 \mathrm{t}}\tag{1}\]

\[\mathrm{dW} _ {1 \mathrm{t}} \mathrm{dW} _ {2 \mathrm{t}} = \rho \mathrm{dt}\]

\[\lambda (\mathrm{S}, \mathrm{V}, t) = \lambda \mathrm{V} _ {t}\]

where is the instantaneous expected rate of return of the underlying asset, is the instantaneous stochastic variance, θ is the long-term mean of the variance, κ governs the rate at which the variance converges to this mean, and σ represents the volatility of the variance process. The parameters of the variance process, θ κ σ , ,and are all strictly positive constants. and are each a standard Brownian motion allowed to be instanteneously correlated. Thus increases (decreases) in volatility could be related to the level of the underlying asset. Finally, the volatility risk premium, , is assumed to be proportional to the instantaneous variance, and its sign arises from the (sign of) correlation between the Brownian processes assumed for the instantaneous variance and the (aggregate) consumption.

futures market is at least as liquid as the spot market in terms of comparable measures of trading volume.

Under the risk-neutral probabilities (or -measure), the model is given by:

\[\mathrm{d} S _ {t} = r S _ {t} \mathrm{d} t + \sqrt {V _ {t}} S _ {t} \mathrm{d} W _ {1 t} ^ {*}\]

\[\mathrm{dV} _ {t} = \kappa^ {*} \left(\theta^ {*} - \mathrm{V} _ {t}\right) \mathrm{dt} + \sigma \sqrt {\mathrm{V} _ {t}} \mathrm{dW} _ {2 t} ^ {*}\tag{2}\]

\[\mathrm{dW} _ {1 \mathrm{t}} ^ {*} \mathrm{dW} _ {2 \mathrm{t}} ^ {*} = \rho \mathrm{dt}\]

where

Let be the value of a European call option where and to abbreviate. Heston´s formula is given by:

\[\mathrm{c} (\mathrm{S}, \upsilon , t) = \mathrm{S} _ {t} \mathrm{P} _ {1} - \mathrm{Ke} ^ {- \mathrm{r} (\mathrm{T} - t)} \mathrm{P} _ {2}\tag{3}\]

where, and are two risk-neutralized probabilities having the same interpretation as in the standard BS expression.

In the application below, we use future options so that the actual version of the formula we employ is given by:

\[\mathrm{c} (\mathrm{F}, \upsilon , t) = \mathrm{e} ^ {- \mathrm{r} (\mathrm{T} - t)} \left(\mathrm{F} _ {t} \mathrm{P} _ {1} - \mathrm{KP} _ {2}\right)\tag{4}\]

where F is the future price on the underlying spot price, and

\[\mathrm{P} _ {\mathrm{j}} \big (\mathrm{x}, \upsilon , \mathrm{T} - t; \ln [ \mathrm{K} ] \big) = \operatorname{Prob} \big (\mathrm{x} _ {\mathrm{T}} \geq \ln [ \mathrm{K} ] | \mathrm{x} _ {t} = \mathrm{x}, \upsilon_ {t} = \upsilon \big)\tag{5}\]

where, (the probability of the event depends on whether we chose the future for j = 1, or the riskless rate for j = 2), and is, therefore, the conditional probability that the option expires in-the-money. These probabilities depend on the vector of parameters, given by the processes assumed by Heston under the original probability. It is important to note that the expressions for these probabilities are slightly different from the original values given by Heston (1993) since we are using futures. The actual formulae employed in this paper are provided in Appendix A.

In the empirical implementation of the model, we could always go to option prices and implicitly infer the parameters under the risk neutral probabilities. Given the limited number of option prices available at each time, the cross-sectional approach is just not possible. Our approach employs the original asset data, so that we estimate the parameters from the true process by adjusting the discretization biases throughout the indirect inference estimation methodology. This is discussed in the following section.

5. The indirect estimation of the stochastic variance process

5.1 The estimation framework

To estimate the process under the original probability measure given by (1), we have to assume that available data are discrete-time observations of a continuous time process. If we apply regular econometric methods to discrete-time approximations, we would have a serious estimation -discretization- bias in our results. In order to avoid this bias, we employ the indirect estimation proposed by Gouriéroux, Monfort and Renault (1993). The procedure consists of two steps. First of all, by maximum likelihood techniques, we estimate an appropriate auxiliary model. Secondly, the estimates of the auxiliary model are compared with estimators based on simulations of the path of the continuous time process given by (1). More specifically, we have to introduce a discrete time analogue of (1) corresponding to a small time unit τ, such that 1/τ is an integer. This is done by either Euler approximation of (1) or, alternatively, by using the discretization suggested by Nowman (1997). Then for a given value of the parameters, we simulate the process, and obtain simulated values for the observation dates by merely selecting the values corresponding to integer indexes. This yields an accurate simulation of V as long as τ is sufficiently small.

Next, we precisely describe the steps necessary for the indirect estimation procedure: 1. Let t

Then, equation (1) becomes

\[\mathrm{dx} _ {\mathrm{t}} = \left(\mu - \mathrm{V} _ {\mathrm{t}} / 2\right) \mathrm{dt} + \mathrm{V} _ {\mathrm{t}} ^ {1 / 2} \mathrm{dW} _ {1 \mathrm{t}}\tag{6}\]

\[\mathrm{dV} _ {\mathrm{t}} = \kappa (\theta - \mathrm{V} _ {\mathrm{t}}) \mathrm{dt} + \sigma \mathrm{V} _ {\mathrm{t}} ^ {1 / 2} \mathrm{dW} _ {2 \mathrm{t}}\tag{7}\]

\[\mathrm{dW} _ {1 \mathrm{t}} \mathrm{dW} _ {2 \mathrm{t}} = \rho \mathrm{dt}\]

The expressions (6) and (7) are used to obtain the following set of estimators:

\[\Rightarrow \Omega \equiv (\mu , \kappa , \theta , \sigma , \rho)\]

2. Let us consider next the discretization of both (6) and (7) with frequency τ: For (6):

\[\mathrm{x} _ {t} = \mathrm{x} _ {t - \tau} + \mu \tau - \frac {\mathrm{V} _ {t - \tau}}{2} \tau + \mathrm{V} _ {t - \tau} ^ {1 / 2} \tau^ {1 / 2} \eta_ {1 t}\tag{8}\]

For (7), we employ two alternative discretizations:

a) Euler discretization:

\[\mathrm{V} _ {t} = \kappa \theta \tau + (1 - \kappa \tau) \mathrm{V} _ {t - \tau} + \varepsilon_ {t}\tag{9}\]

where,

\[\varepsilon_ {t} = \sigma \tau^ {1 / 2} V _ {t - \tau} ^ {1 / 2} \left[ \rho \eta_ {1 t} + (1 - \rho^ {2}) ^ {1 / 2} \eta_ {2 t} \right]\]

and,

\[\left(\eta_ {1 t}, \eta_ {2 t}\right) ^ {\prime} \approx \mathrm{N} (0, \mathrm{I})\]

b) Nowman discretization:

\[\mathbf {V} _ {t} = \theta \Big (1 - e ^ {- \kappa \tau} \Big) + e ^ {- \kappa \tau} \mathbf {V} _ {t - \tau} + \varepsilon_ {t}\tag{10}\]

where,

\[\varepsilon_ {t} = \sigma \left(\frac {1 - e ^ {- 2 \kappa \tau}}{2 \kappa}\right) ^ {1 / 2} V _ {t - \tau} ^ {1 / 2} \left[ \rho \eta_ {1 t} + (1 - \rho^ {2}) ^ {1 / 2} \eta_ {2 t} \right]\]

\[\left(\eta_ {1 t}, \eta_ {2 t}\right) ^ {\prime} \approx \mathrm{N} (0, \mathrm{I})\]

3- Our auxiliary model is given by the Nagarch(1,1) specification⁹:

Let

\[\mathrm{R} _ {\mathrm{t}} = \mu + \xi_ {\mathrm{t}}; \xi_ {\mathrm{t}} = \mathrm{h} _ {\mathrm{t}} ^ {1 / 2} \varepsilon_ {\mathrm{t}}; \varepsilon_ {\mathrm{t}} \stackrel {\mathrm{iid}} {\approx} \mathrm{N} (0, 1)\tag{11}\]

\[\mathbf {h} _ {t} = \boldsymbol {\omega} + \beta \mathbf {h} _ {t - 1} + \alpha \bigg (\xi_ {t - 1} + \gamma \mathbf {h} _ {t - 1} ^ {1 / 2} \bigg) ^ {2}\]

where γ represents the relation between the shocks and the conditional variance.

The set of paramaters to be estimated by maximum likelihood is given by:

\[\Psi \equiv (\mu , \omega , \alpha , \beta , \gamma)\]

4. Now we discuss the steps for the indirect estimation itself:

[4.1] Estimation of the auxiliary Nagarch(1,1) model with the observed data

\[\Rightarrow \hat {\Psi} \equiv (\hat {\mu}, \hat {\omega}, \hat {\alpha}, \hat {\beta}, \hat {\gamma})\]

[4.2] We give initial values for

⁹ See León and Mora (1998) for the behavior of alternative specifications within the GARCH family in the Spanish stock market.

[4.3] We simulate with

\[\Rightarrow \text { we have: } \frac {\mathrm{T}}{\tau} \text { observations } \Rightarrow 1 0 \mathrm{xTobservations}\]

( ) τ 1 is a dimensional vector T x

\[\Theta_ {\tau} \equiv \left\{\left(\eta_ {1 t}, \eta_ {2 t}\right) ^ {\prime}: t = 1, 2, \dots , T / \tau \right\}\]

[4.4] Simulate with Euler (Nowman) discretization

Given [4.3] and , go to (9) (or (10)) to generate and go to (8) to generate:

\[\mathrm{X} \left(\Omega_ {0}, \Theta_ {\tau}\right) \equiv \left\{\widetilde {\mathrm{x}} _ {\mathrm{t}}; \mathrm{t} = 0, \tau , 2 \tau , \dots , \mathrm{T} / \tau \right\}\]

[4.5] In order to work with the same frequency as the real data, take from the following values:

\[\Gamma \big (\Omega_ {0}, \Theta_ {\tau} \big) \equiv \left\{\widetilde {\mathrm{x}} _ {0}, \widetilde {\mathrm{x}} _ {1 0 \tau}, \widetilde {\mathrm{x}} _ {2 0 \tau}, \dots , \widetilde {\mathrm{x}} _ {\mathrm{T}} \right\}\]

[4.6] Now, we go back to the auxiliary model and with the data from [4.5], we have a new set of in (11)

⇒ we estimate the Nagarch with the simulated data at real frequency, and get

\[\tilde {\Psi} \big (\Omega_ {0}, \Theta_ {\tau} \big)\]

which is the vector of ML estimators of the Nagarch with for

[4.7] We have the same number of parameters in Ω and , so that we, in fact, minimize a distance with I as the weighting matrix. Then, if

\[\hat {\Psi} = \tilde {\Psi} \left(\Omega_ {0}, \Theta_ {\tau}\right) \Rightarrow \text { indirect estimators are } \Omega_ {0} \Rightarrow \text { END }\]

[4.8] If

\[\hat {\Psi} \neq \tilde {\Psi} \big (\Omega_ {0}, \Theta_ {\tau} \big) \Rightarrow \mathrm{GOTO[4.2]}\]

Give new values for

Continue the process and stop if,

\[\hat {\Psi} = \tilde {\Psi} \big (\Omega_ {1}, \Theta_ {\tau} \big)\]

otherwise come back to [4.2] and repeat the process until convergence.

5.2 Empirical results

We first estimate the in-sample period from January 2, 1994 to January 2, 1996 with continuously compounded hourly returns from the Spanish IBEX-35 index. The calculation of returns is based on the last recorded logarithmic index levels over consecutive hourly intervals. It is well known that the first hourly return incorporates adjustments to the information which has arrived overnight, and therefore presents a higher average return variability than any other hourly return. This basically implies that this first return is not an hourly return, and we consequently delete it from the estimation. Our final sample length consists of 2,450 hourly observations.

The results for the in-sample period are reported in Table 1. We consider four alternative combinations of simulations and frequencies, and two different discretization specifications. The results contained in the first column are obtained simulating the process once (N = 1), and 2,450x10 data points since the frequency used is equal to . In the second column of Table 1, we simulate 10 times , and we generate 10 series of size 2,450x10 given that the frequency is again . Hence, our final estimates are based on the following minimization: . In the third column, the process is simulated once (N = 1), but the calibration of the Nagarch(1,1) model is performed with more data. In particular, the length of the vector used in the estimation is 2,450x10. Given that the frequency is τ = 1/10, we have a total of 2,450x10x10 data points. Note, of course, that once we go back to the Nagarch(1,1) using simulated data of equal frequency as the real data, we have 2,450x10 observations. This is the methodology employed in the rolling procedure for the out-of-sample estimation which will be used in testing Heston´s option pricing model under Euler discretization. Finally, the results reported in the fourth column simulate the process once (N = 1), but the frequency is established at τ = 1/50.

The estimations contained in Table 1 suggest that, independently of the alternative procedure employed, the results tend to be quite similar. In particular, whether we discretize the process either by Euler or by Nowman does not make any difference at all. There seems to be some minor difference in the estimator of the volatility of the variance process, and the estimator of the correlation coefficient between the shocks when we use the frequency at τ = 1/50. In any case, independently of the procedure employed, the estimate of the correlation coefficient is, surprisingly, positive and close to zero. Given the asymmetry coefficient found in the Nagarch (1,1), we would have expected a negative correlation coefficient to reflect the asymmetry generally observed in literature to reflect that agents seem to react more to bad news than to good news¹⁰.

For the out-of-sample estimation, the same process is estimated 80 times using systematically 2,450 past observations. This rolling procedure of the indirect inference is necessary to yield estimates of the parameters involved in Heston´s expression for each day between January 3, 1996 and April 30, 1996. Thus, the daily changing estimates of these parameters are used as inputs in Heston´s option pricing formula. Figure 1 presents the evolution of the parameters associated with the stochastic variance process throughout the out-of-sample period. As we can observe from the figure, the long-term volatility (the standard deviation of θ ), the rate of convergence of the instantaneous variance to the long-term average, κ , and the volatility of the variance process, σ , remain quite stable over the out-of-sample period. However, the coefficient of correlation between the shocks for the stock and the variance, ρ , increases continuously. As before, it is interesting to observe that the correlation remains positive throughout the out-of-sample period.

¹⁰ Duan (1997) shows that both the Glosten, Jagannathan and Runkle (1993) -GJR (1,1)- and Nagarch (1,1) models converge to the same limiting diffusion process for the stochastic volatility, i.e. the meanreverting Geometric Brownian motion. León and Sentana (1998) shows that the relationship between the parameters of this Brownian process and the ones corresponding to the Nagarch (1,1) model is exactly

ρ = + γ 2 1 2γ ²

given by: . Then, it must be the case that sign(ρ) = sign (γ). Of course, this limiting result does not hold for the square-root mean reverting process. However, it seems reasonable to expect the same sign for both parameters.

5.3 Skewness, kurtosis and the parameters of the stochastic volatility process

The time-varying behavior of the correlation coefficient found above may have a serious impact on the capacity of Heston´s model to explain option pricing data. It suggests that, in this case, we may have a similar problem than the one we have when assuming constant volatility in the BS context. If correlation between prices and volatility changes continuously over time, skewness may also exhibit time-varying behavior. This is clearly a potential and relevant problem for models with stochastic volatility.

On the other hand, according to the estimates shown in Figure 1, it seems that the behavior of the volatility of volatility is rather stable over time. This suggests that accounting for changing kurtosis may not be as crucial as taking into consideration changing skewness.

Das and Sundaram (1998) obtain closed-form expressions for conditional and unconditional skewness and kurtosis under Heston´s stochastic volatility model. Their expressions, for a given frequency of ∆t = 1, are as follows:

a) The conditional case:

\[\mathrm{SKEW} = \left(\frac {3 \sigma \rho e ^ {\kappa / 2}}{\sqrt {\kappa}}\right) \left[ \frac {\theta (2 - 2 e ^ {\kappa} + \kappa + \kappa e ^ {\kappa}) - v (1 + \kappa - e ^ {\kappa})}{\left[ \theta (1 - e ^ {\kappa} + \kappa e ^ {\kappa}) + v (e ^ {\kappa} - 1) \right] ^ {3 / 2}} \right]\]

\[\mathrm{KURT} = 3 \left[ 1 + \sigma^ {2} \left(\frac {\theta \mathrm{A} _ {1} - v \mathrm{A} _ {2}}{\mathrm{B}}\right) \right]\tag{12}\]

where,

\[\mathrm{A} _ {1} = \left(1 + 4 \mathrm{e} ^ {\kappa} - 5 \mathrm{e} ^ {2 \kappa} + 4 \kappa \mathrm{e} ^ {\kappa} + 2 \kappa \mathrm{e} ^ {2 \kappa}\right) + 4 \rho^ {2} \left(6 \mathrm{e} ^ {\kappa} - 6 \mathrm{e} ^ {2 \kappa} + 4 \kappa \mathrm{e} ^ {\kappa} + 2 \kappa \mathrm{e} ^ {2 \kappa} + \kappa^ {2} \mathrm{e} ^ {\kappa}\right)\]

\[\mathrm{A} _ {2} = 2 \left(1 - \mathrm{e} ^ {2 \kappa} + 2 \kappa \mathrm{e} ^ {\kappa}\right) + 8 \rho^ {2} \left(2 \mathrm{e} ^ {\kappa} - 2 \mathrm{e} ^ {2 \kappa} + 2 \kappa \mathrm{e} ^ {\kappa} + \kappa^ {2} \mathrm{e} ^ {\kappa}\right)\]

\[\mathrm{B} = 2 \kappa \biggl [ \theta \Bigl (1 - \mathrm{e} ^ {\kappa} + \kappa \mathrm{e} ^ {\kappa} \Bigr) + \upsilon \Bigl (\mathrm{e} ^ {\kappa} - 1 \Bigr) \biggr ] ^ {2}\]

and where, as before,

b) The unconditional case:

\[\mathrm{SKEW} = 3 \left(\frac {\rho \sigma}{\sqrt {\kappa \theta}}\right) \left[ \frac {1 - \mathrm{e} ^ {\kappa} + \kappa \mathrm{e} ^ {\kappa}}{\kappa^ {3 / 2} \mathrm{e} ^ {\kappa}} \right]\tag{13}\]

\[\mathrm{KURT} = 3 \left[ 1 + \frac {\sigma^ {2}}{\kappa \theta \kappa^ {2} e ^ {\kappa}} \left(1 - e ^ {\kappa} + \kappa e ^ {\kappa} + 4 \rho^ {2} \left[ 2 - 2 e ^ {\kappa} + \kappa + \kappa e ^ {\kappa} \right]\right) \right]\]

Given our rolling estimates of ρ and σ from January 1996 to April 1996, we can daily estimate (12) and (13), so that we may observe how the characteristics of the distribution of the underlying asset -skewness and kurtosis- change with both the correlation coefficient and the volatility of volatility.

Figure 2 depicts the conditional and unconditional skewness over the sample period. As we can easily observe, their behavior closely follows the pattern of the correlation coefficient of Figure 1. As expected, skewness mainly arises from the correlation between changing prices and stochastic volatility of the underlying asset. The problem is that, of course, Heston´s model assumes a constant (unconditional) skewness over time¹¹.

In the Appendix B of this paper, it is shown that, all else being constant, the relationship between skewness and the correlation coefficient is positive; i.e. for both the conditional and unconditional cases. This explains the behavior of skewness in Figure 2. Therefore, if (as is in fact the case) the correlation coefficient tends to be rather unstable, an option pricing model with stochastic skewness would be welcomed¹². Given this evidence, we should not expect to find a good performance of the stochastic volatility model when we compare observed market prices with theoretical prices. We will come back to this issue later in the paper.

Figure 3 contains the same evidence for the kurtosis case. In principle, the impact of time-varying kurtosis on the misspecification of Heston´s model seems to be much less severe than the influence of the correlation coefficient. Its pattern over time is much more stable than ρ. This is related to the behavior of the volatility of volatility, σ, in Figure 1. This is the parameter that allows for kurtosis in the stochastic volatility option pricing model, and it does not seem to change much over time. Once again, in Appendix B, we show that, in general and for reasonable values of the parameters, the relationship between conditional and unconditional kurtosis and σ is positive:

unconditional kurtosis:

conditional kurtosis: as long as

¹¹ Given the similarity between the behavior of unconditional and conditional skewness, the impact of the instantaneous stochastic variance on the conditional skewness must be negligible relative to the effect of the correlation coefficient.

¹² See Nandi (1998) for an exhaustive discussion of the importance of the correlation coefficient on option pricing behavior.

As a summary, time-varying skewness may be the key issue to analyze if we want to understand the failure of the tests we report below. Its consequences may be much more serious than the potential effects of changing kurtosis. However, this point will be clarified in the next section.

6. The empirical performance of Heston´s option pricing model

6.1 Introductory empirical evidence

Empirical tests of Heston´s stochastic volatility model are based on all available call options transacted over the 45-minute interval from 16:00 to 16:45 during the period January 3, 1996 to April 30, 1996.

Table 2 describes the sample properties of the call option prices employed in this work. Average prices, average relative bid-ask spread and the number of available calls are reported for each moneyness category. Moneyness is defined as the ratio of the exercise price to the futures price. A call option is said to be deep out-of-the-money if the ratio K/F belongs to the interval (1.03, 1.08); out-of-the-money if ; at-themoney when ; in-the-money when ; and deep-in-themoney if . As we already discussed, there are 768 call option observations, with OTM, ATM and ITM options respectively representing 51%, 32% and 17%. The average call price ranges from 12.88 pesetas for deep OTM options to 185.42 pesetas for deep ITM options. The average relative bid-ask spread goes exactly in the opposite direction to the average price. In particular, it ranges from 0.39 for deep OTM options to 0.10 for deep ITM calls.

Any reasonable alternative model to BS must be able to properly price deep OTM and deep ITM call options. Of course, Heston´s stochastic volatility model, given by equations (4) and (5), is a potential and particularly interesting candidate. Given that we do not back out the implicit parameters of the stochastic variance process from option prices, we must take explicitly into account the volatility risk premium. Note that, once we have estimated for each day the parameters of the variance process, κ θ σ , , , and , ρ using the rolling indirect inference procedure, we still need to get the volatility risk premium, λ, and the instantaneous variance, V, before we can actually price a given call option¹³.

In principle, we now have just two parameters to be estimated, λ and V. Therefore, it seems reasonable to expect that we may use cross-sectional data to implicitly inferred the parameters that minimized the sum of squared errors (SSE) in a given day of the sample.

Given the set of parameters of the indirect estimation procedure obtained for a particular day t in the sample, , and for each option, and each day t, we define the pricing error as:

\[\mathbf {e} _ {\mathrm{it}} \left(\mathbf {V} _ {\mathrm{t}}, \lambda ; \hat {\boldsymbol {\Omega}} _ {\mathrm{t}}\right) = \hat {c} _ {\mathrm{it}} \left(\mathbf {K} _ {\mathrm{i}}\right) - c _ {\mathrm{it}} \left(\mathbf {K} _ {\mathrm{i}}\right)\tag{14}\]

where is the theoretical price of call i in day t, and is the corresponding observed market price.

We then want to find the instantaneous variance, and the risk premium parameter, , to solve:

\[\mathrm{SSE} _ {\mathrm{t}} \equiv \min _ {\left\{\mathrm{V} _ {\mathrm{t}}, \lambda \right\}} \sum_ {\mathrm{i} = 1} ^ {\mathrm{n}} \left[ \mathrm{e} _ {\mathrm{it}} \left(\mathrm{V} _ {\mathrm{t}}, \lambda ; \hat {\Omega} _ {\mathrm{t}}\right) \right] ^ {2}\tag{15}\]

Surprisingly, however, we are not able to find a global minimum for and that simultaneously solves (15). This is a regular finding independently of the number of options available for a given day. This is a particularly annoying result, since we are now forced to impose a given risk premium parameter before solving for the instantaneous variance, V, within a simplified version of (15)¹⁴.

¹³ We have estimated the original process and not the risk neutral process.

¹⁴ A similar problem is found in Guo (1998) who employs a summation over the number of options and the number of trading days. He is forced to employ a two-step iteration approach to solve for the minimization problem. Note that, for a given day, we are just summing over the number of available options.

As discussed before, the sign of the volatility risk premium arises from the (sign) correlation between the Brownian motion of the variance process and the Brownian motion of the consumption process. Since we do not want to assume a priori a particular sign for λ¹⁵, and given the evidence about the historical risk premium associated with the stock market index which may suggest an upper bound for the covariance between stochastic variance and the growth rate of aggregate consumption, we estimate Heston´s theoretical option prices with three alternative lambdas: and λ . In any case, more will be said about these magnitudes later.

For each of the three levels of chosen lambdas, we can solve equations (14) and (15), and obtain a daily estimate of instantaneous variance from January 2, 1996 to April 29, 1996. In this way, for each assumed lambda, we have 80 daily estimates of the instantaneous variance which can be used as inputs in Heston´s expression for each of the subsequent days in the testing sample period. Hence, we will be able to calculate 80 daily prediction errors from January 3, 1996 to April 30, 1996 for each of the three versions of Heston´s model and also, of course, for the BS formula. In the latter case, the implied volatility estimated in the corresponding BS´s versions of (14) and (15) is employed.

Figure 4 contains the evolution for each of the four series of implied volatilities from January 3, 1996 to April 30, 1996. In general, the larger the required volatility risk premium imposed, the (very slightly) larger the instantaneous volatility. Moreover, it can be easily appreciated that Heston´s volatility, independently of the volatility risk premium assumed, tends to be higher than BS´s volatility. A priori, this may have serious implications for pricing.

To obtain a general picture of the potential misspecifications of the option pricing models employed in this research, we report the average pattern of implied volatilities across degree of moneyness. The results are shown in Figure 5.

¹⁵ In this regard, recent papers by Kapadia (1998) and Guo (1998) suggest that a negative volatility risk premium means that equity options act as a hedge to the market portfolio, and that this negative sign is consistent with stock returns reacting negatively to positive volatility shocks.

¹⁶ From 1963 to 1997 the annualized average market risk premium of the Spanish value-weighted index is 6.8% with a standard deviation of 18.5%.

In the BS case, we back out the implied volatility of each call option and for each day between January 3, 1996 to April 30, 1996 using the procedure discussed in Section 3. Then, the equally-weighted implied volatility for each moneyness category and each day in the sample period is calculated. There is a U-shaped pattern with a hump in the middle. This suggests that the BS model tends to underprice deep OTM and deep ITM calls. This is the typical smile pattern of the Spanish option market analized by Peña, Rubio, and Serna (1998).

At the same time, note that, given the parameters from the indirect estimation, the observed market price and the assumed level of lambda, we can back out the implied instantaneous variance for each of the 768 available calls. Hence, for each level of lambda, we may analyze the pattern of implied volatilities across alternative degrees of moneyness. The evidence reported in this regard in Figure 5 suggests a rather asymmetric smile in Heston´s model independently of the volatility risk premium imposed. It seems that Heston´s approach tends to underprice (deep) ITM calls and overprice (deep) OTM calls.

We now turn to formal tests of the alternative option pricing specifications considered in this paper.

6.2 Out-of-sample pricing performance for alternative option pricing models

In order to test the out-of-sample pricing performance for each model analyzed in this work, we employ two years of rolling data to estimate, by indirect inference, the parameters of the stochastic variance process assumed in (1). Given these estimates, and a chosen volatility risk premium, λ, we use all call options available in our 45-minute window to compute for each day from January 2, 1996 to April 29, 1996 the instantaneous variance that minimized the squared error between the theoretical value and the market price of the call options according to (14) and (15). We then compute the theoretical price of each option using the previous day´s instantaneous variance and the corresponding parameters of the stochastic volatility process. For the BS case, the previous day´s implied volatility that minimized the squared error between the theoretical value and the market price of the options is used to obtain the theoretical BS price of each option in the sample.

In this way, we have 768 pricing errors for each of the calls available from January 3, 1996 to April 30, 1996, and for each of the models analyzed. These pricing errors are the basis for our analysis. Table 3 reports two measures of performance for the alternative model specifications. Panel A contains the absolute pricing error which is the sample average of the absolute difference between the model price and the market price for each call in the testing sample period. This statistic is reported for each moneyness category and for all calls in the sample. In Panel B, the reported percentage pricing error is the sample average of the theoretical price minus the market price, divided by the market price. Again, this statistic is calculated for each moneyness category and for all calls in the sample.

Overall, in absolute terms, Heston´s option pricing model tends to value slightly better than BS. The absolute pricing error over all calls is approximately 2.8 pesetas for Heston´s model independently of the volatility risk premium assumed¹⁷, and 3.2 pesetas for the BS model. It is quite important to notice that the level of the volatility risk premium does not seem to have any influence on the performance of Heston´s stochastic volatility model. The pricing errors obtained under alternative lambdas are practically identical. There might be some evidence in favor of a positive risk premium, but we can safely conclude that option prices do not seem to be sensitive to the volatility risk premium. This is an important empirical result. Kapadia (1998) shows that the expected value of the delta-hedged gain, under a stochastic volatility model where volatility risk is not priced, is equal to zero. This is exactly the same result that holds under BS. Consequently, if we may assume that volatility risk is not priced, the analysis of the dynamic hedging performance of both models is clearly facilitated¹⁸.

It should be pointed out that the overall slightly better performance of Heston´s model is not maintained throughout all moneyness categories. In particular, Heston´s model tends to value ATM and ITM calls better than BS. However, the opposite result holds for

¹⁷ The last column for λ = -15 will be discussed later.

18 On the other hand, if volatility risk is priced, the average delta-hedged gain is proportional to the magnitude and sign of the volatility risk premium. Comparisons of the dynamic hedging performance between Heston´s model and BS may be much more complicated than the analysis carried out by Bakshi, Cao, and Chen (1997).

OTM calls. This is also the case when we analyze the percentage pricing error in Panel B. Heston´s model, regardless of the volatility risk premium imposed, tends to overvalue OTM calls and, at the same time, the model undervalues ITM calls. However, the percentage pricing for ATM calls is practically zero. In fact, ATM calls are clearly more consistent with a stochastic volatility model than with the well known lognormal assumption. BS, on the other hand, tends to undervalue deep OTM and deep ITM calls. This is consistent with a U-shaped volatility smile. In Heston´s case, the evidence points towards a sneer rather than a regular smile.

6.3 A further and brief discussion on the volatility risk premium

It is of course difficult to justify the chosen magnitudes for the volatility risk premium. To clarify this issue, we employ the Nagarch specification, given by (11), to obtain an estimation for the instantaneous variance for each day in our sample period. Unfortunately, as we pointed out before, the discrete Nagarch model does not converge, even in the limit, to the square root mean-reverting diffusion process assumed for the instantaneous variance. Therefore, this exercise should be understood as a simple and approximate way of obtaining the volatility risk premium embedded in the option pricing data over our sample period. Figure 6 presents the evolution of the Nagarch daily estimated volatility. It resembles quite closely to the implied volatilities shown in Figure 4.

Given the daily variance estimate from the Nagarch model, we can solve for that λ that, on daily basis, minimizes the sum of squared errors as we already did in expression (15) in Section 6.1. Hence, we are now able to obtain an approximate volatility risk premium for each day in our sample which is consistent with option market prices.

Figure 7 shows the behavior of our estimate of the volatility risk premium from January 1996 to April 1996. As we can easily observe, this premium seems to be quite volatile throughout the four months of available data¹⁹. It turns out that, for most days, the volatility risk premium is negative and its average magnitude is approximately -15. Figure 8 shows the daily implied volatilities for the BS case, and for Heston´s model with either , and . The last column of Table 3 reports the absolute pricing errors and the percentage pricing errors for all options and for options across alternative degrees of moneyness when . Once again, the results are quite robust relative to our previous discussion, so that we should be confident about the relatively little impact caused by the volatility risk premium on the performance of Heston´s stochastic volatility model.

19 Guo (1998) obtains a similar result using an ad hoc procedure to estimate the implicit average volatility risk premium per year in the currency options market.

6.4 The statistical significance of the out-of-sample performance

Overall, at least over the sample period studied and regardless of Heston´s model seems to overvalue, while BS tends to undervalue call options. Unfortunately, however, the simple statistics reported above do not help in making inferences in terms of the statistical significance of improvement when we contrast one model versus another.

In this paper, the statistical significance of performance for out-of-sample pricing errors is assessed by analyzing the proportion of theoretical prices lying outside their corresponding bid-ask spread boundaries²⁰. The following Z-statistic for the difference between two proportions is employed in the tests. The statistic is given by:

\[Z = \frac {p _ {1} - p _ {2}}{\sqrt {p _ {1} (1 - p _ {1}) / n _ {1} + p _ {2} (1 - p _ {2}) / n _ {2}}}\tag{16}\]

where is always the proportion of BS prices outside the bid-ask boundaries, and is the equivalent proportion for alternative Heston´s specifications. and are sample sizes corresponding to these proportions. The statistic is asymptotically distributed as a standardized normal variable.

The empirical results are reported in Table 4. Given that we are also interested in knowing whether a given theoretical valuation model undervalues or overvalues market prices, the Z-statistic is also calculated to obtain the proportion for which the theoretical model yields a price below the bid quote, and the proportion for which the model gives a price above the ask quote. If a theoretical model tends to undervalue market prices, it would yield a higher proportion of prices below the bid quote. If, on the other hand, the model tends to overvalue market prices, it would have a higher proportion of prices above the ask quote.

²⁰ See Corrado and Su (1996).

When we consider all call options together, the p-value for statistiscal improvement of Heston´s model over the BS formula is equal to 0.068. The proportion of BS prices lying outside the bid-ask boundaries is 48%, while the proportion of Heston´s model, regardless of the volatility risk premium, is about 42%. As we see, there is a slight improvement, but we might interpret the results as rather disappointing. It is not clear at all that, given the costs of implementation, Heston´s approach is worthwhile in terms of practical applications.

It is also the case that 35% of BS prices are below the bid quote. Given that Heston´s model yields approximately 18% of prices below the bid, we can conclude that BS, on average, significantly undervalues market prices relative to Heston´s approach. At the same time, Heston´s prices are 24% of total observations above the ask quote. We can also conclude that Heston´s model tends to significantly overvalue market prices relative to the BS formula. Thus, mispricing associated with BS is basically related to the tendency of the model to yield prices below market values. However, Heston´s mispricing is a consequence of the model´s tendency to offer prices above their corresponding market values.

When we classify all call options by moneyness, similar conclusions are obtained. A call option is said to be OTM if K/F > 1, and a call is classified as ITM when K/F < 1. In general, within a given category of moneyness, neither model is statistically superior to the other. However, once again, BS mispricing comes from the tendency of the model to undervalue either OTM or ITM calls relative to Heston´s model (and relative to market prices). The opposite is true for Heston´s formula relative to BS (and market prices). It should be pointed out that in the latter case, the main problem arises when we value OTM with Heston´s formula. There seems to be a strong tendency in Heston´s model to overvalue this type of options. This was also reflected in Table 3.

It was mentioned above that the percentage pricing error of Heston´s model for ATM calls turns out to be quite small. It may be the case that Heston´s formula works particularly well for ATM options. Calls are said to be ATM when The exercise described above is repeated for this type of options. Overall, as expected, Heston´s model tends to yield lower proportions of prices lying outside the bid-ask spread than in previously analyzed cases. However, the p-value for the difference relative to BS proportions is just 0.078. As before, BS tends to yield a statistically significant higher proportion of prices lower than the bid quote relative to Heston´s prices, and Heston´s model presents a statistically significant higher proportion of prices above the ask quote relative to BS prices. Now, however, Heston´s formula has similar proportions above the ask or below the bid. Regarding ATM calls, Heston´s misspecification cannot be explained by either overvaluation or undervaluation of call prices.

6.5 The structure of pricing errors

Given the poor empirical performance of both BS formula and Heston´s stochastic volatility model, a further analysis trying to understand the structure of pricing errors of these models would seem to be called for. Following the evidence reported by Peña, Rubio, and Serna (1998) and Bakshi, Cao, and Chen (1997), we use a simple regression framework to study the relation between the percentage pricing errors and factors that are either option-specific or market dependent. We first take as given an option pricing model, and let be the i-th call option´s percentage pricing error on day t. Finally, we run the following regression for the whole sample period:

\[\mathrm {e_ {it}} = \alpha + \beta_ {1} \mathrm {X_ {it}} + \beta_ {2} \tau_ {\mathrm{it}} + \beta_ {3} \mathrm {SP_ {t}} + \beta_ {4} \mathrm {VOL_ {t}} + \beta_ {5} \mathrm {TERM_ {t}} + \beta_ {6} \mathrm {SKEW_ {t}} + \beta_ {7} \mathrm {KURT_ {t}} + \omega_ {\mathrm{it}}\tag{17}\]

where X is the moneyness of the ith call at time t as defined by the ratio between the exercise price (K) and the futures price (F); is the annualized time to expiration of the ith call on day t; SP is the average relative bid-ask spread of all calls and puts transacted between 16:00 and 16:45 on date t; is the annualized standard deviation of the IBEX-35 index returns computed from 1-minute intradaily returns; is the yield differential between the annualized ten-year government bond and the annualized onemonth repo Treasury bill; SKEW_t is the conditional skewness, and KURT_t the conditional kurtosis. They are given by expression (12)²¹.

Table 5 contains the regression results based on the entire sample period and 768 call options, and where the standard error for each coefficient estimate is adjusted by the White (1980) heteroskedasticity-consistent estimator.

The explanatory variables employed in the regressions tend to be statistically significant. However, there are important differences between the percentage errors associated with either BS or Heston.

In particular, a key point of the results is the statistical significance of the coefficient estimates related to skewness and kurtosis, when we consider Heston´s model under any of the three volatility risk premia used in the analysis. This is a very important result. As we argued in Section 5.3, the assumption of constant correlation between stochastic variance and price changes, and even the assumption of constant volatility of variance under Heston´s model do not seem to be the appropriate assumptions to adequately explain the behavior of option prices even allowing for stochastic volatility.

Note that, on the other hand, the coefficient associated with kurtosis is not statiscally significant in the BS case. However, as in Heston´s case, the skewness bias is also relevant to explain the BS pricing errors.

Table 5 also shows that the annualized standard deviation of the IBEX-35 index slightly explains the percentage pricing errors independently of the model employed in the estimation. On average, pricing errors tend to be lower the higher the volatility of the index. However, the statisitical significance of the coefficients is very weak.

Traditional biases are not corrected for any of the models. The bias associated with moneyness has the opposite sign in both types of models. As expected, given previous results, Heston tends to price OTM options worse than ITM calls. However, on average, the opposite result is found for BS. Moreover, the bias related to time to expiration seems to be relevant for both types of models. On average, the percentage pricing errors are larger the longer the time to expiration.

21 The same regression was run using one day lagged values for the market dependent variables. Very similar results were found. The actual regressions employ excess kurtosis as an explanatory variable.

The influence on percentage pricing errors of both the yield differential between interest rates and the average spread for all calls and puts transacted over the 45 minute window are, interestingly enough, different for Heston and BS expressions. In the BS case, the higher the long-term yield relative to the short-term rate, the lower the percentage pricing error. However, this result disappears when we allow for stochastic volatility under any of Heston´s specifications.

Let us analyze the spread variable. Regressions of a similar type were run including the relative bid-ask spread at date t of the call i. This is, contrary to the results reported in Table 5, a contract-specific variable. Surprisingly, the estimated coefficients are never significant regardless of the model considered. By doing this, we are really incorporating a transaction cost variable directly associated with the liquidity of each individual option. Again, this variable does not seem to be significant in explaining percentage pricing errors. On the other hand, however, when we include the average spread over all call and put options for a particular day t, the estimated coefficient becomes positive and significant in Heston´s case. This aggregate variable may indicate the average consensus about the uncertainty of trading throughout the option market. It may be understood as the average adverse selection confronting market makers in trading options on the Spanish index. As we see from Table 5, the larger the average spread -larger average adverse selection among traders- the higher the percentage pricing error in Heston´s stochastic volatility models. The impact of this type of uncertainty does not seem to be relevant in explaining the percentage pricing errors of BS.

In short, the pricing errors from Heston´s framework have some moneyness, maturity, average (aggregate) bid-ask spread, skewness and kurtosis related biases. On the other hand, the BS case presents some moneyness, maturity, yield differential and skewness associated biases. Neither model seems to capture appropriately the underlying distribution characteristics of the underlying asset. Further research is clearly justified.

8. Conclusions

This is one of the first papers that has investigated the behavior of Heston´s stochastic volatility model in pricing options with actual market data²². It is however, the first paper employing a time-series framework. We suggest that this is a more flexible approach since it allows estimation with limited cross-sectional samples. This should be particularly useful in thin markets where a single cross-sectional approach may be difficult to implement. Moreover, to employ just a cross-sectional procedure may ignore relevant information that may be included in the original series but not in the option prices.

The empirical results, however, are quite disappointing. On average, over all call options available in our sample, Heston´s model improves the (poor) performance of BS just marginally. It is clear that this extremely limited improvement cannot justify the implementation costs involved in the estimation of Heston´s approach²³. The overall rejection of Heston´s model coincides with recent findings by Bakshi, Cao and Chen (1997) and Chernov and Ghysels (1998) for options written on the S&P 500 index.

We are quite convinced that the ultimate reasons behind the performance failure of Heston´s model are closely related to the time-varying skewness and kurtosis found in the data. In particular, the assumption of a constant correlation coefficient between returns and stochastic volatility should be relaxed if we really want to have a richer model. Unfortunately, of course, the complexities needed to price options seem to increase without bounds. It may be the case that simple nonparametric methodologies are able to incorporate the missing (realistic) factors in our option pricing models.

It is also found that the daily volatility risk premium presents a quite volatile behavior over time. However, our evidence suggests that the volatility risk premium has a negligible impact on the pricing performance of Heston´s model.

A potentially interesting area of research might be related to endogenously incorporating liquidity costs in option pricing models with either stochastic volatility, stochastic jumps or both. Once again, this approach may be extremely demanding from a theoretical point of view but it would probably be welcomed.

22 Bates (1996) uses deutschmark foreign currency options, whereas Bakshi, Cao and Chen (1997), Nandi (1998) and Chernov and Ghysels (1998) employ S&P 500 options.

APPENDIX A Heston´s stochastic volatility model using futures options

Given that we use futures options, the actual version of the formula we employ is given by:

\[\mathsf {c} \big (\mathrm{F}, \upsilon , t \big) = \mathsf {e} ^ {- r (\mathrm{T} - t)} \big (\mathrm{F} _ {t} \mathrm{P} _ {1} - \mathrm{KP} _ {2} \big)\]

²³ Hedging performance of alternative models is not analyzed in this paper. It is possible that the hedging improvement under Heston´s stochastic volatility model might be clearly superior to BS.

where F is the future price on the underlying spot price, K is the exercise price and the probabilities are given by:

\[P _ {j} = \frac {1}{2} + \frac {1}{\pi} \int_ {0} ^ {\infty} R e \left(\frac {e ^ {- i \phi \ln [ K ]} f _ {j}}{i \phi}\right) d \phi ; j = 1, 2\]

where is the real part of the function i is the imaginary number , and

\[\mathrm{f} _ {\mathrm{j}} \big (\mathrm{x}, \upsilon , \mathrm{T} - \mathrm{t}; \phi \big) = \exp \big [ \mathrm{C} (\mathrm{T} - \mathrm{t}; \phi) + \mathrm{D} (\mathrm{T} - \mathrm{t}; \phi) \upsilon + \mathrm{i} \phi \mathrm{x} \big ]\]

where²⁴,

\[\mathrm{C} (\mathrm{T} - \mathrm{t}; \phi) = \frac {\mathrm{a}}{\sigma^ {2}} \left\{\left(\mathrm{b} _ {\mathrm{j}} - \rho \sigma \phi \mathrm{i} + \mathrm{d}\right) (\mathrm{T} - \mathrm{t}) - 2 \ln \left[ \frac {1 - \mathrm{ge} ^ {\mathrm{d} (\mathrm{T} - \mathrm{t})}}{1 - \mathrm{g}} \right] \right\}\]

\[D (T - t; \phi) = \frac {b _ {j} - \rho \sigma \phi i + d}{\sigma^ {2}} \left[ \frac {1 - e ^ {d (T - t)}}{1 - g e ^ {d (T - t)}} \right]\]

\[\mathrm{g} = \frac {\mathrm{b} _ {\mathrm{j}} - \rho \sigma \phi \mathrm{i} + \mathrm{d}}{\mathrm{b} _ {\mathrm{j}} - \rho \sigma \phi \mathrm{i} - \mathrm{d}}\]

\[\mathrm{d} = \sqrt {(\rho \sigma \phi \mathrm{i} - \mathrm{b} _ {\mathrm{j}}) ^ {2} - \sigma^ {2} (2 \mathrm{u} _ {\mathrm{j}} \phi \mathrm{i} - \phi^ {2})}\]

\[\mathbf {a} = \kappa \theta\]

\[b _ {1} = \kappa + \lambda - \rho \sigma\]

\[\mathbf {b} _ {2} = \kappa + \lambda\]

\[\mathbf {u} _ {1} = 1 / 2; \mathbf {u} _ {2} = - 1 / 2\]

C(T-t;φ)

²⁴ As we have already pointed out, the expression for C(T-t;φ) below is slightly different than the original value given by Heston (1993) since we are using futures.

verifying that , and where and (and therefore the probabilities depend on the vector of parameters, given by the processes assumed by Heston under the original probability.

APPENDIX B

(A) The relation between conditional and unconditional skewness and the correlation between changing prices and stochastic volatility for a given frequency

A.1) The unconditional case:

\[\mathrm{SKEW} = 3 \left(\frac {\rho \sigma}{\sqrt {\kappa \theta}}\right) \left[ \frac {1 - \mathrm{e} ^ {\kappa} + \kappa \mathrm{e} ^ {\kappa}}{\kappa^ {3 / 2} \mathrm{e} ^ {\kappa}} \right]\]

\[\frac {\partial \mathrm{SK}}{\partial \rho} = \frac {3 \sigma}{\sqrt {\kappa \theta}} \left[ \frac {1 - \mathrm{e} ^ {\kappa} + \kappa \mathrm{e} ^ {\kappa}}{\kappa^ {3 / 2} \mathrm{e} ^ {\kappa}} \right] \cong \frac {3 \sigma \kappa^ {2}}{(\kappa \theta) ^ {1 / 2} \kappa^ {3 / 2} \mathrm{e} ^ {\kappa}} > 0\]

A.2) The conditional case:

where,

\[\mathrm{B} _ {1} \equiv \theta \left(2 - 2 \mathrm{e} ^ {\kappa} + \kappa + \kappa \mathrm{e} ^ {\kappa}\right) - v \left(1 + \kappa - \mathrm{e} ^ {\kappa}\right)\]

\[\mathrm{B} _ {2} \equiv \left[ \theta \left(1 - \mathrm{e} ^ {\kappa} + \kappa \mathrm{e} ^ {\kappa}\right) + v \left(\mathrm{e} ^ {\kappa} - 1\right) \right] ^ {3 / 2}\]

Let us simplify the expressions for and to show that they are strictly positive:

\[\mathrm{B} _ {1} = \theta \Big (2 - 2 \mathrm{e} ^ {\kappa} + \kappa + \mathrm{ke} ^ {\kappa} \Big) - \upsilon \Big (1 + \kappa - \mathrm{e} ^ {\kappa} \Big) \cong \theta \kappa^ {2} - \upsilon (0) = \theta \kappa^ {2} > 0\]

\[\mathrm{B} _ {2} = \left(\theta \Big [ 1 - \mathrm{e} ^ {\kappa} + \kappa \mathrm{e} ^ {\kappa} \Big ] + \upsilon \Big [ \mathrm{e} ^ {\kappa} - 1 \Big ]\right) ^ {3 / 2} \cong \left(\theta \Big [ \kappa^ {2} \Big ] + \upsilon \Big [ \kappa \Big ]\right) ^ {3 / 2} = \left(\theta \kappa^ {2} + \upsilon \kappa\right) ^ {3 / 2} = \kappa^ {3 / 2} \big (\upsilon + \theta \kappa \big) ^ {3 / 2} > 0\]

\[\Rightarrow \frac {\partial \mathrm{SK}}{\partial \rho} = \frac {3 \sigma \mathrm{e} ^ {\kappa / 2}}{\sqrt {\kappa}} \frac {\mathrm{B} _ {1}}{\mathrm{B} _ {2}} > 0\]

(B) The relation between conditional and unconditional kurtosis and the volatility of volatility for a given frequency

B.1) The unconditional case:

\[\frac {\partial \mathrm{K}}{\partial \sigma} = \frac {6 \sigma}{\kappa^ {3} \theta \mathrm{e} ^ {\kappa}} \left(1 - \mathrm{e} ^ {\kappa} + \kappa \mathrm{e} ^ {\kappa} + 4 \rho^ {2} \left[ 2 - 2 \mathrm{e} ^ {\kappa} + \kappa + \kappa \mathrm{e} ^ {\kappa} \right]\right) \cong \frac {6 \sigma}{\kappa \theta \mathrm{e} ^ {\kappa}} \left(1 + 4 \rho^ {2}\right) > 0\]

B.2) The conditional case:

\[\mathrm{KURT} = 3 \left[ 1 + \sigma^ {2} \left(\frac {\theta \mathrm{A} _ {1} - v \mathrm{A} _ {2}}{\mathrm{B}}\right) \right]\]

where and B are given in (12) and do not depend on σ, and

\[\frac {\partial \mathrm{K}}{\partial \sigma} = 6 \sigma \left(\frac {\theta \mathrm{A} _ {1} - v \mathrm{A} _ {2}}{\mathrm{B}}\right)\]

Let us analyze the sign of this derivative:

\[\cong 8 \kappa^ {2} + 4 \rho^ {2} \kappa^ {2} (9 + \kappa) = 4 \kappa^ {2} [ 2 + \rho^ {2} (9 + \kappa) ] > 0\]

\[\cong 4 \kappa^ {2} + 8 \rho^ {2} \kappa^ {2} (3 + \kappa) = 4 \kappa^ {2} \Big [ 1 + 2 \rho^ {2} (3 + \kappa) \Big ] > 0\]

\[\begin{array}{r l} & {\theta \mathrm{A} _ {1} - \upsilon \mathrm{A} _ {2} \cong 4 \theta \kappa^ {2} \Big [ 2 + \rho^ {2} (9 + \kappa) \Big ] - 4 \upsilon \kappa^ {2} \Big [ 1 + 2 \rho^ {2} (3 + \kappa) \Big ]} \\ & {= 4 \kappa^ {2} \left\{2 \theta + (9 + \kappa) \theta \rho^ {2} - \upsilon - 2 (3 + \kappa) \upsilon \rho^ {2} \right\} = 4 \kappa^ {2} \left\{\big [ \theta (9 + \kappa) - 2 (3 + \kappa) \upsilon \big ] \rho^ {2} + 2 \theta - \upsilon \right\} > 0} \end{array}\]

iff

\[\left[ \theta (9 + \kappa) - 2 (3 + \kappa) v \right] \rho^ {2} + (2 \theta - v) > 0\]

Let,

\[\mathbf {C} _ {1} = (9 + \kappa) \theta - 2 (3 + \kappa) v\]

\[\mathbf {C} _ {2} = 2 \theta - \upsilon\]

\[\Rightarrow \frac {\partial \mathrm{K}}{\partial \sigma} > 0 \text {iff} \mathrm{C} _ {1} \rho^ {2} + \mathrm{C} _ {2} > 0\]

What is the sign of C₁?

\[C _ {1} = (9 + \kappa) \theta - 2 v (3 + \kappa) > 0 \text { iff } (9 + \kappa) \theta > 2 v (3 + \kappa)\]

\[\Rightarrow \frac {v}{\theta} < \frac {9 + \kappa}{6 + 2 \kappa} \Rightarrow \frac {v}{\theta} \in \left(0, \frac {9 + \kappa}{6 + 2 \kappa}\right)\]

\[\Rightarrow \mathrm{h} (\kappa) = \frac {9 + \kappa}{6 + 2 \kappa} > 1 \text { iff } \kappa \in [ 0, 3 ]\]

\[\mathrm{h} ^ {\prime} (\kappa) = \frac {6 + 2 \kappa - 2 (9 + \kappa)}{(6 + 2 \kappa) ^ {2}} = \frac {6 + 2 \kappa - 1 8 - 2 \kappa}{(6 + 2 \kappa) ^ {2}} = \frac {- 1 2}{(6 + 2 \kappa) ^ {2}} < 0\]

is a decreasing function

Now,

\[\mathrm{h} (0) = 3 / 2; \mathrm{h} (3) = 1\]

\[\kappa \in [ 0, 3 ] \Rightarrow \mathrm{h} (\kappa) \in [ 1, 3 / 2 ] \Rightarrow \frac {\upsilon}{\theta} \in [ 1, 3 / 2 ]\]

It must therefore be the case that,

\[C _ {1} = \left\{ \begin{array}{l l} & > 0 \text { iff } \frac {\upsilon}{\theta} \in [ 1, 3 / 2 ] \\ < 0 \text { iff } \frac {\upsilon}{\theta} \in (0, 1) \cup (3 / 2, + \infty) \end{array} \right.\]

What is the sign of

\[\mathrm{C} _ {2} = 2 \theta - v < 0 \text { iff } v > 2 \theta \Rightarrow v / \theta > 2\]

Therefore,

\[C _ {2} = \left\{ \begin{array}{l} > 0 \text {iff} \frac {\upsilon}{\theta} \in (0, 2 ] \\ < 0 \text {iff} \frac {\upsilon}{\theta} \in (2, + \infty) \end{array} \right.\]

Hence, we should consider 4 cases:

\[\text {(i)} \quad v / \theta \in (0, 1) \Rightarrow C _ {1} < 0, C _ {2} > 0\]

Let the function g( ) ρ be equal to,

\[\mathbf {g} (\boldsymbol {\rho}) = \mathbf {C} _ {1} \boldsymbol {\rho} ^ {2} + \mathbf {C} _ {2}\]

Then,

\[\mathrm{g} (0) = \mathrm{C} _ {2}\]

\[\mathrm{C} _ {1} \rho^ {2} + \mathrm{C} _ {2} = 0 \Rightarrow \rho^ {2} = - \mathrm{C} _ {2} / \mathrm{C} _ {1}\]

\[\mathrm{g} ^ {\prime} (\rho) = 2 \mathrm{C} _ {1} \rho = 0 \Rightarrow \rho = 0\]

\[\mathrm{g} ^ {\prime \prime} (\rho) = 2 \mathrm{C} _ {1}\]

Therefore, in this particular case,

\[\mathrm{g} ^ {\prime \prime} (\rho) = 2 \mathrm{C} _ {1} < 0 \Rightarrow \text { maximum when } \rho = 0\]

Then,

\[\rho^ {2} = - \mathrm{C} _ {2} / \mathrm{C} _ {1} > 0 \Rightarrow \rho = \pm \sqrt {\mathrm{C} _ {2} / \mathrm{C} _ {1}}\]

\[\Rightarrow \rho < \left| \sqrt {\mathrm{C} _ {2} / \mathrm{C} _ {1}} \right| \Rightarrow \mathrm{g} (\rho) > 0 \Rightarrow \theta \mathrm{A} _ {1} - v \mathrm{A} _ {2} > 0 \Rightarrow \frac {\partial \mathrm{K}}{\partial \sigma} > 0\]

\[\text {(ii)} \mathrm{v} / \theta \in [ 1, 3 / 2 ] \Rightarrow \mathrm{C} _ {1} > 0, \mathrm{C} _ {2} > 0\]

In this second case,

\[\mathrm{g} ^ {\prime \prime} (\rho) = 2 \mathrm{C} _ {1} > 0 \Rightarrow \text { minimum when } \rho = 0\]

As before, let us check what happens when . Given that now , there does not exist any intersection with the real line. We can then conclude that,

\[\forall \rho , \text { we have that, } g (\rho) > 0 \Rightarrow \theta A _ {1} - v A _ {2} > 0 \Rightarrow \frac {\partial K}{\partial \sigma} > 0\]

\[\text {(iii)} \mathrm{v} / \theta \in (3 / 2, 2 ] \Rightarrow \mathrm{C} _ {1} < 0, \mathrm{C} _ {2} > 0\]

Therefore, this is exactly the same as (i):

\[\rho^ {2} = - \mathrm{C} _ {2} / \mathrm{C} _ {1} > 0 \Rightarrow \rho = \pm \sqrt {\mathrm{C} _ {2} / \mathrm{C} _ {1}}\]

\[(\mathrm{iv}) \upsilon / \theta \in ] 2, + \infty [ \Rightarrow \mathrm{C} _ {1} < 0, \mathrm{C} _ {2} < 0\]

\[\Rightarrow \mathrm{g} ^ {\prime \prime} (\rho) = 2 \mathrm{C} _ {1} < 0 \Rightarrow \text { maximum when } \rho = 0\]

And, once again, there is no intersection with the real line. Relative to case (ii) however,

for all

Summing up:

(I) For unconditional kurtosis:

(II) For conditional kurtosis:

\[\mathrm{(II.1)} \frac {\partial \mathrm{K}}{\partial \sigma} > 0 \text {iff} \frac {\upsilon}{\theta} \in (0, 2 ]\]

\[\text {(II.1.1)} \text {If} \frac {\upsilon}{\theta} \in (0, 1) \cup (3 / 2, 2 ] \Rightarrow \frac {\partial K}{\partial \sigma} > 0 \text {iff} \rho < \left| \sqrt {C _ {2} / C _ {1}} \right|\]

\[\text {(II.1.2)} \text { If } \frac {\upsilon}{\theta} \in [ 1, 3 / 2 ] \Rightarrow \frac {\partial K}{\partial \sigma} > 0 \quad \forall \rho\]

\[\text {(II.2)} \frac {\partial K}{\partial \sigma} < 0 \text {iff} \frac {v}{\theta} \in (2, + \infty) \Rightarrow (g (\rho) < 0, \forall \rho)\]

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FIGURE 1 INDIRECT INFERENCE (January 96-April 96)

Conditional Kurtosis ■Unconditional Kurtosis

•BS ■HES (lambda =0) □HES (lambda = 0.5) HES (lambda = -0.5)

FIGURE 7 VOLATILITY RISK PREMIUM (January 96-April 96)

FIGURE 8 IMPLIED VOLATILITIES (January 96-April 96)