Retrieving Risk Neutral Moments and Expected Quadratic Variation from Option Prices

Transcription

1 Retrieving Risk Neutral Moments and Expected Quadratic Variation from Option Prices by Leonidas S. Rompolis and Elias Tzavalis Abstract This paper derives exact formulas for retrieving risk neutral moments of future payoffs of any order from generic European-style option prices. It also provides an exact formula for retrieving the expected quadratic variation of the stock market implied by European option prices, which nowadays is used as an estimate of the implied volatility, and a formula approximating the jump component of this measure of variation. To implement the above formulas to discrete sets of option prices, the paper suggests a numerical procedure and provides upper bounds of its approximation errors. The performance of this procedure is evaluated through a Monte Carlo exercise. The paper provides clear cut evidence that ignoring the jump component of the underlying asset can lead to seriously biased estimates of the new volatility index suggested by the Chicago Board Options Exchange (CBOE). This is also confirmed by an empirical exercise based on market option prices written on the S&P 5 index, which shows that the jump component of quadratic variation is significant and varies substantially during financial crises. Keywords: Risk neutral moments, characteristic function, expected quadratic variation. JEL: C4, G, G2 The authors would like to thank Karim Abadir and Jens Carsten Jackwerth for useful comments on an early version of the paper. Department of Accounting and Finance, Athens University of Economics and Business. 76 Patission street, 434 Athens, Greece. rompolis@aueb.gr Department of Economics, Athens University of Economics and Business. 76 Patission street, 434 Athens, Greece. e.tzavalis@aueb.gr.

2 Introduction There is recently growing interest in the option pricing literature to retrieve model-free estimates of risk neutral moments (RNM) of the future payoff or return of an underlying asset (stock) implied by cross-sectional sets of generic European-style call/put options. These moments do not rely on any option pricing model and thus, have a number of interesting applications in practice. For instance, they can be used to study the information content of option prices about future stock market volatility or any other higher-order risk neutral moment (see, e.g. Canina and Figlewski (993), Christensen and Prabhala (998), Dennis and Mayhew (22), Jiang and Tian (25) and Bollerslev and Zhou (26)). Secondly, they can be employed to examine the relationship between higherorder risk neutral moments of asset returns and their physical counterparts, and/or to estimate the implied risk aversion coefficient of the stock and option markets participants (see, e.g. Bakshi, Kapadia and Madan (23), Bakshi and Madan (26a,b) and Rompolis and Tzavalis (2)). Thirdly, they can be applied to estimate the risk neutral density (RND) of the underlying asset return based on an approximation density method (see, e.g. Corrado and Su (996), Rompolis and Tzavalis (28) and Rompolis (2)). There are two generally accepted methods of retrieving model-free estimates of risk neutral moments (RNM) of future asset returns from option prices. As shown in this paper, these are theoretically consistent with each other. The first relies on nonparametric estimates of the risk neutral density (RND) of the future payoff of the underlying asset from cross-sectional option prices in a first step, exploiting the Breeden-Litzenberger (978) relationship. From this density, we can then obtain estimates of RNM of any order. However, as is well-known in the literature, in order to be successful nonparametric density estimation methods used to retrieve RNDs require large sets of cross-section option prices, which are not often available in practice (see Pagan and Ullah (999), for a survey). 2 The second method of retrieving model-free estimates of RNM of future asset returns from option prices relies on the work of Bakshi and Madan (2) showing that risk neutral asset price payoffs are spanned by a continuum of out-of-the-money (OTM) generic European-style options. Based on this work, Bakshi, Kapadia and Madan (23) have provided See, e.g. Ait-Sahalia and Lo (998), Bliss and Panigirtzoglou (22), Ait-Sahalia and Duarte (23). 2 To improve over their performance, they have been suggested semi-parametric methods, but these depend on the correct specification of the underlying asset price process, see, e.g. Corrado and Su (996) and Melick and Thomas (997). 2

3 model-free formulas for retrieving the first four non-central RNM of the underlying asset return from OTM cross-sectional sets of European call and put prices directly, without having first to recover the correct RND. Using an analogous methodology, Demeterfi et al. (999) and Britten- Jones and Neuberger (2) have derived a formula of implied volatility based on OTM options. This formula relies on the assumption that the underlying asset price follows a pure continuous process. In this paper, we extent the above works and derive exact, closed-form formulas for retrieving RNM from OTM generic European-style option prices. In particular, the paper extents the work of Bakshi, Kapadia and Madan (23) in two main dimensions. First, it provides RNM formulas of any order and for any random variable taken as a function of the asset payoff. These higher-order RNM are necessary to better approximate the RND. Second, it derives formulas which can be applied to retrieve RNM of more complicated payoffs using generic European-style options, such as Asian or basket, which can be used in applied work. To this end, we first derive the risk neutral characteristic function of options payoffs. From this function, we can directly obtain RNM formulas of any order. The RNM formulas derived by the paper are also used to obtain an exact analytic formula of the expected quadratic variation of the stock market implied by European options, referred to as the new-vix index. 3 This measure of volatility does no longer rely of the Black-Scholes (BS) model. It has been introduced by the Chicago Board Options Exchange (CBOE) to provide model-free estimates of the stock market volatility and facilitate the trading of volatility derivatives. The exact formula that the paper derives for the VIX index indicates that, if the underlying asset price process includes random jumps, then the CBOE formula used to calculate the VIX is only an approximation of the exact formula. The latter depends on the jump component of the underlying asset price. As shown in the paper, ignoring this component will lead to biased estimates of the new- VIX. To estimate this component of expected quadratic variation, the paper derives a model-free approximation formula based on higher-order RNM. To implement the above formulas to finite, discrete sets of option prices, the paper suggests a numerical method which relies on an interpolation-extrapolation smoothing procedure. We use cubic spline functions to interpolate implied volatilities between available strike prices. To extrap- 3 The original VIX now is known as the VXO. 3

4 olate these volatilities beyond the minimum and maximum strike prices, we employ a linear or, alternatively, a constant function. We also consider the case of not extrapolating them. To control for the approximation errors of the above numerical method, the paper provides theoretical upper bounds of them and, more importantly, it shows how these bounds vary at the endpoints of the extrapolation scheme. To assess the performance of this method, the paper conducts an extensive simulation study. This study examines the degree of accuracy of the method with respect to strike price intervals met in practice, the number of observations of the call or put prices available and, finally, alternative extrapolation schemes. Our simulation study is also interested in examining if the model-free formula of the jump component of the expected quadratic variation suggested by the paper provides accurate estimates of it. To carry out our simulation study, we generated data from the stochastic volatility with jumps (SVJ) model suggested in the literature to improve over the performance of the BS and stochastic volatility models (see, e.g. Bakshi, Cao and Chen (997)). The results of our simulation study provide a number of interesting results with important portfolio management implications. They show that the suggested by the paper numerical method for implementing the RNM formulas to option price data sets works very satisfactory even for small intervals of strike prices, often available in practice. This is true independently on the number of the call and put option prices available. As was expected, the magnitude of the approximation errors reduce as the strike price interval increases. This is more apparent for higher than second-order moments. Another interesting result of our simulation study is that ignoring the jump component of the underlying asset price process clearly leads to biased estimates of the VIX index. This component is adequately approximated by our model-free suggested formula. Applying this formula to actual option price data, the paper provides clear cut evidence that the bias of the VIX index due to the omission of the jump component of asset price process is significant and it varies substantially, especially during financial crises. The paper is organized as follows. Section 2 derives the RNM formulas for different categories of European-style option prices, while Section 3 derives the exact formula for retrieving the expected quadratic variation from option price data, taking into account possible jumps in the underlying asset price process. This section also derives the approximation formula of the jump component of the expected quadratic variation. Section 4 presents the numerical method for implementing 4

5 the above formulas to option price data and provides upper bounds of their approximation errors. Section 5 conducts our simulation exercise. Section 6 carries out our empirical exercise. Section 7 concludes the paper. All of the derivations are given in a technical appendix. 2 Risk neutral moment formulas Consider a generic European-style option contract with expiration date and strike price. Denote the random variable upon which the future payoff of the contract is written as ( ). This variable is a function of a stochastic process ( ) [ ] R, defined over the time interval [ ], with. For a plain vanilla option contract, ( ) is defined as ( ),where can be the payoff of any asset, e.g. stock, bond, index, futures or swap, at date. The random variable ( ) can also represent the payoff of an Asian option contract with payoff ( ) R. Under no arbitrage conditions, the price of a generic European call option contract at time, with maturity interval =, defined as ( ), can be written as follows: ( ) = {[ ( ) ] + } = Z = [ ( ) ] [ ( )], () where is the instantaneous riskless interest rate, denotes the risk neutral probability measure, [ ( ) ] + defined as [ ( ) ] + max { ( ) } is the payoff of the option at the expiration date, [ ( )] is the conditional risk neutral density function (RND) of ( ), attime. The price of a generic European put option contract at time, with maturity interval, denoted as ( ), isdefined as ( ) = {[ ( )] + } = Z = [ ( )] [ ( )] (2) where [ ( )] + max { ( )} is the payoff of this option at date. Let ( ) be defined as the following general function of payoff ( ): ( ) [ ( )] ( ),where : R R is a twice continuously differentiable function under measure and 5

6 is a constant. Then, based on Bakshi and Madan s (2) fundamental option pricing theorem, we can derive an analytic formula of the risk neutral characteristic function (RNCF) of function ( ) implied by generic European-style option prices, at time This formula is given in the next proposition. Proposition Let (Ω F ) be the probability space restricted to the time interval [ ], with its filtration F = {F ; [ ]}, ( ) be the conditional on F characteristic function of the random payoff ( ) under measure, where : R R is a twice continuously differentiable function. Then, the analytic form of ( ) is given as ( ) = + ( ) ³ [ ( )] + ½Z + + ( ) 2 ( ) 2 [ ( ) ( )] ( ) + Z + ( ) 2 ( ) 2 ¾ [ ( ) ( )] ( ) (3) where i denotes the imaginary number. The proof of the proposition is given in the Appendix. As expected by Bakshi and Madan s theorem, the RNCF of future payoff ( ) given by Proposition is spanned by a continuum of out-of-the-money (OTM) generic European-style option call and put prices over different strike prices. From this RNCF, we can derive the well known Breeden and Litzenberger (978) formula often used in practice to derive the RND of ( ) from option prices. This theoretical result implies that retrieving risk neutral moments (RNM) directly from OTM generic European-style option prices is theoretically consistent with deriving them from the well known Breeden and Litzenberger formula. This result is established in the next corollary. Corollary Proposition implies that the conditional on F risk neutral density of ( ), denoted [ ( )], can be calculated based on the Breeden-Litzenberger formula: [ ( )] = ½ 2 ( ), for 2 ( ) 6 2 ( ), for 2 ( ), (4) where = [ ( )]. 6

7 The proof of the corollary is given in the Appendix. The RNCF ( ) given by Proposition has a theoretical interest. It can be employed to obtain analytic, exact formulas of the non-central RNM of the future payoff ( ) of any order based on generic European-style option prices. These formulas are given in the next proposition. Proposition 2 Given the results of Proposition, the conditional on filtration F th-order noncentral risk neutral moment of payoff ( ) is given as for =and = ( ) ³ [ ( )] + Z + + ( ) ( ) + Z ( ) ( ), (5) = ½Z + ( ; ) ( ) + Z ¾ ( ; ) ( ), (6) for 2, where ( ; )=[ ( ) ( )] 2 [ ( )( ( ) ( )) + ( ) ( ) 2 ] The proof of this proposition is given in the Appendix. As a corollary of Proposition 2, next we give the th-order central RNM of future payoff ( ), for 2. Corollary 2 Proposition 2 implies that the conditional on filtration F th-order central risk neutral moment of ( ) is given by the following formula: e = ( ) " Z + ( ; ) ( ) + Z # ( ; ) ( ), (7) for 2, where = [ ( )] and ( ; ) =( ) 2. 7

8 The RNM formulas given by Corollary 2 can be obtained by assuming [ ( )] = ( ) and = [ ( )] in relationships (5)-(6). They indicate some distributional features of therndofthefuturepayoff ( ), implied by OTM call or put prices, which have both theoretical and practical interest. In particular, for =2and =4they show that the values of both the conditional variance (referred to as the volatility) and kurtosis of ( ), attime, increase with the prices of OTM calls and puts. This can explain the very high degree of positive correlation between these two central RNM of future payoffs implied by European option prices, which has been found in many empirical studies (see, e.g. Corrado and Su (997)). For =3, formula (7) implies that the third-order central moment determining the skewness of the RND, denoted as e 3, has negative sign when the OTM puts are more expensive that the OTM calls. This result was first noticed by Bates (99), based on a different framework of analysis. Based on Proposition 2, in next corollary we provide exact RNM formulas of the future asset log-return, defined as ( )=ln³, which are functions of plain vanilla European option prices. As mentioned in the introduction, these formulas have many interesting applications in the option pricing literature. Corollary 3 Proposition 2 implies that the conditional on filtration F th-order non-central RNM of the asset log-return ( ) is given as = µ Z + 2 ( ) + Z 2 ( ), (8) for =,and = ( Z + + Z 2 ln 2 ln µ 2 ln µ ( ) µ 2 µ ) ln ( ), (9) for 2. 8

9 To obtain the RNM formulas of the above corollary, given by (8)-(9), we have set ( )=ln( ) and = in equations (5)-(6), since ( ) for plain vanilla European option prices. Some interesting remarks on these formulas are given bellow. Remark The RNM formulas (8)-(9) clearly indicate that only the higher than second-order RNM of the log-return ( ) can be solely derived from OTM plain vanilla European call and put prices. For = {2 3 4}, the formulas of the RNM given by (9) for the log-return ( ) are consistent with those derived by Bakshi, Kapadia and Madan (23). Remark 2 As relationship (8) clearly indicates, the first order RNM depends also on the h i risk neutral expectation of the underlying asset price considered. Below, we present two different cases of assets where this expectation can be analytically derived. The first is when the underlying asset is a stock, or a stock index, which pays a constant dividend yield. Then, we have h i = ( ) and thus, equation (8) becomes = ( ) Z + 2 ( ) + Z 2 ( ) () The second case is when the underlying asset is a futures or a swap contract. Then, we will have h i =which implies that formula (8) becomes Z + Z = 2 ( ) + 3 Retrieving implied volatility from option prices 2 ( ). () Recently, there is a growing interest in deriving a model-free measure of the asset price process ( ) [ ] volatility implied from plain vanilla European options. This measure of volatility is referred to as the annualized expected quadratic variation (see, e.g., Demeterfi et al. (999), Britten- Jones and Neuberger (2), Jiang and Tian (25) and Carr and Wu (29)) and it is defined i hh i,whereh i is the quadratic variation of log-price process =ln over as time interval [ ]. It nowadays has become the benchmark for measuring stock market volatility. On September 22, 23, the Chicago Board Options Exchange (CBOE) started calculating a new stock market volatility index from option prices based on it. This index is known as the VIX 9

10 and is widely accepted by market participants. It also constitutes the underlying asset of volatility derivatives. In the next proposition, we derive an exact formula of the above volatility measure i hh i basedonthernmformulasprovidedinthe previous section (see Appendix). This is done under the more general assumption that the underlying asset price process contains discontinuous components which are nowadays considered very important in capturing the price dynamics of many financial assets (see, e.g. Eraker (24), and Barndorff-Nielsen and Shepard (26)). Proposition 3 Let the stochastic process of the underlying asset price ( ) [ ] be a semimartingale. Then, the annualized expected period quadratic variation i hh i, denoted as, can be analytically written as follows: = hh i i Z + Z + X 6 where =ln is the log-price process. Z 2 ( ) ( ) + µ + 2 ( ) 2 (2) The proof of this proposition is given in the Appendix. Theanalyticformulaof, given by equation (2), provides a direct link between RNM and the expected quadratic variation, which have been studied separately in the literature. According to this formula, the annualized expected -period quadratic variation,, depends on three terms. The first, as shown in the proof of Proposition 3, comes along the risk neutral mean. The second h R term, given as + i, depends on the underlying asset considered. If, for example,

11 this is a stock (or a stock market index) that pays a constant dividend yield, then it becomes 4 Z + =+( ) ( ). h R i Examples of assets where term + does not have any impact on formula (2) are futures or swap contracts. In this case, we can easily show the following result: Z + =. The third term of equation (2), defined as 2 X ( ) 2, is due to the discontinuous component of the asset price process ( ) [ ]. As is explained in the CBOE white page (see CBOE (23)), the CBOE procedure calculates the expected quadratic variation based on the following formula: Z + 2 ( ) + Z 2 ( ) + h +( ) ( ) i (3) Once an estimate of the expected quadratic variation is obtained, the VIX index is calculated as thesquarerootof multiplied by. Proposition 3 implies that if the asset price has no discontinuous component, then formula (3) coincides with the exact formula (2). In any other case, it constitutes an approximation formula, which ignores the discontinuous component of the 4 Note that if = =,then + =. In this case, formula (2) reduces to that provided by Britten-Jones and Neuberger (2) under the assumption that the underlying asset price follows a diffusion process, without discontinuous component. 5 Note that there is a small difference in the formula employed by the CBOE and equation (3). Instead of, the CBOE formula uses the first strike price below the futures price, denoted as, as integration bound. This means that the second term of can be written as 2 ln ' 2 (see also Jiang and Tian (27)) so as for the two formulas to be consistent with each other.

12 asset price ( ) [ ],defined by. The latter is not observed in the market. In the next proposition, we provide a model-free formula which can be used to approximate. Proposition 4 Let the stochastic process of the underlying asset price ( ) [ ] be a semimartingale. If where =ln, then 2 X 6 ( ), forall N +, (4) n P h io ( ) 2 can be approximated as ' 2 µ 3! 3 + 4! 4 (5) where 3 and 4 are respectively the third and fourth-order RNM, at time. The proof of this proposition is given in the Appendix. The approximation of, given by formula (5) of Proposition 4, indicates that one can calculate the jump component of the annualized quadratic variation using higher-order RNM 3 and 4. The latter can be estimated by OTM call and put option prices in a model-free way, using formula (9). The practical implication of these results is that one can correct the implied volatility estimate, calculated by the CBOE, for the jump component based on market data, using formula (5). The accuracy of formula (5) to capture the jump component of the expected quadratic variation will be examined in a simulation study, presented in Section 5. 4 Implementation of the RNM formulas To implement the RNM and the expected quadratic variation formulas derived in the previous sections to observed option prices, we need to rely on an efficient numerical method which fits them into finite, discrete cross-sectional sets of option prices. Following recent literature, 6 amethod which can be used for this purpose relies on an interpolation-extrapolation procedure of option prices implied volatilities. 7 More specifically, this method first interpolates implied volatilities 6 See Campa, Chang and Reider (998), Bliss and Panigirtzoglou (22), Dennis and Mayhew (22) and Jiang and Tian (25), inter alia. 7 Note that interpolation and extrapolation of implied volatilities, instead of option prices, has been recently suggested in many studies (see, e.g. fn 6). This is done in order to avoid numerical difficulties in fitting smooth 2

13 over the observed closed interval of strike prices [ min max ],where min and max denote the minimum and maximum strike prices observed, respectively. Then, it extrapolates these volatilities over intervals ( min ] and [ max + ), where option prices are not available. The interpolation of the implied volatilities is often carried out by using cubic splines, since these functions give smooth and accurate values over interval [ min max ]. On the other hand, the extrapolation of the implied volatilities over intervals ( min ] and [ max + ) can be done either by using a linear function (see, e.g., Bliss and Panigirtzogou (22)) or a constant (see, e.g., Jiang and Tian (25)). Both of these functions are truncated at the strike prices (which is considered as an approximation of a zero strike price) and. The latter is considered as an approximation of a strike price which tends to infinity. The strike prices and are calculated so as to correspond to put and call option prices, respectively, which are very close to zero, e.g., smaller than 3. Instead of these two extrapolation schemes, a third choice would be not to extrapolate the implied volatility function, i.e. to set = min and = max (see CBOE (23)). If the length of the available strike price interval [ min max ] is sufficiently large, then this can be considered as a natural choice, which can provide accurate estimates of RNM and expected quadratic variation from observed option prices. The above suggested numerical methods for implementing the RNM or expected quadratic variation formulas to option prices lead to an approximation error, whose significance must be investigated. This error is related to the curve-fitting scheme of implied volatilities over the interval [ ]. As is expected, it will depend on the number of option prices available, the length of the interval [ min max ] and the extrapolation scheme chosen. As is shown by equations (6) (or (9)), the impact of this approximation error on the estimates of the RNM will depend on the moment-specific function ( ; ), entering into the integrals of the RNM formulas employed. Given that this error can not be measured in practice, due to the fact that the true theoretical implied volatility function is not known, we will appraise its size effects on the estimates of the RNM and the expected quadratic variation formulas by deriving upper and lower bounds of it. This is done in the next proposition. The results of this proposition refer to the risk neutral moments of the asset log-return ( ), given by equations (9) and (), and the expected quadratic variation approximation formula (3), used by CBOE. functions into option prices. To convey the observed option prices into implied volatilities and vise versa, we use the Black-Scholes (BS) formula. This methodology does not require the BS model to be the true option pricing model. 3

14 Proposition 5 Let ( ) (or ( )) denote the true call (or put) pricing function with respect to variable =ln( ),while b ( ) (or b ( )) denote the call (or put) pricing function implied by our suggested smoothing method for implementing the RNM formulas to finite sets of European call and put prices. If b and d denote the approximated values of the true risk neutral moments and the approximated expected quadratic variation,, respectively, obtained by our numerical method, then the following approximation error bounds for b and d can be respectively derived: and where = b 6 ( ( ; )+ ( ; )+ ) for N + (6) ( ) ( ) d 2 6 ( ( ;)+ ( ;)+ ), (7) max ( ) b ( ) and = max ( ) b ( ) constitute upper bounds of the call and put option pricing functions approximation errors, respectively, = ln( ), =ln( ),and ( ; ) and ( ; ) are two functions which are evaluated at endpoints and and they are defined as follows: for = ( ; ) = (2( ) ( ) ) if 6 for > 2 if and for = ( ; ) = ( ) for > 2, respectively. is the truncation error of strike prices at endpoints and, respectively, defined as where ( ; )= Z + = ( ; ) ( ) + 2 hln ³ i 2 h ln Z ( ; ) ( ). for = 2 ³ i. for > 2 4

15 The proof of the proposition in given in the Appendix. The results of Proposition 5 imply that the upper bounds of the approximation errors of the RNM and the approximated expected quadratic variation, defined as b and d, respectively, depend critically on the option pricing function approximation errors and, multiplied with functions ( ; ) and ( ; ), respectively. These functions indicate that the approximation errors b and d depend on the order of the RNM,. investigate more analytically the effects of functions ( ; ) and ( ; ) on b and d, nextwederivethelimitsofthemfor + (or ),whileinfigurewe plot their values over, fordifferent RNM orders. Notethatthisfigure gives the values of the joint function of ( ; ) and ( ; ), which is written as ( ; ) if ( ; ) = ( ; ) if > To By taking limits of, we can prove the following result: and lim ( ;) = for = + lim ( ; ) = 2 ( ) ( ) for > 2 + lim ( ; ) =+ for N + We can also prove that ( ; ) 6 ( ) R + and N + where ( ) is given as for = ( ) = 2 ( ) ( ) for > 2 The above results indicate that function ( ; ), which multiplies the call option pricing approximation error, is bounded by function ( ). Although the values of ( ) increase with the 5

16 order of RNM, 8 the effects of ( ; ) on b and d can be controlled. This can be confirmed by the plots of Figure which clearly indicate that, for the empirically plausible values of endpoint [6], the values of function ( ; ) are small and close to each other independently of. In contrast to ( ; ), the above results indicate that function ( ; ), which multiplies the put option pricing approximation error, is not bounded. This function tends to infinity as. Although this result theoretically implies that the size-effects of the error on b and d maynotbecontrolled,inpracticethismaynotbetrue.thismay happen because, for empirical plausible values of endpoint [ ], function ( ; ) does not take very large positive values. This can be confirmed by the inspection of the plots of Figure. 9 Taking together the results of Figure and Proposition 5 indicate that, when implementing the RNM or formulas to option prices, the extrapolation scheme, and the endpoints and chosen lead to a trade-off effect between truncation error and option pricing approximation errors and. As and tend to zero and infinity, respectively, decreases, while and increase, since both intervals ( ) and ( ) increase. The value of function ( ; ) controlling the effect of on bounds b and d will also increase. If the negative effect caused by and on dominates its positive effect on, then extrapolating implied volatilities through linear or constant functions will increase the magnitude of the above bounds. The effects of,and and of the alternative extrapolation choices considered on these bounds are investigated by our a Monte Carlo simulation study, presented in the next section. 5 Simulation study In this section, we assess the performance of the numerical method suggested in the previous section for implementing the model-free RNM formulas (9)-() and the expected quadratic variation 8 The function ( ) takes the following values for = 2 6 : (2) = 47, (3) = 3 24, (4) = 75, (5) = and (6) = Note that, in our simulation experiments presented in the next section, almost all the strike price intervals are found to be between the values of the endpoints = and = 57. In iterations, it was found only one case where = 3 which lies outside the above interval [ = = 57]. This happened when a linear function was used in the extrapolation scheme, instead of a constant. 6

17 formula (3) to plain vanilla European option prices. This is done by conducting an extensive Monte Carlo simulation exercise. This exercise will also show if our numerical method is robust to two sources of errors. The first is due to the interpolation of implied volatilities by cubic splines at observed strike price intervals. The second is related to the extrapolation scheme adopted. This error is associated with truncation error. Apart from these two sources of errors, a third one can be attributed to the numerical integration technique chosen. However, as noted recently by Jiang and Tian (25) and it can be confirmed by our simulation exercise, this third type of error has not significant effects on numerical procedures used for the calculation of integrals of OTM option prices. Our simulation study consists of eighteen in total different experiments. These can evaluate the performance of our numerical method with respect to the length of the strike price interval [ min max ], the number of observed call and put prices (denoted, respectively as and ) and, finally, the extrapolation scheme employed. The latter considers also the choice of not extrapolating, as a non-extrapolation scheme. The different strike intervals that we consider in our study are the following: [ ], [ 735 8] and [ ], whilethedifferent pairs of call and put number of prices ( ) assumed are given as: ( =5 =7), ( = = 8) and ( =2 = 34). The above intervals of strike prices and pairs of ( ) are chosen so as to make our simulation study as close as possible to reality. They constitute representative strike intervals [ min max ] and pairs of ( ) of small, medium and large size taken from a cross-sectional sets of European option prices written on the S&P 5 index with maturity interval one month which covers the period from January 996 to October 29. This sample of option and strike prices is used in our empirical exercise, which follows in the next section. The theoretical European call and put prices used in our simulation study are generated from the stochastic volatility with jumps (SVJ) model. This model is frequently used in practice to This source of error is known as discretization error. By choosing a large number of knot points in the numerical integration technique, this error can be proved negligible. In our simulation study, to calculate the integrals we employ the Gaussian quadrature numerical procedure. The full-specification of the diffusion-jump asset price process () [ ] under the risk neutral measure for the SVJ model is given as follows: = ( ) + + with = ( ) + 2 ( = ) =, ln( + ) 2, 7

18 improve upon the performance of the BS and stochastic volatility models, as it implies high levels of negative skewness and kurtosis of the RND. These two features of the RND are consistent with the pattern of the implied volatility across different strike prices observed in reality (see, e.g. Bakshi, Cao and Chen (997)). To generate option prices closed to the market ones, we are based on values of the structural parameters of the SVJ model found in the literature (see Bakshi, Cao and Chen (997)). That is, we have assumed the following values for the structural parameters of the model under the risk neutral measure: =, = 4, =3 93, = 52, = 6, = 4 and = 4. To be consistent with the intervals of strike prices considered in our simulation study, the current values of the two state variables of the SVJ model and are taken to be the mean of the S&P 5 index and the variance of its log-return for the sample period from January 996 to October 29. These are given as = 32 and = 26, respectively. For the interest rate, we used the average level of the three-month US Treasury bill for the above period given as = 5. For the dividend yield, we used its average estimate over the same period given as = 5. This was calculated based on the OptionMetrics data set. Based on the above values of the SVJ option pricing model parameters and state variables, we generated European call and put option prices with one-month to maturity interval (i.e. = 83). We have chosen this short maturity interval, given that the biggest failures of most of the European option pricing models occur for it. From these sets of option prices, we derived the implied volatilities which are then used to fit the functions of our interpolation-extrapolation scheme. Following Ait-Sahalia and Duarte (23), we added a noise term to our generated set of implied volatilities drawn from the uniform distribution, with interval support [ 2525]. The above experiment is repeated times. The noise term added to the implied volatilities can be taken to reflect random effects of the bid-ask spread and the different degree of liquidity on call and put prices. 2 These perturbated implied volatilities werethenusedtoobtainestimatesofthe RNM using formulas (9)-() and of the expected quadratic variation ignoring the jump component using formula (3) based on the numerical method suggested in the previous section. where =ln(+ ) 2 2,and and 2 are two correlated Brownian motions with correlation coefficient. 2 Note that the perturbated implied volatilities or their associated option prices do not violate the arbitrage conditions, i.e. monotonicity and convexity. 8

19 5. RNM To assess the performance of our numerical method in retrieving RNM from option price data, we will employ the root mean squared error (RMSE) metric. This is defined as the following distance between the RNM estimates retrieved by our method b and their true values: r h b i 2 RMSE = The RMSE criterion can be decomposed as follows: RMSE 2 = RSB 2 + RV 2, where q r h 2 b RSB = b and RV = i 2 b stand for the root squared bias (RSB) and the root variance (RV) (i.e. the standard deviation) of b, respectively. The RMSE metric can be thought of as a measure of the overall performance of a numerical method to retrieve accurate estimates of RNM from option prices. The RSB and RV metrics can be considered as more appropriate for assessing properties of the method such as unbiasedness and efficiency, respectively. These two properties are important for good density and moment estimators. The true values of the RNM and expected quadratic variation used in the calculation of the above all metrics were derived by differentiating the moment-generating function of the risk neutral distribution of the log-return ( ) implied by the SVJ model. 3 Table presents the results of our Monte Carlo simulation study for the case that the extrapo- 3 The moment-generating function of the SVJ model is given as: ( ) = ( ) + ( )+ ( ) + ( ) where = and with ( ) = ( + ) 2ln 2 ( ) = + 2 ( ) = = + and = ( ) 2 2 ( ) 9

20 lation scheme s function is constant, while Table 2 gives the results for the case that this function is linear. Finally, Table 3 reports the results for the case that non-extrapolation scheme is chosen, i.e., = min and = max. More specifically, the three tables report the theoretical and approximated by our numerical method values of the first six non-central RNM of the asset return ( ), as well as the values of the following three metrics RMSE, RSB and RV for them. The approximated values of the RNM reported in the tables constitute average values of their estimates over the iterations of our Monte Carlo exercise. In Tables to 3, we also provide average values of the estimates of the upper bounds of the approximation error given by equation (6) over the iterations. These are based on the values of ( ) and ( ) generated by the SVJ model. The results of the above tables lead to a number of interesting conclusions which have important practical implications. First, they clearly show that the theoretical RNM formulas (9)-() can be successfully implemented through our suggested numerical method to discrete sets of option prices. The values of the RNM provided are very close to their theoretical ones, as the values of the three metrics reported in the tables are very small. This is true for all different ( )-pairs of call and put prices considered. The results of the tables indicate that the accuracy of higher-order RNM estimates (i.e. for 2) increases with the length of the strike interval [ min max ].This result was expected, as more information of OTM options is required to obtain accurate estimates of higher-order RNM. As the values of the RSB metric reveal, the increase in the accuracy of our numerical method with the length of the strike price interval can be attributed to the smaller magnitude of the bias encountered. Further support for the accuracy of the method can be obtained see Bates (996). Based on this moment-generating function, we derived the theoretical values of the th-order RNM by numerically calculating the th-order derivative of ( ) at =, i.e. ( ) = = The annualized expected quadratic variation implied by the SVJ can be calculated analytically based on the following formula: h i =2( ) Thus the jump component is given as: =

21 by the inspection of the estimates of the upper bounds of the approximations error b, reported in the tables. These are found to be very small and, as was expected, they decrease with the length of the strike price interval. Regarding the extrapolation schemes chosen, the results of the tables reveal that, with the exception of the first two moments, our numerical method performs better both in terms of accuracy and efficiency for strike intervals of smaller length (i.e. for [ ]) when the constant function is used, instead of the linear. When the length of the strike interval increases, the two functions are found to perform similarly. The worse performance of the extrapolation scheme employing a linear function for small strike price intervals can be attributed to the fact that this scheme can lead to a lower value of the integral bound (or ), compared to that with a constant function. Such value of implies that function ( ; ), multiplying the effects of the put pricing error on the RNM estimates, will take a large value which, in turn, will lead to a large in magnitude approximation error of our numerical implementation procedure, as noted in the previous section. This can be confirmed by the results of Table 2, see Subpanel C, which covers the case where endpoint takes its lowest value given by = 3. Compared to the non-extrapolation scheme, the scheme based on a constant function performs better in terms of accuracy, while the opposite is true in terms of efficiency. The linear function extrapolation scheme outperforms the non-extrapolation one in terms of accuracy only for small sample sizes and large strike price intervals. In terms of efficiency, again the non-extrapolation scheme outperforms the linear one. The approximation error bounds reported in the tables reveal that the non-extrapolation scheme outperforms both the constant and linear ones, for the majority of sample sizes and strike price intervals examined. As mentioned in the previous section, this canbeexplainedbythefactthatthenegativeeffect caused by and on dominates its positive effect on, which implies that extrapolating the implied volatility function increases the approximation error bounds. This result has important practical implications. As the true RNM are unknown, it means that a constant or a linear function extrapolation scheme may increase the approximation error bounds b, for all. Thus, it may be better to choose not to extrapolate implied volatilities over intervals ( min ] and [ max + ). The magnitude of the truncation error encountered in this case can be assessed by fitting an option pricing formula into option price data. 2

22 5.2 Expected quadratic variation The results of our simulation exercise evaluating the performance of our numerical method to retrieve estimates of the expected quadratic variation from option prices are reported in Table 4. The table presents average values of, implied by the formula used by CBOE which ignores the jump component term, over the iterations, and average values of term, approximated through relationship (5). As in Tables -3, this is done for different sample sizes and strike price intervals. The sum of the above values of and provides estimates of. For reasons of space, the table presents results only for the case of the non-extrapolation scheme. Instead of values of the RMSE, RSB and RV metrics, the table reports estimates of the error bounds of, and. These are obtained based on the values of the call and put prices generated by the SVJ model. The results of the table clearly indicate that the estimates of and,aswellasthoseof obtained through our numerical procedure are very close to their theoretical values, predicted bythesvjmodel. Thelatteraregiveninthefirst row of the table. The estimates of and converge to their true values, as the strike price interval [ min max ] and sample size increase. These results indicate that the model-free formula (5) provides accurate estimates of. As is predicted by the theory, these can explain the downward bias of, due to the omission of the jump component from formula (2) used by the CBOE. The theoretical value of term is given as = 4, which can be translated into an underestimation of the true VIX index (given as )by33basispoints. Aseachindexbasispointsisworth$perVIXfutures contract, this means an undervaluation of $33 per contract. 6 Empirical application Based on the theoretical formulas derived in the previous sections, in this section we retrieve estimates of the jump component term of expected quadratic variation from option price data. This can show how important is the jump component of quadratic variation, ignored by CBOE s formula calculating the VIX index, i.e.,, based on (3). To carry out this exercise, we rely on call and put option data written on the S&P 5 index. More specifically, we use the implied volatility surface (IVS) of the S&P 5 index provided by OptionMetrics Ivy DB database. This volatility surface is constructed from implied volatilities 22

23 with kernel smoothing technique, which is described in details in OptionMetrics data manual. This data set contains implied volatilities (of both calls and puts) on a grid of fixed maturities and option deltas. Using these Black-Scholes delta values OptionMetrics also calculate the implied strike price of each implied volatility value. We select the IVS of -month time-to-maturity options every third Wednesday of each month from January 996 to October 29. As the risk-free interest rate and dividend yield we use the estimates employed in the OptionMetrics calculations. The interest rate is derived from British Banker s Association LIBOR rates and settlement prices of Chicago Mercantile Exchange Eurodollar futures. The dividend yield is estimated by the put-call parity relation of at-the-money option contracts. Table 5 reports average values of and over our whole sample, based on formulas (3) and (5), respectively. t-statistics are in parentheses. In addition to these values, the table also reports average values of RNM 3 and 4, needed by formula (5). The estimates of and across all points of our sample are graphically presented in Figures 2 and 3, respectively. To retrieve the above estimates, we rely on the numerical interpolation-extrapolation procedure which assumes a non-extrapolation scheme. As shown in the previous section, this leads to smaller error bounds. Since these mainly depend on the truncation error, Table 5 reports estimates of them for, 3, 4 and, as well as the percentage of these bounds due to. These bounds are calculated by fitting the SVJ model into our option price data and using the formulas of Proposition 5 to calculate errors, and. 4 Figure 2 presents upper and lower intervals of adjusted by its error bound d. The upper interval of canbeconsideredasan estimate of it which is net of truncation error, since this is found to explain almost all (about 97.76%) of the error bound. The results of Table 5, and Figures 2 and 3 clearly indicate that constitutes a significant component of expected quadratic variation, which implies that provides downward biased estimates of. moves closely with during our sample and it takes its highest values during the periods of financial crises, i.e., the Asian crisis, the 22 US corporate crisis and Lehman Brothers collapse in year 28, respectively. As was expected, investors expectations for possible random jumps in the market are intensified during these periods. The negative bias of 4 Note that we have chosen to calculate errors, and, and, hence, bounds and, given that this model fits better into our data and is used very frequently in the empirical literature. 23

24 implies an undervaluation of a VIX futures contract written on the implied volatility.for example, a negative bias of 5 basis points (observed immediately after Lehman Brothers default, according to Figure 3) implies an undervaluation of $5 per VIX futures contract. 5 7 Conclusion This paper provides analytic formulas for retrieving model-free risk neutral moments (RNM) of any order of future asset payoffs implied by generic out-of-the-money European-style option call/put prices, such as Asian or plain vanilla. Based on these formulas, next the paper derives an exact formula of the expected quadratic variation which nowadays is used as a new measure of implied volatility by the Chicago Board Options Exchange (CBOE), known as the VIX index. It also derives a formula which enables us to approximate the jump component of the expected quadratic variation in a model-free way, based on the third and fourth-order RNM. This component is not taken into account by the formula used by CBOE to calculate the VIX index. To implement the above formulas to discrete sets of option prices, the paper suggests a numerical procedure based on an interpolation-extrapolation, curve-fitting scheme of implied by option prices volatilities. Interpolation is used between the observed strike prices interval, while extrapolation is carried out for those which are unobserved. To control for the approximation errors of this numerical method, the paper derives upper bounds of these errors. These may be proved very useful, in practice, to assess the relative magnitude of approximation errors encountered when retrieving RNM and/or expected quadratic variation from option price data. The performance of our numerical method is assessed through an extensive Monte Carlo exercise, assuming that the stochastic volatility with jumps (SVJ) model characterizes our data. The results of this exercise clearly indicate that this method can be successfully employed to retrieve efficient estimates of RNM of any order and expected quadratic variation from option price data. This is true even for strike price intervals of small length, which are often available in practice, and independently on the number of call or put prices. They also indicate that calculating the expected quadratic variation based on the CBOE s VIX formula can lead to downward biased estimates of it, if the underlying asset price process includes a jump component. This bias matches the estimates 5 Note that this can be thought of as a conservative estimate of, given that a non-extrapolation scheme is employed. If we add the unspecified integral parts of higher-order RNM, controlling for the truncation error,to this estimate then the value of will increase. 24

25 of the model-free approximation function of the jump component term of quadratic variation, suggested by the paper. In an empirical application, the paper shows that the jump component term of expected quadratic variation is significant and varies substantially across time, especially during financial crises. In such periods, it is shown that the CBOE estimates of the VIX are substantially undervalued. This finding has important asset pricing implications. It means that the VIX futures contracts written on this index can be severely undervalued, due to the jump component bias. A Appendix In the Appendix we provide proofs of the main theoretical results of the paper. A. Proof of Proposition To prove Proposition, we are based on the result that, for a twice continuously differentiable function ( ) of the payoff ( ), the following result holds: [ ( )] = ( )+ ( )[ ( ) ]+ + Z + ( )[ ( ) ] + + Z ( )[ ( )] +, (8) for some constant (see Bakshi and Madan (2) and Carr and Madan (2)). Suppose that : R C : ( ) 7 [( )], R. Then, equation (8) implies [( )] = ( ) + ( ) ( ) ( ( ) )+ Z + h + ( ) ( ) 2 ( ) 2 ( )i [ ( ) ] Z h ( ) ( ) 2 ( ) 2 ( )i [ ( )] +. Multiplying both sides of the last relationship with ( ) and taking the conditional expec- 25