Empirical pricing kernels for the UK equity market are derived as the ratio between risk-neutral densities, inferred from FTSE 100 index options, and historical real-world densities, estimated from time series of the index. The kernels thus obtained are almost compatible with a risk averse representative agent, unlike similar estimates for the US market

Liu, X

Shackleton, M B

Taylor, S J

Xu, X

English

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           Lancaster University Management School Working Paper  2006/049      Empirical pricing kernels obtained from the UK index options market      Xiaoquan Liu, Mark Shackleton, Stephen Taylor and  Xinzhong Xu    The Department of Accounting and Finance           Lancaster University Management School Lancaster LA1 4YX UK   © Xiaoquan Liu, Mark Shackleton, Stephen Taylor and Xinzhong Xu All rights reserved. Short sections of text, not to exceed two paragraphs, may be quoted without explicit permission, provided that full acknowledgement is given.  The LUMS Working Papers series can be accessed at http://www.lums.lancs.ac.uk/publications/ LUMS home page: http://www.lums.lancs.ac.uk/ Empirical Pricing Kernels Obtainedfrom the UK Index Options MarketXiaoquan Liu∗Mark B Shackleton†Stephen J Taylor‡Xinzhong Xu§October 16, 2006∗Department of Accounting, Finance, and Management,University of Essex, ColchesterCO4 3SQ, UK. Email: liux@essex.ac.uk.†Department of Accounting and Finance, Lancaster University, Lancaster LA1 4YX,UK. Email: m.shackleton@lancaster.ac.uk.‡Department of of Accounting and Finance, Lancaster University, Lancaster LA1 4YX,UK. Email: s.taylor@lancaster.ac.uk.§Guanghua School of Management, Peking University, Peking and Cardiff BusinessSchool, Cardiff University, UK. Email: xux@gsm.pku.edu.cn.AbstractEmpirical pricing kernels for the UK equity market are derived asthe ratio between risk-neutral densities, inferred from FTSE 100 indexoptions, and historical real-world densities, estimated from time seriesof the index. The kernels thus obtained are almost compatible witha risk averse representative agent, unlike similar estimates for the USmarket.JEL classifications : G12; G13Keywords : Pricing kernels; Risk-neutral densities; Index options; Risk aver-sion11 IntroductionThe pricing kernel assumes a central role in asset pricing literature, as itsuccinctly summarizes investors’ risk and time preferences. With a correctlyidentified pricing kernel, asset pricing becomes a straightforward discountingof future payoffs by the kernel.In this article, we estimate empirical pricing kernels from the options market,as option prices have been shown to contain incremental information in fore-casting future volatilities and price distributions compared to the time seriesof asset prices1. We express the empirical pricing kernels as a ratio betweenrisk-neutral densities (RND), inferred from FTSE 100 index options, andhistorical densities obtained from time series of the index, averaged acrosstime to minimize the impact of measurement errors.In particular, we assume that risk-neutral densities follow certain distribu-tions. They can be either a mixture of two lognormal densities (MLN), ageneralized beta distribution of the second kind (GB2), or a flexible splinefunction, all of which are easy to estimate and able to capture the stylizedfacts of negative skewness and excess kurtosis that are associated with indexdistributions.Using a sample of 126 months data from July 1993 to December 2003, wefind that the average empirical pricing kernel for the UK equity market isgenerally downward sloping and does not exhibit the puzzling hump shapedocumented by Jackwerth (2000), Rosenberg and Engle (2002), and Brownand Jackwerth (2002). This work is also related to Brennan et al (2006),which adopts the traditional asset pricing approach by identifying state vari-ables that fully describe the investment opportunities and specifying flexiblefunctional forms for the pricing kernels.The rest of the paper is organized as follows. Section 2 introduces the risk-1A vast literature has documented that option-implied volatility provides better fore-casts of future volatilities than realized volatility. Poon and Granger (2003) and Taylor(2005) provide recent survey evidence. In terms of density forecasts, Liu et al (2006)demonstrates that distributions from option prices are better forecasts than those ob-tained from asset price histories.2neutral and historical densities and the estimation procedures. Section 3discusses data. Section 4 presents empirical results and describes the empir-ical pricing kernels for the UK. Finally, Section 5 concludes.2 Risk-neutral and historical densities2.1 Mixture of lognormal densities (MLN)Following Ritchey (1990) and Melick and Thomas (1997), the risk-neutraldensity of the asset price when options expiry can be defined as a mixtureof lognormal densities. The MLN densities are flexible and easy to estimate,with the possibility of attaching an economic interpretation to the parameterswhen the component densities are determined by specific states of the worldwhen the options expire. The MLN density function in this study is thefollowing weighted average of two lognormal densities gLN ,gMLN(x|θ) = wgLN(x|F1, σ1, T ) + (1− w)gLN(x|F2, σ2, T ) (1)withgLN(x|F, σ, T ) = 1xσ√2piTexp[−12(log x− (logF − σ2T/2)σ√T)]. (2)The parameter vector is θ = (F1, F2, σ1, σ2, w), with 0 ≤ w ≤ 1 and F1, F2, σ1, σ2 >0. The parameters F1, σ1, and w denote the mean, volatility, and weight ofthe first lognormal density, while F2, σ2, and 1− w are the mean, volatility,and weight of the second lognormal density.The density is risk-neutral when its expectation equals the current futuresprice F , i.e. when wF1 + (1− w)F2 = F . The theoretical Euruopean optionpricing formula is then simply the weighted average of two option prices givenby the Black (1976) formula, denoted by cB(.),3c(X|θ, r, T ) = wcB(F1, T,X, r, σ1) + (1− w)cB(F2, T,X, r, σ2). (3)2.2 Generalized beta distribution (GB2)The generalized beta distribution of the second kind (GB2) was first proposedby Bookstaber and McDonald (1987) and utilized by Anagnou-Basioudiset al (2005). The GB2 density incorporates four positive parameters θ =(a, b, p, q) that permits general combinations of the mean, variance, skewness,and kurtosis of a positive random variables. The GB2 density function isdefined as,gGB2(x|a, b, p, q) = axap−1bapB(p, q)[1 + (x/b)a]p+q(4)with B(j, k) = Γ(j)Γ(k)/Γ(j + k). The density is risk-neutral whenF =bB(p+ 1a, q − 1a)B(p, q). (5)The parameter b is seen to be a scale parameter, while the product of a andq determines the maximum number of moments and hence the asymptoticshape of the right tail.The theoretical option pricing formula depends on the cumulative distribu-tion (c.d.f.) function of the GB2 density, denoted GGB2, which is a functionof the c.d.f. of the beta distribution, denoted Gβ,GGB2(x|a, b, p, q) = GGB2((x/b)a|1, 1, p, q) = Gβ(y(x, a, b)|p, q) (6)with y(x, a, b) = (x/b)a/(1 + (x/b)a). If the density is risk-neutral, so that(5) applies, the European call option prices are given byc(X|θ) = e−rT∫ ∞X(x−X)gGB2(x|a, b, p, q)dx (7)4= Fe−rT[1−Gβ(y(X, a, b)|p+ 1a, q − 1a)]−Xe−rT [1−Gβ(y(X, a, b)|p, q)] .2.3 Flexible densitiesTo make sure that the parametric densities discussed above are not inferiorto more flexible density curves, we also infer risk-neutral densities definedby spline functions, as estimated by Bliss and Panigirtzoglou (2004). Weapply their methodology to obtain implied volatilities, denoted by σ(∆|θ),which are a function of option delta, ∆, and a parameter vector θ. Numericalmethods then give call prices c as functions of strikes X and hence define therisk-neutral densities as g(X) = erT∂2c/∂X2. For options on futures, ∆ isdefined as a function of the Black-Scholes at-the-money volatility σA∆(X) = e−rTN(d1(X)) (8)d1(X) =log(F/X) + σ2AT/2σA√T.The spline function σ(∆|θ) is defined over 0 ≤ ∆ ≤ exp(−rT ). It is composedof linear pieces and cubic polynomials, defined on intervals determined byknot points ∆1 < ∆2 < · · · < ∆N . Each cubic is defined over an intervalfrom ∆i to ∆i+1, while the function is linear for ∆ ≤ ∆1 and ∆ ≥ ∆N .The coefficients of the lines and cubics are constrained by the requirementthat σ(∆|θ) and its first two derivatives are continuous functions. The splinefunction has N free parameters [Lange (1998, page 104)] and there is a uniquespline with the required properties that passes through a given set of Npoints (∆i, σi). Taking the knot points as given, the parameter vector is thecorresponding set of implied volatilities θ = (σ1, · · · , σN).2.4 Estimation of the RND parametersThe RND parameter vector θ is estimated once a month with 4-weeks to ma-turity so that the densities are non-overlapping for each of the three density5functions. For the MLN and GB2 densities, θ is obtained by minimizing thefollowing average squared difference between observed market call prices andtheoretical option prices:1NN∑i=1(cmarket(Xi)− c(Xi|θ))2 (9)withc(Xi|θ) = e−rT∫ ∞Xi(x−Xi)g(x|θ)dx, 1 ≤ i ≤ N.In these equations, N is the number of European option prices used for aparticular day, g(x|θ) is a parametric density function, and c(X|θ) is theassociated theoretical option pricing formula, given by either equation (3) orequation (7).The estimates of θ for the flexible densities are obtained by minimizing afunction that combines two criteria, namely the accuracy and the smoothnessof the fitted spline function σ(∆|θ). From N market prices, we derive impliedvolatilities and hence co-ordinates (∆i, σmarket(Xi)). Then for a set of weightswi, we select σ1, · · · , σn to minimizeηm∑i=1wi(σmarket(Xi)− σi)2 + (1− η)∫ ∆M∆1σ′′(∆|σ1, · · · , σm)2d∆. (10)There is a straightforward solution to this optimization problem [Lange(1998, page 111)]. Implementation requires making a subjective choice forthe parameter η, that controls the trade-off between accuracy and smooth-ness. Following Bliss and Panigirtzoglou (2002, 2004), appropriate weightsare proportional to option vega, so we use wi = exp(−d1(Xi)2/2)/√(2pi) withd1(Xi) given by equation (8).2.5 Historical densitiesARCH models for daily index returns are estimated and simulated to providehistorical real-world densities. The simulated ARCH models must accommo-6date the stylized facts documented in the literature, including a time-varyingconditional mean, a persistent conditional volatility, and an asymmetric re-sponse of volatility to positive and negative returns. We choose the GJR-GARCH(1,1)-MA(1)-M specification, following Glosten, Jagannathan andRunkle (1993) and Engle and Ng (1993). The conditional mean µt and theconditional variance ht of the daily index return rt are as follows,ht = ω + βht−1 + (α1 + α2Dt−1)(rt−1 − µt−1)2, (11)µt = ξ + λ(ht−1)1/2 +Θ(rt−1 − µt−1),Dt = 1 if rt ≤ µt,Dt = 0 otherwise.Ten years of daily index returns prior to each estimation date ti are used toestimate the ARCH parameters θ = (ω, β, α1, α2, ξ, λ,Θ), by maximizing thequasi-log-likelihood function which assumes the conditional distributions arenormal. These estimates are consistent even when the normality assumptionis false [Bollerslev and Wooldridge (1992)].The parameters obtained from the returns information up to selected timesti are used to simulate the ARCH equations for 4-week periods that end onoption expiry dates. A large number,M , of simulations of the final asset levelST,i are obtained for each month i. The historical real-world density g˜i is thenthe smooth function obtained by using the Gaussian kernel with bandwidthH = 0.9ψ/ 5√M and with ψ the standard deviation of the simulated finallevels. We have set M equal to 100,000. Our formula for the bandwidth His recommended by Silverman (1986, page 48) and used by Rosenberg andEngle (2002).3 DataThe futures and options contracts are written on the FTSE 100 index andthey are traded at the London International Financial Futures and Options7Exchange (LIFFE). Futures and options have the same expiry month andshare a common expiry time, 10:30 on the third Friday. European optionscan then be valued by assuming that they are written on the futures contract,and hence spot levels of the index are not needed.Daily settlement values for futures and options prices with 4-week to maturityare obtained from LIFFE, for 126 consecutive expiry months from July 1993to December 2003 inclusive. The call and put implieds for the same contractare almost identical, as should be expected from put-call parity. We averagethe call and put implieds and use them to calculate European call prices foreach contract.On average, 37 exercise prices are available for each month. The exerciseprices are always separated by 50 index points. Table 1 provides summaryinformation about the option prices.There are twelve expiry dates per annum for the options but the futurescontracts are traded for only one expiration date each quarter. Syntheticfutures prices must be calculated for the remaining eight months. Fair futuresprices, F , are the future value of the current spot prices S minus the presentvalue of dividends expected during the life of the futures contractF = erT (S − PV (dividends)). (12)We have obtained actual dividend payments for the 100 component com-panies of the index from DataStream, and computed the present value ofdividends by assuming that future expected dividends can be approximatedby realized dividend payments.Risk-free interest rates are collected from DataStream. We prefer the LondonEurocurrency rate to the UK treasury bill rate, because the Eurocurrencyrate is a market rate accessible to AA corporate borrowers.Figure 1 shows typical estimates of the risk-neutral densities, which exhibitmarked negative skewness that has already been extensively documented.The MLN and GB2 densities are almost identical, while the spline density8differs in the left tail. The historical real-world density is very different fromthe risk-neutral ones, with much less skewness and kurtosis.4 Empirical pricing kernelsEmpirical pricing kernels are estimated by using RNDs obtained from theoptions market and historical real-world densities from index returns. Threeempirical pricing kernels M(x) = e−rTg(x)/g˜(x) are constructed for eachoption expiry date, with g(x) either the MLN, GB2 or spline RND and g˜(x)the historical density from GARCH simulations. The geometric mean ofeach set of kernels is computed, across expiry dates, to reduce the impact ofthe noise created firstly by fitting the RNDs and secondly by using differentsources of information to find g and g˜. We plot the geometric means of thethree sets of ratios g(yF )/g˜(yF ) against the moneyness variable y = x/Fin Figure 2. All three graphed kernels are generally decreasing functions ofx/F , although they are almost flat between 0.98 and 1.05.None of our empirical pricing kernels for the UK equity market resemblesthose of Ait-Sahalia and Lo (2000), Jackwerth (2000), Brown and Jackwerth(2002) and Rosenberg and Engle (2002) for the US market. These researchersestimate very clear hump-shaped kernels, using S&P 500 data that ends in1995, which challenges economic theory and indicates that the representativeagent has a risk-seeking utility function in some wealth region. The risk-seeking range obtained by Jackwerth (2000) is 0.96 ≤ x/F ≤ 1.01 whileRosenberg and Engle (2002) obtain 0.96 ≤ x/F ≤ 1.02. Brennan et al (2006)also use FTSE 100 index options data to estimate the option pricing kernel,approximated by a three-term Chebyshev polynomial whose state variablesare the real interest rate, the maximum Sharpe ratio, and volatility. Thepricing kernel is upward sloping in the region 1.03 ≤ x/F ≤ 1.05.95 ConclusionUsing a dataset from July 1993 to December 2003, we derive three seriesof risk-neutral densities, namely the mixture of two lognormal densities, thegeneralized beta distribution, and a flexible spline distribution from FTSE100 index options data. We also fit GARCH models to the time series ofFTSE 100 index returns and simulate historical real-world densities. Theempirical pricing kernel obtained from the two sets of densities is broadlydownward sloping and therefore consistent with economic theories.10ReferencesAit-Sahalia, Y. and A. W. Lo, 2000, Nonparametric risk management andimplied risk aversion, Journal of Econometrics, 94, 9-51.Anagnou-Basioudis, I., M. Bedendo, S. D. Hodges, and R. Tompkins, 2005,Forecasting accuracy of implied and GARCH-based probability densityfunctions, Review of Futures Markets, 11, 41-66.Black, F., 1976, The pricing of commodity contracts, Journal of FinancialEconomics, 3, 167-179.Bliss, R. and N. Panigirtzoglou, 2002, Testing the stability of implied prob-ability density functions, Journal of Banking and Finance, 26, 381-422.Bliss, R. and N. Panigirtzoglou, 2004, Option-implied risk aversion estimates,Journal of Finance, 59, 407-446.Bollerslev, T., and J. M. Wooldridge, 1992, Quasi-maximum likelihood es-timation and inference in dynamic models with time varying covariances,Econometric Reviews, 11, 143-172.Bookstaber, R., and J. MacDonald, 1987, A general distribution for describ-ing security price returns, Journal of Business, 60, 401-424.Brennan, M. J., X. Liu, and Y. Xia, 2006, Option pricing kernels and ICAPM,Unpublished manuscript, UCLA and University of Essex.Brown, P., and J. Jackwerth, 2002, The pricing kernel puzzle: reconciling in-dex option data and economic theory, Unpublished manuscript, Universityof Wisconsin.Engle, R. F., and V. K. Ng, 1993, Measuring and testing the impact of newson volatility, Journal of Finance, 48, 1749-1778.Glosten, L. R., R. Jagannathan, and D. Runkle, 1993, Relationship betweenthe expected value and the volatility of the nominal excess return on stocks,Journal of Finance, 48, 1779-1801.Jackwerth, J., 2000, Recovering risk aversion from option prices and realizedreturns, Review of Financial Studies, 13, 433-451.11Lange, K., 1998, Numerical Analysis for Statisticians, Springer Verlag.Liu, X., M. B. Shackleton, S. J. Taylor, and X. Xu, 2006, Closed-form trans-formations from risk-neutral to real-world distributions, Journal of Bank-ing and Finance, forthcoming.Melick, W. R., and C. P. Thomas, 1997, Recovering an asset’s implied PDFfrom option prices: an application to crude oil during the Gulf crisis,Journal of Financial and Quantitative Analysis, 32, 91-115.Poon, S.-H., and C. W. J. Granger, 2003, Forecasting volatility in financialmarkets, Journal of Economic Literature, 41, 478-539.Ritchey, R. J., 1990, Call option valuation for discrete normal mixtures,Journal of Financial Research, 13, 285-296.Rosenberg, J. and R. F. Engle, 2002, Empirical pricing kernels, Journal ofFinancial Economics, 64, 341-372.Silverman, B. W., 1992, Density Estimation for Statistics and Data Analysis,Chapman & Hall.Taylor, S. J., 2005, Asset Price Dynamics, Volatility, and Prediction, Prince-ton University Press.12Table 1. Summary statistics for the dataset of FTSE 100 index option pricesThe average number of option prices is 37 per month for the 126 expiry monthsfrom July 1993 to December 2003 inclusive. The moneyness of a call option isdefined by X/F − 1, with X the exercise and F the futures price.Moneyness No. of options Percentage (%)≤ −0.20 19 0.4(-0.20, -0.10] 392 8.3(-0.10, -0.03] 849 18.0(-0.03, 0.03] 702 14.9(0.03, 0.10] 731 15.5(0.10, 0.20] 818 17.3(0.20, 0.30] 572 12.1(0.30, 0.40] 303 6.4> 0.40 335 7.1Totals 4721 100.013Figure 1. Three risk-neutral densities and the historical density on March21, 19973000 3500 4000 4500 5000 550000.511.522.533.5x 10−3Index level at expiryDensityGB2MLNsplinehistorical14Figure 2. Empirical pricing kernels, as geometric averages across all expirymonths0.9 0.94 0.98 1.02 1.06 1.101234x/FGeometric meanGB2/historicalMLN/historicalspline/historical15

Empirical pricing kernels obtained from the UK index options market

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Empirical pricing kernels obtained from the UK index options market

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