A New Methodology For Measuring and Using the Implied Market Value of Aggregate Corporate Debt in Asset Pricing:

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1 A New Methodology For Measuring and Using the Implied Market Value of Aggregate Corporate Debt in Asset Pricing: Evidence from S&P 500 Index Put Option Prices By Robert Geske* and Yi Zhou The Anderson School at UCLA* The University of Oklahoma January 2007 This revision September 2008 Abstract The primary purpose of this paper is to introduce a new methodology for measuring the implied market value of aggregate market debt and analyzing the resultant risk effects of stochastic market leverage on asset prices in the economy. To our knowledge this is the first paper to attempt to directly isolate and analyze the effects of the implied market value of aggregate market debt on equity index option prices. We present what we believe are the first implied measures of the market value of aggregate corporate debt. We derive the implied market value of aggregate debt from option theory using only two contemporaneous market prices for the index price level and index option price. We demonstrate that the inclusion of the implied market value of aggregate debt results in significant statistical and economic improvements in the pricing of S&P 500 index put options relative to more complex models which omit leverage. JEL Classification: G12 Key Words: Derivatives, Options, Leverage, Stochastic Volatility We thank Richard Roll, Subra Subramanyam, Walt Torous, Mark Grinblatt, Mark Rubinstein, Rajna Gibson, Michael Brennan, and especially Hayne Leland for comments on earlier drafts of this paper. This paper has also benefited from comments by seminar participants at UCLA, the FMA in Orlando, 2007, IBC in Hawaii, 2007, and at the CIFC in Dalian, China, As this research is preliminary please do not quote or distribute without the author s permission. *Contact Geske by mail at The Anderson School at UCLA, 110 Westwood Plaza, Los Angeles, California 90095, USA, by telephone at , or by rgeske@anderson.ucla.edu.

2 1. Introduction In this paper we focus on whether the time series variation in implied market value (not book) of aggregate market debt is sufficient to cause significant price effects in options on a market index. 1 For this purpose we consider the market to be the 500 firms comprising the S&P 500 index. S&P 500 equity index options are the world s most widely traded index option. In the year 2004 when our data ends, the S&P 500 index option (SPX) volume was about 50 million contracts compared to about 16 million contracts for the next most active equity index option, the S&P 100 (OEX), respectively. 2 The SPX options are European and the OEX options are American. The fact that the SPX options cannot be exercised early makes them good candidates for examining if there are any time series leverage effects in index option prices, independent of the American early exercise premium. When S&P 500 index option trading began in 1983, it was initially thought that the Black-Scholes model (1973) should do better pricing these options on a portfolio of stocks instead of pricing the individual options in the portfolio. The reason was SPX options conformed more to the Black-Scholes assumptions because they are European and because the limit distribution of the sum of returns on a large number of random variables is more likely to be the normal distribution. 3 Since that time much research has shown that the Black-Scholes model has many biases and does not do as well pricing these S&P 500 index options as initially hoped. It is often thought that the biases in Black-Scholes arise because the underlying state variable is not normally distributed and does not exhibit the assumed constant or deterministic volatility. Heston 1 We imply the market value of aggregate debt from the index level and an index option price using methodology identical to implied volatility which has been widely used in option theory and practice. Christie (1982) shows with regression there is a significant time series correlation between the book value of individual firm leverage and the individual equity volatility. We show this relation is true for the aggregate market. 2 See CBOE Market Stats For perspective, 50 million contracts per year is about 200,000 contracts per trading day, characteristic of the most active global equity index option market. 3 See Khintchine (1938) and Gnedko and Kolmogorov (1954). 2

3 (1993) develops a closed-form stochastic volatility model 4 with arbitrary correlation between volatility and asset returns and demonstrates that this model has the ability to improve on the Black-Scholes biases when the correlation is assumed to be negative. Dupire (1994) is cited with the first development of a lattice approach to best fit the cross-sectional structure of option prices wherein the volatility can depend on the asset price, strike price, and time. Dumas, Fleming and Whaley (1998) describe the lattice approaches of Dupire and others as deterministic volatility functions (DTV), and find that these implied tree approaches work no better than an ad hoc version of Black-Scholes where the implied volatility is modified for strike price and time. Bakshi, Cao, and Chen (1997) model and test stochastic volatility (SV), stochastic volatility and stochastic interest rates (SVSI), and stochastic volatility and jump (SVJ) models compared to Black-Scholes with a focus on hedging errors, and demonstrate significant improvements. Bates (2000) describes the post 1987 crash period and volatility skew, and he documents Black-Scholes biases when pricing S&P 500 index futures options. He tests a model with stochastic volatility and jumps that shows improvements, however, Bates states, the parameter estimates cannot match the market option prices (or implied volatilities) for different option strike prices and expirations. Heston and Nandi (2000) develop a closed-form GARCH option valuation model with assumed negative correlation between price and volatility which also shows relative improvements. Recently, Pan (2002) tests a version of the Bates (2000) model with either constant or state dependent jump intensity using an integrated strategy which relies on implied-state, generalized method of moments estimation (IS-GMM) in order to examine more explicitly and measure the risk premia in option prices. Pan finds more support for jump risk premia rather than volatility risk premia. 5 4 Wiggins (1978), Hull White (1987) and others have also developed arbitrary stochastic volatility models. 5 Bakshi, Cao, and d Chen (1997), Bates (2000), and Pan (2002) all say the stochastic volatility models (SV) require implausible volatility-return correlations and volatility variation. The stochastic volatility with jumps models (SVJ) result in more reasonable parameters, but still appear to be inconsistent with the observed distribution. 3

4 More recently, Liu, Pan, and Wang (2005) attempt to further disentangle the rare-event premia by separating the premia into diffusion premia and jump premia, driven by risk aversion, and then adding an intuitive component driven by imprecise modeling and subsequent uncertainty aversion. In order to fit observed option prices all of these more complex models require the assumption of a negative correlation between the stock price and the stock volatility because this correlation does not occur directly from the economics of these models. As already mentioned much of the literature described above is based on the idea that Black-Scholes (1973) distributional assumption of stock returns being instantaneously normally distributed with a constant or deterministic instantaneous conditional volatility is not realistic. For example, the skew observed in the equity distribution with a fat left tail and thin right tail cannot occur with the normality assumption underlying the Black-Scholes model. There exists extensive empirical evidence of a persistent inverse relation between the level of equity prices and instantaneous conditional volatility, observed for both individual firms and for indexes (Nelson, 1991). This inverse relation which results in the observed skew has long been thought to be leverage related (Black, 1976), and more recently has been described as volatility feedback effects (Campbell and Hentschel, 1992) where changes in volatility may affect the discount rates for future cash flows and dividends. In this paper we focus on the financing choices (stock, bonds, and thus capital structure) of the collection of firms which result in the market value of aggregate market leverage when the market is defined as the 500 firms that comprise the S&P 500. We investigate whether changes in implied market value of aggregate market debt have an important effect on index option prices by altering the distribution of aggregate equity price changes, inducing the observed stochastic volatility that is inversely related to equity returns and resulting in the observed equity index distribution skew. Thus, we are concerned with the time series effects of debt and the resultant leverage, since there are no cross- 4

5 sectional differences in debt or leverage for the aggregate index, as there would be when investigating the differential effect of the market value of debt on options on the individual firm s that comprise the index. Most equity option pricing models do not consider the influence of a firm s choice of financing securities on option prices. In a recent paper, Eom, Helwege, and Huang (2004) examine five structural models of the firm s financing choices, and taking the stock price as a given input, they report how accurate the models are for estimating corporate bond prices, yields, and spreads. The five models they implement are those of Merton (1973), Geske (1977), Longstaff-Schwartz (1984), Leland-Toft (1994), and Colin-Dufresne-Goldstein (1997). For the sake of brevity when referring to these papers, we adopt their notation convention and refer to these names by their initials M, G, LS, LT, and CDG, and we also use the initials TP, BCC, P, and BS to refer to the papers by Toft-Prucyk (1997), Bakshi, Cao and Chen (1997), Pan (2002), and Black-Scholes (1973). 6 Of these models only those of G (1977) and LT (1996) have been extended by G (1979) and TP (1997) to directly analyze the effects of the firm s financing choice (leverage) on equity options. If, as we show in this paper, the time series variations in the implied market value of aggregate leverage cause significant effects on the total volatility risk of equity in the index, then leverage should also exhibit significant cross-sectional effects on the individual firms in the index because the cross section of firms will contain greater leverage extremes. 7 The potential cross-sectional affects of firm s differential leverage on options has been described. However, to the best of our knowledge there has been no published demonstration of empirical pricing improvements resulting from tests of comparable 6 See Eom, Helwege, Huang (RFS, 2004), for notation, page 500, and accuracy, Table 3, page The extremes of cross-sectional variation in leverage for the individual firms in the index will be much greater than the time series leverage variation of the aggregate index because the index is an average of the 500 individual stocks. 5

6 models with and without leverage for pricing either equity index options or individual stock options. Furthermore, to our knowledge there no papers which present a methodology that uses only two consistent equations for two contemporaneous, liquid prices to measure the implied market value of corporate debt, and then use this measure to examine the effects of the market value of leverage on option prices. 8 Ericsson and Reneby (2005) use a maximum likelihood methodology developed by Duan (1999) to measure the value and risk of the firm. However, Duan s approach is not an implied methodology using only two contemporaneous prices. Instead it requires a time series of historical prices to derive the likelihood function for the historical distribution which then allows measurement of the firm s value and expected future risk. This of course assumes that the historical distribution will repeat itself. Duan both stated and showed that this was an improvement because the methodology he critiqued was based on a volatility restriction which assumed that the volatility of the equity was constant. This constant volatility assumption is inconsistent with both the equations and with the data. However, a stochastic volatility estimate could be made using historical prices that would not be inconsistent. In summary, these criticisms of Duan, Ericsson and Reneby to not apply to the implied and thus forward looking methodology presented here which is consistent with stochastic volatility. Toft and Prucyk (TP) (1997) adapt a version of Leland and Toft (1996) to individual (not index) stock options, and using ordinary regression in cross-sectional tests they demonstrate significant correlations between their model s variables and the volatility skew for a 13 week period in 1994 for 138 firms in their final sample. However, TP do not investigate the extent of option pricing improvement attributable 8 Ericsson and Reneby (ER), Journal of Business, 2005, v. 78, no. 2, use historical prices to derive a likelihood function which they maximize to allow them to solve for the firm s value and risk. ER do not imply the firm risk and market value of firm debt in the same sense that one implies the underlying asset volatility using Black-Scholes. Here, using Geske (1979) we imply the two important variables (the asset volatility and the market value of aggregate debt) using two liquid, contemporaneous prices for the equity index and an option on the index and two equations which are both consistent with each other and with the equity volatility being stochastic. 6

7 to leverage by comparison to more complex models which omit leverage. Instead they examine the cross-sectional correlations between volatility skew for individual stocks and their model variables which are: (i) LEV, the ratio of book value (not implied market value) of debt and preferred stock to debt plus all equity, and (ii) CVNT, ratio of short maturity debt (less than 1 year) to total debt, as a proxy for a protective covenant. Geske and Zhou (2008) using G (1979) show, as expected, that cross-sectional variations in the market value of debt/equity ratios are much greater for individual firms than for the index during their 10 year sample period ( ) which includes 1683 firms. They demonstrate variations in individual firm leverage cause significant pricing improvements when compared to the models of BS (1973), BCC (1997), and Pan (2002) which omit leverage. By comparison, the more complex models which omit leverage have many more parametric degrees of freedom in order to model the non-normality and inverse relation between returns and total risk of the underlying equity distribution. The idea that the implied market value of the aggregate market leverage resulting from the financing choices of individual firms might have significant effects on the prices of index options on aggregate equity has many important distinctions from the idea that differences in individual firm leverage might have significant effects on the prices of options on individual stocks. For example, the distinctions that TP (1997) adapt from Leland (1994) between individual firm s coupon payments, proportional payouts, firm volatilities, tax rates, and bankruptcy costs are not important as they are averaged for equity options on the aggregate index. Furthermore, the Leland (1994) (and Geske (1977)) distinction between endogenous and exogenous bankruptcy is not relevant for index options. TP state that in Leland (1994), firms with a large amount of short term debt may result in exogenous, protective covenant bankruptcy, and firms financed mostly with longer term debt exhibit endogenous, stockholder decided bankruptcy. As already mentioned, TP compute a ratio of the book value of short term debt due in 1 year to book 7

8 value of total debt to serve as a covenant proxy. They suggest that when this proxy is large it may trigger short term exogenous bankruptcy. In this paper we find that the daily average of aggregate short term debt in year 1 in the index, after netting accounts receivable, is small relative to total debt in years The resultant average daily duration of aggregate debt in the aggregate balance sheet for the 500 firms in the S&P index is 4.71 years during the 100 month period in Thus, for equity index options there will be neither short term bankruptcy, nor long term stockholder decided bankruptcy. In fact, for an index like the S&P, the notion of collective bankruptcy happening at some specific time or being decided by the collection of stake-holders in the 500 firms in the S&P is not relevant. Furthermore, when a firm currently included in the S&P 500 does enter distress, Chapter 11, or default it is immediately replaced by a stable, healthy firm. 11 Consequently, the S&P 500 index has a survivor bias because it does not contain companies in serious financial trouble. In fact we think the index has very small but non-zero probability of default. 12 So, stylized models which attempt to include the effects of the aggregate debt in an underlying index on the prices of options on the aggregate index equity have to be as or more concerned with modeling the leverage effects on total risk of the aggregate equity, and less concerned with aggregate default details. Thus, if the market value of aggregate debt is relevant to index option pricing, it is probably because changes in the total risk of aggregate equity via 9 See paragraph 1, p. 18 herein, for a more complete description of how we treat the debt reported in years 1 to Guedes and Opler (JF,v51, 5, 1996), p. 1818, provide evidence that the mean duration of 7,362 bond issues of long term US corporate debt is 7 years during the 12 year time period Since outstanding bonds are a long term part of total debt in the corporate balance sheet, it is possible that the duration of corporate balance sheets during the Guedes-Opler period would have been similar to our average of 4.71 years. We also find that the variation in aggregate debt duration in our 100 month sample is bounded between and minimum and maximum of 4.5 and 5.1 years, respectively. 11 See Campbell, Stuart, 2004, Price Effects Surrounding Composition Changes in the S&P 500, Stanford Economics. AIG was replaced with Kraft in the S&P 500 in September, 2008, the day it was insolvent. This replacement rule for troubled firms suggests a survivor bias, so the observed distribution of the S&P 500 understates the total risk, and especially the left tail risk, of aggregate market equity. 12 From a model perspective, it is both theoretically and empirically possible, if not very probable, for the total market value of the 500 firms to be less than the face value of debt due at a payoff date. All we need here is for this default probability to be small but not zero. Systemic credit risk is present in markets as demonstrated in the depression in 1930 s and in the Fall, 2008 credit crisis. 8

9 changes in market value of aggregate leverage is important to index option prices. 13 The next few paragraphs describe our approach to provide evidence for this idea. We believe that the implied market value of aggregate corporate debt in an economy consisting of the 500 firms in the S&P should exhibit an expected leverage effect on both the total risk of the underlying equity index and consequently on the price of options on this index. It is of course necessary to first compute the implied market value of aggregate debt in order to compute the market value of aggregate leverage, D/E. The size of this leverage effect might be less insignificant when the market value of aggregate market leverage is less than 1 (D/E < 1), and more significant when the market value of aggregate market leverage greater than 1 (D/E > 1). Analysis is required to determine how different aggregate leverage ratios may affect index option prices. The purpose of this paper is to introduce a new methodology to measure the magnitude of the implied market value of aggregate debt in the index, and then test whether including the resultant market value of stochastic leverage as an explanatory variable can substantially improve the pricing of index options when compared to more complex models which omit leverage. 14 As previously stated this new methodology for estimating the market value of aggregate debt requires no historical price data. Instead this method relies only on the most liquid contemporaneous prices of the underlying equity and options on the equity. Thus, it is not necessary to estimate the volatility of the equity index from historical data. Instead, the volatility of the total market value V=D+E of the 500 firms in the index and the market value of aggregate debt can be implied directly from contemporaneous prices. Then, if desired, the stochastic volatility of the equity component, E, is immediately known from Ito s lemma, as σ E = ( E/ V) (V/E) σ V = ( E/ V) (1 + D/E) σ V. This equation shows it is the market 13 The default component of credit spreads has been shown to be small by Geske and Delianedis (2001). 14 Alternative techniques can be used to solve for the leverage and volatility, such as maximizing a likelihood functions, minimizing the sum of squared errors, or using historical volatility but to our knowledge no one has used the well known implied technique from liquid, contemporaneous prices similar to implied volatility methodology. In Section 4 we demonstrate how the differences in the implied market value of aggregate leverage (D/E) significantly improve index option pricing by altering the total risk of the underlying equity. 9

10 value of the aggregate debt, D, the resultant debt/equity ratio,d/e, and the equity sensitivity that is directly related to the total risk of the aggregate equity. To be more explicit, in this paper we first find the implied market value of aggregate market debt by solving simultaneously the leverage based equations for option prices from Geske (1979) and stock prices from Merton (1973). As originally derived both Merton s stock as an option and Geske s compound option model require the current total (total means market debt + market equity) market value, V, of the 500 firms in the index, and the instantaneous volatility of the rate of growth of this total market value, σ V, and both are not directly observable. Here this problem is circumvented by observing the known and traded market equity index price, and observing the known and traded market price of a put on the index, and then solving three simultaneous equations for the total market value, V, market return volatility, σ V, and the critical total market value, V τ *, for the option exercise boundary. As explained in the footnote below, since the total market value of the aggregate S&P index of 500 firms is V= D + E, where D represents the unobserved market value of the aggregate debt and E represents the observed, traded market value of the aggregate equity, solving for V is equivalent to solving for D, the implied market value of aggregate debt. 15 These three equations can be solved using contemporaneous market prices (or market prices lagged a day to be consistent for comparisons with this BCC requirement), and the volatility can be implied from a single liquid at-the- money option (ATM) or for BCC comparison by minimizing the sum of squared pricing errors for all options on a given day. 16 Neither of these implementation choices has any effect on our conclusions. We demonstrate that when the implementation methodology is identical to BCC (1997), G (1979) is superior to both BCC and BS. 15 By Modigliani and Miller, V=D+E, all in market values, and since E is known, solving for implied V is the same as solving for implied D. Obviously D+E can be substituted for V making D, the implied market value of debt, the relevant unknown. 16 We believe the implied methodology is best. Solving for multiple and different volatilities from each option price is inconsistent with a single true volatility, and uses the much less liquid away from the money strikes whose option prices must have greater errors. Furthermore, choosing as best a volatility number that minimizes the sum of squared pricing errors may not be consistent with any observed option price. 10

11 Before leaving this introduction we stress both the theoretical consistency and the importance of separating the potential measurable economic effects of the actual market value of aggregate leverage and its resulting induced stochastic equity volatility, from any other assumptions. The existence and effects of leverage are not an assumption, but instead are known to be present in equity index options. Whether or not aggregate market leverage alters the distribution of aggregate market equity in the index sufficiently enough to be important to pricing options on the index is an empirical question. We reason that if leverage is an important variable that is not properly included prior to assuming stochastic characteristics for other parameters such as volatility, interest rates, or jumps, then the additional assumed stochastic parameters will be estimated with error because of a relevant omitted variable. In the following pages, Section 2 and Appendix II describe the models of Geske (G), Merton (M), Black-Scholes (BS), and Bakshi, Cao, and Chen (BCC), and discusses how these models are implemented. Section 3 describes our data, Section 4 describes our results, and Section 5 concludes the paper. 11

12 2. Models The assumptions underlying the BS (1973) and M (1974) model and their resulting equations are very similar and are better known than the same for G s (1979) compound option model. This is especially true for this application of G to index options since this idea has not been considered previously in the literature. Thus, to review, recall that G s model applied to listed equity index options must transform the state variable underlying the option from the equity index level, E, to the total market value, V, of the 500 firms comprising the index (V = market debt D + market equity E). In this case the volatility of the equity index will be random and inversely related to the value of the total market equity. This interpretation of G introduces a new methodology to measure the market value of aggregate market debt and leverage (D/E). This model is consistent with Modigliani and Miller, and the BS model is a special nested case of G s model, which will reduce to the BS equation when either the dollar amount of leverage is zero or when the leverage is perpetuity. The boundary condition for exercise of an index option must also be transformed from depending on the index level, E, and strike price, K, to depending on the total market value of the firms in the index, V, and critical market value for option expiration, V*. If the conditions BS assume for the equity distribution are instead assumed for the total market value, V, then this above interpretation results in G s equation for pricing S&P 500 index put options: 17 rf 2 ( T2 t) rf1 ( T1 t) P = V N ( h + σ T t) N ( h + σ T t, h + σ T t; ρ)] + Fe [ N ( h ) N ( h, h ; ρ)] + Ke (1 N ( h)) (1) [ 1 2 v v 1 2 v where h 1 ln( V / V*) + ( r = σ F1 v 1/ 2σ T t 1 2 v )( T t) 1 17 See Geske (1979) for more detail. This application of G, where V equals the total market value of both the aggregate debt and the aggregate equity for 500 firms each day, results in a new methodology to measure market values for aggregate debt and leverage (D/E), and a new measure of aggregate equity volatility dependent on the market value of aggregate leverage. In this compound option application the index put option is on aggregate equity which is modeled as a call option on the aggregate total market value. 12

13 h 2 ln( V / F) + ( rf = σ v 2 1/ 2σ T 2 t v 2 )( T 2 t) and ρ = ( T t)/( T t 1 2 ) Here V* at option expiration date t=t 1 is the critical total market value at which the equity index level equals the strike price, E T1 = K. E T1 is deduced from Merton s application of the Black-Scholes equation which treats stock as an option: rf 2 ( T2 t ) E = V N h + σ T t ) Fe N ( h ) (2) and thus at t=t 1 where E T1 = K, 1( 2 v E T1 = V * T1 rf 2 ( T2 T1 ) N ( h + σ T T ) Fe N ( h ) = K (3) 1 2 v and h 2 is given above. The face value of all 500 firm s debt outstanding is F and T 2 is the duration of this debt. For Geske s compound option there are two correlated exercise opportunities at T 1 for the index put option exercise and at T 2 for the firm debt repayment. Their correlation is measured by ρ = ( T1 t)/( T2 t) where the index option expiration T 1 is less than or equal to debt duration T 2. In the special case when the firm has no debt or when the debt is perpetuity, V = E and σ V = σ E, and equation (1) reduces to the well known Black-Scholes put equation: rf 1 ( T1 t ) P = E( 1 N ( h + σ T t )) + Ke (1 N ( h )) (4) s 13

14 The notation for these models can be summarized as follows: P = current market value of an index put option E = equity index level net of dividends d, D = current market value of aggregate debt in the 500 firms in the S&P V = current total (debt D+ equity E) market value of 500 firms in the S&P 500, V* = critical total market value where V T1 V T1 * implies S T1 K, F K = face value of market debt (debt outstanding for S&P 500 firms), = strike price of the option, r Ft = the risk-free rate of interest to date t, σ v = the instantaneous volatility of the total market return, σ E = the instantaneous volatility of the equity index return, t T 1 T 2 = current time, = expiration date of the option, = duration of the market debt, N 1 (.) = univariate cumulative normal distribution function, N 2 ( ) = bivariate cumulative normal distribution function, ρ = correlation between the two exercise opportunities at T 1 and T 2. d = dividends Because of leverage the volatility of an option is always greater than or equal to the volatility of the underlying state variable, and given these assumptions the exact relation between the volatility of the 14

15 equity index and the volatility of the index total market value is expressed as follows: 18 σ E E V E D = σ = σ V 1 + (5) V V E V E Thus, while BS assume the equity s return volatility is deterministic and not dependent on the equity level, G s model implies that the volatility of the equity s return is stochastic, inversely related to the equity index level, and directly related to the market value of the aggregate debt/equity ratio. When the equity index level drops (rises), assuming the market does not react, the market value of leverage rises (falls), and the equity index volatility also rises (falls). For a comparison to more complex models of the underlying equity distribution which omit leverage, we implement the three versions of the BCC (1997) models (SV, SVSI, SVJ) using identical techniques described in their paper. We sketch the details of the BCC models in our Appendix II (p. 55). Here we show that while BS must solve for 1 implied parameter, σ s, and G must solve for 2 implied parameters, σ v and D (or V), the more complex BCC models of the equity distribution must solve for 5 required parameters for SV, 8 required parameters for SVSI, and 9 required parameters for SVJ. BCC cannot imply these parameters from two contemporaneous market prices, but instead must use many historical option prices that are more noisy because they are away from the liquid at-the-money strike. Also for comparison we implement the seminal BS model. In all implementations we adjust the equity index level for dividends, and we use the same normalization factor for the index debt that we compute for the index equity. In the next section we describe the data and calculations necessary to both test for the presence of leverage affects in index put option prices and to compare G s prices to the prices BCC and BS models. 18 See Geske (1979) for details. If dv= μ v Vdt + σ v Vdz v, and equity E is a function of V and t, E(V,t), then from Ito s lemma, de / E = {[( E / V ) μ E + E / t +.5σ V ( E / V )] / E} dt + ( E / V )( V / E) σ dz. E V E 15

16 3. Data In order to test for the effects of leverage on S&P 500 index put options we need option price data, stock price data, stock dividend data, interest rate data, and balance sheet information. We also require the composition of the S&P 500 firms on a daily basis. We collect daily closing stock prices, daily shares outstanding, and the daily composition of the S&P 500 index from CRSP. The interest rate data are daily from the Federal Reserve for government securities with maturities ranging from 1 month to 10 years, which we adjust to use discount factors for the market debt, option strike prices, and dividends. 19 The option prices we use are from Option Metrics from January 4, 1996 through April 30, This 100 month sample period covering 8 1/3 years contains about 200,000 index put options and 2080 observation days. The data are the daily closing prices, if there was a trade at the close, or the closing best bid and best ask as a spread, which we average for the closing option price. 20 We also collect the option volume and open interest data and dividend data for the S&P 500 from Option Metrics. The at the money (ATM) S&P 500 index options have the highest daily volume of all traded equity options and thus they should not exhibit much non-synchronicity. However, in order to further minimize nonsynchronous problems, first we check to see if there was an option trade on that day. Next we check to see if arbitrage bounds are violated (c.f. P E + K e r T T ) and eliminate these option prices. If nonsynchronicity occurred because the stock price moved up after the less liquid in or out of the money put option last traded, then option over-pricing would be observed. If non-synchronicity occurred because the stock price moved down after the less liquid in or out of the money option last traded, then option under-pricing would be observed, and some of these options would be removed by the above arbitrage 19 Option theory requires a risk-less interest rate. However, the over-night call money rate which is used for option market maker margin or the libor rate might also be used. Option prices are not too sensitive to the choice of interest rates, especially if the options expire in less than one year. 20 Option Metrics takes the best bid and ask from the exchange (CBOE, Phlx, Amex, Ise) that has the trade closest to the closing stock price which for the active at the money S&P 500 index options will almost always be synchronous. Using the mid-point of the bid/ask avoids bid/ask bounce problems. 16

17 check. Because we cannot perfectly eliminate non-synchronous pricing for the in and out of the money options with this data base, we keep track of the amount of under and over-pricing in order to relate this mis-pricing to the resultant over (under) pricing of in (out of) the money index put options for all models tested. The balance sheet information we collect from S&P s annual and quarterly Compustat. This book value of debt data is categorized as due in years 1 through 5 (Data 44, 91,92,93,94), and greater than 5 (Data 9 minus items (91-94)), which we place at 7 years. To these categories we add current liabilities (Data 5), deferred charges (Data 152), accrued expenses (Data 153), short term notes payable, deferred federal, foreign, and state taxes (Data 206,269,270,271), all payable in year 1. All long-term debt tied to prime (Data 148) and debentures (Data 82), we place in year 7, respectively. 21 The debt due on each day in each quarter of each year for the S&P 500 firms is the sum of the debt due for all 500 firms for that day in that quarter of that year. This structure of the S&P 500 debt outstanding permits the computation of the daily duration of the book value of market debt and the daily amount due at the duration date. Next we calculate the daily market value (cap) of the S&P 500 (stock price times shares outstanding for 500 firms), and we find the factor f which is used to normalize the index, and we confirm that we match the reported index level each day during our sample. We use this same normalization factor for the daily S&P 500 debt outstanding. This procedure produces daily for the aggregation of firms in the S&P 500, the exact market value of the aggregate equity and the face value and duration of the aggregate debt outstanding. 21 This follows from Guedes and Opler (JF,v51, 5, 1996), p. 1818, who provide evidence that the mean (of 7,362 issues) duration of long term US corporate debt is 7 years during the time period

18 Now we have the data defined on page 15 as P, E, F, K, r FT, t, T 1, T 2, and ρ, and are prepared to compute D, V, V*, and σ v. In order to compute D, V, V *, and σ v, we numerically solve simultaneously equations (1), (2), and (3), given market values for P, E, and the contracted strike price K. First we test these models using the methodology which allows a term structure of volatility, possibly different for different option expirations, but the same volatility for all strikes of the same option expiration. 22 We compute this term structure of volatility daily. All the tests use forward looking implied volatilities for all models. Since we know from the open interest and volume data that the most at-the-money (MATM) options are the deepest and most liquid we base the volatility term structure on the most at-the-money options. Thus, daily we compute the implied volatilities for the equity index, σ E, using BS and for the total market value of the index, σ v, using G for each time to expiration for the liquid most at the money option, given the stock price, option price, and strike price. Daily, for different times to expiration we hold the observed equity index level, E, and the computed total market value of the debt and equity, V, constant, and allow the implied volatility for the most at the money option to produce this option s market price. Given the observed market prices of index put options this methodology produces the well documented BS pricing biases observed for S&P 500 index put options. Relative to the market price, the Black-Scholes model over values the vast majority of in the money(itm) put options and under values the vast majority of out of the money(otm) put options. As is the case with many of the more recent models discussed in our Introduction, the three versions of BCC models, SV, SVSI, and SVJ, have many additional parameters to be estimated for the stochastic processes assumed. To estimate these additional parameters it is necessary for researchers to use most of the options present on each day in order to find the volatility that day that minimizes the sum of 22 It is our understanding that most derivative professionals and academics accept the concept of a term structure of volatility. However, in order to compare models with identical implementation methodologies the Appendix shows none of the results change when G is implemented exactly as reported in BCC (JF, 1997), finding the volatility from the previous days set of option prices that minimizes the SSE. 18

19 squared errors across all those options. 23 Thus, in order for these parameter estimates to remain out of sample, researchers following BCC typically estimate the required parameters from prices lagged one day, and then use these parameter estimates to price options the next day. We follow the BCC implementation technique exactly, and compare their prices with G and BS using both the MATM term structure volatility and the 1day lagged estimate of volatility that minimizes the sum of squared errors. 24 Given the data and estimates described, we can now examine what improvement, if any, G s leverage based option model may provide when compared to BCC and BS models which omit any leverage measure. Depending on the definition of at the money as being plus or minus 5% or 0% for MATM (1 option), after screening we examine between 115,000 and 190,000 matched pairs of options over the 2080 trading days. In the next section we present the results of these comparisons. 4. The Results In this section we present evidence about the size and variation of the market value of aggregate leverage in the S&P 500 firms derived directly from option theory, and details about the model matched pair comparison results for BCC, BS, and G. We present graphs and detailed tables of each model s pricing errors. The results illustrate both the statistical and economic significance of the BCC and BS pricing errors, the relation of these errors to their omission of leverage, and G s relative improvements with respect to matched pairs of options categorized by time to expiration, leverage, moneyness, and calendar year. For ease of reading we briefly summarize the main results here. We start with Table 8a, where the number of OTM matched pair comparisons of options arising from the two ATM definitions (+/- (5%) or (0%)), is 75,300 (109,301). G s model is closer to the market price than the BS model for 23 The use of the MATM options which are daily the world s most liquid options avoids non-synchronicity. Also, the necessity of using away from the money options which are much less liquid than the ATM options may introduce measurement error in the parameter estimates of these more complex models. 24 In the Appendix we show in detail that volatility estimation based on minimizing the sum of squared errors and lagged one day has no effect on our conclusions. 19

20 75,052 (108,437) of these matched pairs. Remarkably, G is closer than BS on 99% of all matched pairs of OTM index put options. The number of OTM matched pairs of options compared in Table 8d, Panels A and B, arising from the two ATM definitions of 5% (0%) is 95,186 (139,016), respectively. G s model is closer to the market price than the BCC s SVJ model for 58,466 (90,230) of these matched pairs (about 65%) and BCC is closer on 36,720 (48,786) pairs. So, by comparison we see that G is closer than BS to the OTM put market prices for 99% of the matched pairs, but G is only closer than BCC best model, SVJ, on about 65% of the matched pairs. However, because BCC produces many extreme values, the economic (basis point bp) improvement of G relative to BCC is greater than G s bp improvement relative to BS. Similarly, the number of ITM matched pairs of options compared for BS and G in Table 7a, Panels A and B arising from the two ATM definitions of 5% (0%) is 22,853 (50,452). G s model is closer to the market price than the BS model for 19,837 (44,812) of these ITM matched pairs (about 90%) and BS is closer on 3,016 (5,640) pairs. Also, the number of ITM matched pairs of options compared for BCC and G in Table 7d, Panels A and B, arising from the two ATM definitions of 5(0)% is 23,792 (59,569), respectively. G s model is closer to the market price than the BCC (SVJ) model for 15,123 (43,733) of these 23,792 (59,569) ITM matched pairs (about 75%) and BCC is closer on 8,669 (15,836) pairs. This summary reveals the conclusive dominance of G relative to both BCC and BS models. Details about this summary are in the discussed in the following sub-sections of Section 4, a - e. 4a. Market Leverage First, using G s compound option model we compute the daily market value of the aggregate debt to equity ratio of the 500 firms in the S&P, where, as previously explained, the market value of aggregate debt is derived from the option pricing structure using daily market prices for the equity index and 20

21 market prices of ATM options on the index. 25 We believe that Figure 1 is the first presentation of implied market value of aggregate debt depicted as a time series of market value of the aggregate D/E ratio for the S&P 500, and we plot D/E along with the level of the S&P 500 equity index. Figure 1 shows that during our sample period January, 1996 to April, 2004, the market value of the aggregate debt/equity ratio for all firms in the S&P 500 has considerable variation, ranging from a minimum of about 0.40 in January 2000, to a maximum of 1.20 in April As expected, market value of the aggregate debt/equity ratio and the S&P 500 equity index level, and index volatility that are inversely related. 26 Figure 1 shows the S&P 500 leverage is highest in years 2002 and 2003, and lowest in years 1999 and Later in this section we demonstrate that models which omit leverage (BCC and BS) exhibit larger (smaller) pricing errors in 2002 and 2003 (1999 and 2000) when the market value of leverage is larger (smaller). D/E Daily Fig 1. S&P 500 D/E & Index Level Daily Graph 04jan jan jan jan jan2004 Date Index Level Daily D/E Daily Index Level Daily 25 We repeat for emphasis this methodology uses only very liquid contemporaneous index prices and index option prices and interest rates, and does not require any historical price inputs such as the computation of historical volatility. 26 In order to keep the graph uncluttered we omit the equity index volatility which is inversely related to the equity level by equation (5). However, on the next page we give an example of the leverage effect on volatility. 21

22 The S&P 500 Stock Index represents about 80% of the market capitalization of all stocks listed on the New York Stock Exchange, and it is often treated as the market in asset pricing tests. For example, in order to characterize the effect of the market value of leverage presented in Figure 1 on equity index volatility, if the duration of the aggregate debt was 5 years, the 5 year risk free rate was 5% per annum, and the total volatility of the market value of the aggregate market, V, was 0.50 (0.20) per annum, then when the market value of the debt/equity ratio was about either 0.40 or 1.20, the low to high D/E for our sample period, the aggregate equity volatility from the leverage effect of equation (5) would be about 0.66 or 0.85 (0.28 or 0.41). 4b. Model Pricing Error Comparison for BS and G We first represent separately BS model values relative to the market prices because this characterizes how the literature and later models (BCC, G, et al) evolved to attempt to mitigate these well known BS problems. Because G uses more parameters than BS we may expect G to be superior to BS if the additional data necessary to implement G is relatively accurate. If we use the same criteria that models with more parametric degrees of freedom might be more accurate, then we would expect BCC to outperform G. BS must estimate 1 parameter, the equity index volatility. G must estimate 2 parameters, the market value of aggregate debt, D (and thus V), and the volatility, σ v, of the aggregate market value, V. BCC omits leverage but assumes much more parametric complexity for the underlying equity index distribution. BCC must estimate multiple volatilities, correlations, and speeds of adjustment, resulting in 5 parameter estimates for SV, 8 parameter estimates for SVSI, and 9 parameter estimates for SVJ. 22

23 Fig 2. Put Option Prices Option Prices 1 K/S BS Market Figure 2 presents a graph of put option market prices, BS model values, and moneyness, K/S, which is representative of most research findings for the S&P 500 index put options. Since the index level, S in the Figures, is the same for all K at any point in time during or at the end of any day, as K varies in Figure 2, the out of the money (OTM) stock index puts (low K) are shown to be under valued and the in the money (ITM) index puts (high K) are shown to be over valued by the BS model relative to the market prices. 27 We show in Figure 3 that G s compound option model has the potential to improve or even eliminate these BS valuation errors because of the leverage effect. The reason for this, once again, is the economic effects of leverage create the necessary negative correlation between the index level and the index volatility. This interaction between the index level and index volatility implies that the index volatility is both stochastic and inversely related to the level of the index, and that the resultant implied index return distribution will have a fatter left tail and a thinner right tail than the BS assumption of a normal distribution. Thus, G s compound option model produces 27 Figure 2 represents the most ubiquitous result from our data. However, for a very small number of index option matched pairs there are (15) different model distance comparisons that are made: both over, both under, one over while the other is under, one equal to the market while the other is either over or under, both equal to each other but either over or under, both equal to each other and equal to the market, and furthermore, there are multiple cases for each situation when the models are not equal to each other. 23

24 option values that are less (greater) than the BS values for in (out of) the money European index put options, and could potentially eliminate the well known BS bias. Fig 3. Put Option Prices Option Prices 1 K/S CO BS Figure 4 presents how we measure the amount of improvement G s model provides relative to the alternative model (here BS) for S&P 500 stock index put options during this sample period. We create thousands of matched pairs of all options for each expiration date and each strike price, and we measure the distance between each model s value and the market price. We compare the distance that each model value is from the market price for each matched pair, find the model that produces the closest distance to the market, and we compute the improvement of one model to the other for that pair. We then net these distances for all matched pairs in order to find which model is closest to the market for all matched pairs on average and how much net improvement, if any, is present. We present this analysis for all matched pairs of options for a variety of categories with different times to expiration, different moneyness, and for the different market leverage exhibited during our sample period. 24

25 Fig 4. Put Option Prices G's Improvment BS Error Option Prices G Error 1 K/S CO BS Market The improvement of Geske s compound option model compared to the Black-Scholes is calculated with the following formula: 28 BS error BS error G error = (Market - BS) - (Market - G) (6) (Market - BS) The tables that follow demonstrate the importance of leverage by presenting both the statistical significance and the economic significance of G s index option pricing improvements relative to the alternative model, here BS. 4c. Tables of BS vs. G by Moneyness, Year, Expiration, and Leverage First we focus on BS and present a detailed analysis of the ITM and OTM option pricing errors of BS and G s relative improvements as depicted in Section 4c, for different times to expiration by calendar year and by market leverage, using two definitions of ATM. We also present the number of matched 28 Care must obviously be taken with the signs of the variety of matched pair errors explained in footnote 12, especially if one model value distance is above and the other distance is below the market price, when computing the average error across all matched pairs. However, the results depicted in Figures 2, 3, and 4 are found for more than 98% of all option pairs. 25

26 pairs of options available in each of these categories during this time period, and we examine both the statistical and economic significance of G s improvements relative to BS. When the ATM option region is considered to be within 5% of the index level a large number of better priced but still mis-priced options are eliminated. If we consider only one option per day per time to expiration as the most at the money option, to be defined as MATM, then all but one previously eliminated 5% ATM s will now be either in or out of the money and priced with some error. This alternate definition of an ATM option will increase the sample size of mis-priced options. We also would expect this ATM definitional change to reduce the average net pricing error because the ATM options exhibit smaller pricing errors. 29 Consider the number of matched pairs of ITM index put options presented in Table 1. Panel A illustrates that if we consider only one option to be ATM each day (the most at the money option), the sample of in the money put index options more than doubles from 23,438 to 57,177 matched pairs. Panel A also shows the most active trading years for ITM put index options during our sample period are 2000, 2001, and 2002, containing 11,349 of the 23,438 (22,381 of the 57,177) total ITM option matched pairs. As expected, Figure 1 shows that during these years the market was the most decreasing, resulting in the highest ITM years. Also as expected, this table shows that the ITM nearer expiration puts (< 120days) are traded more heavily than the longer 29 Near or ATM options are the most liquid and for implied volatilities are generally the best estimates of future realized volatility. Because of this near or ATM volatilities are believed to contain the most accurate price information and be more accurately valued. 26

27 TABLE 1 PUT ITM PANEL A TOTAL NUMBER OF OPTIONS Option Expiration (in Days) Option Expiration (in Days) YEAR Min Max TOTAL YEAR Min Max TOTAL , , , , , , ,866 1,056 2, , , , , , , , , , , , , , , , , , , , , , , , , , ,086 3,094 1,048 2, , , , , , , TOTAL 1,938 7,599 3,007 8,101 2,793 23,438 TOTAL 7,515 22,219 7,650 15,028 4,765 57,177 PANEL B Option Expiration (in Days) Option Expiration (in Days) D/E Min Max TOTAL D/E Min Max TOTAL , , ,242 3,555 1,409 2, , , , , ,287 6,872 2,241 4,643 1,711 17, , , , ,493 4,590 1,791 3,584 1,332 12, , , , , , , , , , , , , , , , ,341 TOTAL 1,938 7,599 3,007 8,101 2,793 23,438 TOTAL 7,515 22,219 7,650 15,028 4,765 57,177 expiration puts (> 120 days) in every year. In Table 1, Panel B, we present these same ITM index put options by time to expiration and now categorized by leverage (D/E) ratio. Recall the market D/E ratio during this time period ranges between 40% and 120%, as was depicted in Figure 1. For the 5% ATM sample, Panel B shows that about 25% of ITM put index option matched pairs traded when the market D/E ratio was in a high range from 80% to 120%. Table 2 presents the net pricing error improvement of G relative to BS by calendar year and by leverage ratio for the various times to expiration and for the two definitions of ATM for all ITM put index option matched pairs during this sample period. Note in Panel A that the improvement of G s model with 27

28 respect to time to expiration varies in total from 12% for the shortest expiration index options to 45% for longest expirations, and is strictly monotonic across all ranges of expiration. 30 G s improvement is greater for the longer time to expiration options because the leverage has a longer effect on their value. Also note that G s improvement over BS is greatest in the complete years 2002 & 2003, averaging between 70% and 90% across all times to expiration. This is as expected because Figure 1 (and Table 2, Panel A) shows that market leverage was highest in 2002 and 2003, with D/E ranging (averaging) between 0.8 (0.72) and 1.2 (1.0). Similarly, G s improvement is smallest but still greater than 20% in the low leverage years of 1999 and 2000, when the D/E ratio ranges (averages) between 0.4 (0.48) and 0.5 (0.5). Table 2 also illustrates, as expected, that when ATM is defined as a single most at the money option, the previously excluded but now ITM options which have smaller pricing errors reduce the net pricing improvement in all years and across all times to expiration except the nearest to expiration. Table 2, Panel B, categorizes options by leverage instead of by year. Here it is shown that relative to BS the improvement of G s model increases with the D/E ratio almost monotonically for every time to expiration, especially when the sample number of options in each category is sufficiently large. As expected, for the highest leverage categories of 0.9 to 1.2, G s improvement is greatest and averages between 76% and 96% for both 5% and 0% ATMs. 30 Note that in a few instances the pricing error correction is greater than 100%. This can happen when the two models errors are on opposite sides of the market price. 28

29 TABLE 2 PUT ITM PANEL A PRICING ERROR IMPROVEMENT Option Expiration (in Days) Option Expiration (in Days) YEAR D/E Min Max TOTAL YEAR D/E Min Max TOTAL % 75% 48% 63% 73% 68% % 34% 37% 56% 71% 53% % 57% 45% 59% 72% 62% % 33% 38% 56% 71% 51% % 23% 21% 30% 40% 31% % 19% 21% 30% 39% 28% % 18% 19% 24% 28% 24% % 13% 17% 23% 28% 20% % 18% 22% 26% 28% 26% % 14% 20% 25% 28% 23% % 40% 38% 42% 52% 45% % 35% 37% 40% 52% 42% % 71% 71% 80% 74% 76% % 68% 73% 80% 75% 75% % 78% 63% 96% N/A 86% % 51% 54% 92% 0% 70% % 152% 62% N/A N/A 421% % 43% 45% 0% 0% 48% TOTAL % 37% 35% 42% 45% 42% TOTAL % 28% 33% 41% 45% 38% PANEL B Option Expiration (in Days) Option Expiration (in Days) D/E Min Max TOTAL D/E Min Max TOTAL % 17% 20% 24% 27% 24% % 13% 18% 23% 27% 21% % 24% 24% 32% 39% 32% % 19% 22% 31% 40% 29% % 40% 37% 48% 57% 49% % 29% 35% 46% 56% 44% % 67% 47% 66% 65% 65% % 51% 46% 62% 65% 58% % 52% 55% 84% 69% 71% % 31% 45% 78% 69% 54% % 74% 67% 85% 88% 83% % 61% 62% 83% 91% 76% % 82% 82% 95% 79% 88% % 93% 96% 102% 88% 96% TOTAL 12% 37% 35% 42% 45% 42% TOTAL 18% 28% 33% 41% 45% 38% Table 3 presents similar data to Table 1 for out of the money (OTM) index put option matched pairs. First consider the number of traded index puts presented in Table 3 for OTM options. In Panel A when ATM is defined as 5% the near expiration index puts are traded much more heavily than the far expiration puts every year. Here the shorter expiration options comprise about 52% (47,605/91,950) of these matched pairs. Panel A also illustrates that when we consider only one option to be ATM each day (the most at the money option), the sample of OTM put index option increases from 91,950 to 132,388 matched pairs. The most active trading years for OTM put index options during our sample period are 1997, 1998, and Figure 1 shows this is the time period when the S&P 500 index level was the most increasing, resulting in the highest OTM years. 29

30 TABLE 3 PUT OTM PANEL A TOTAL NUMBER OF OPTIONS Option Expiration (in Days) Option Expiration (in Days) YEAR Min Max TOTAL YEAR Min Max TOTAL ,105 3,664 1,626 2, , ,880 6,223 2,689 3, , ,583 5,196 2,152 4, , ,564 8,081 3,147 5,604 1,191 20, ,775 5,462 2,054 3, , ,794 8,010 2,825 5,103 1,275 20, ,356 5,005 2,119 4, , ,460 7,295 2,760 5, , ,000 4,051 1,643 2, , ,830 5,746 2,191 3, , ,095 4,010 1,459 2, , ,759 5,527 1,971 2, , ,076 3,880 1,531 2, , ,678 5,343 2,030 3, , ,422 4,329 1,928 2, , ,357 6,452 2,525 3, , , , , ,647 TOTAL 10,906 36,699 14,644 25,134 4,567 91,950 TOTAL 18,180 54,287 20,317 33,297 6, ,388 PANEL B Option Expiration (in Days) Option Expiration (in Days) D/E Min Max TOTAL D/E Min Max TOTAL ,944 6,858 3,174 4, , ,483 10,032 4,127 6,484 1,340 25, ,485 11,986 4,455 8,601 1,783 30, ,701 17,537 6,231 11,133 2,424 43, ,972 7,121 2,941 4,820 1,097 17, ,314 11,120 4,449 6,982 1,562 27, ,359 3,845 1,178 2, , ,233 5,736 1,742 2, , , , ,194 3,181 1, , ,061 3,352 1,337 2, , ,648 4,652 1,703 3, , , , , , , ,136 TOTAL 10,906 36,699 14,644 25,134 4,567 91,950 TOTAL 18,180 54,287 20,317 33,297 6, ,388 In Table 3, Panel B, we present these same OTM index put options by time to expiration and by debt/equity (D/E) ratio for the same ranges of time to expiration and leverage. Here the higher leverage categories (0.8 to 1.2) comprise about 18% of the OTM matched pairs. Table 4 presents the net pricing error improvement by year and by D/E ratio for the various times to expiration and for the two definitions of ATM for all OTM put index options matched pairs during this sample period. In Panel A when ATM is defined as 5%, the high leverage years 2002 and 2003 again exhibit G s greatest pricing improvement of 20% relative to BS for these less valuable OTM matched pairs. 30

31 TABLE 4 PUT OTM PANEL A PRICING ERROR IMPROVEMENT Option Expiration (in Days) Option Expiration (in Days) YEAR D/E Min Max TOTAL YEAR D/E Min Max TOTAL % 7% 13% 22% 38% 18% % 9% 15% 24% 39% 18% % 8% 14% 25% 40% 19% % 9% 15% 25% 41% 19% % 5% 8% 15% 26% 13% % 6% 9% 15% 27% 13% % 5% 8% 14% 23% 12% % 6% 8% 14% 23% 12% % 6% 9% 15% 23% 12% % 6% 10% 16% 23% 12% % 7% 12% 20% 35% 15% % 8% 12% 21% 35% 15% % 9% 13% 24% 42% 20% % 9% 14% 25% 42% 19% % 10% 14% 29% N/A 20% % 11% 15% 30% N/A 20% % 5% 8% N/A N/A 5% % 7% 10% N/A N/A 6% TOTAL % 7% 11% 19% 30% 15% TOTAL % 7% 11% 19% 30% 15% PANEL B Option Expiration (in Days) Option Expiration (in Days) D/E Min Max TOTAL D/E Min Max TOTAL % 5% 8% 14% 22% 12% % 5% 8% 14% 22% 11% % 6% 10% 17% 29% 14% % 7% 11% 17% 30% 14% % 7% 12% 22% 36% 17% % 8% 13% 23% 37% 17% % 7% 12% 22% 40% 17% % 8% 13% 23% 40% 17% % 7% 12% 23% 43% 14% % 8% 14% 24% 43% 15% % 10% 14% 27% 45% 20% % 11% 15% 27% 45% 20% % 14% 17% 32% 42% 24% % 14% 17% 32% 42% 24% TOTAL 2% 7% 11% 19% 30% 15% TOTAL 2% 7% 11% 19% 30% 15% In the lowest leverage years of 1999 and 2000 G exhibits the smallest pricing error improvement of 12% (excluding the small sample partial year 2004). Table 4 also illustrates that when ATM is defined as a single most at the money option, the previously excluded but now OTM options which have smaller pricing errors reduce the net pricing improvement in all years. Table 4, Panel A, again illustrates that G s improvement increases monotonically with time to expiration because these options have a longer lasting leverage effect. Table 4, Panel B, demonstrates that G s improvement also increases with the D/E ratio, again almost monotonically for every time to expiration, especially when the sample number of options is sufficiently large. Again, as expected for these OTM matched pairs, the highest (lowest) D/E categories exhibit G s greatest (smallest) improvement over BS. 31

32 We have also tried a different volatility methodology identical to the implementation by BCC of basing the aggregate net pricing errors on the volatility that minimizes the sum of squared errors. We find that this does not change the characteristics of our results, and this is evident regardless of whether we allow or do not allow a term structure of volatility, or whether we lag the volatility estimate by one day. This result is not surprising because if the volatility that minimizes the sum of squared pricing errors is moved away from the near or ATM volatility toward either the ITM or OTM volatilities, then there will be an off-setting effect from the larger errors in the other moneyness direction. This off-setting effect will be present independent of the definition of ATM (%5 or 0% most at the money). Also, we show a one day lag in the volatility estimation does not change any of our conclusions. 31 Statistical Significance of BS and G Differences Here we use non-parametric statistics to test the significance of the differences between BS and G s model, which is the same as the significance of the reported improvements, using both the 5% ATM and the 0% (most at the money) ATM. As can be seen in Table 5 for ITM options and Table 6 for OTM options, we find G s model improvements are all significant at greater than the 99.99% level except for the very near maturity options. 32 Near maturity when market option prices are converging to the in and out of the money boundaries there is much more noise in the pricing errors, especially for the out of the money options that are approaching zero. 31 See the Appendix for results with implementation identical to BCC (1997). 32 Furthermore, when using a volatility that is minimizing the sum of squared errors these significance results also hold, and the near maturity options remain significantly different as others have reported (see Heston and Nandi (2000). 32

33 TABLE 5 PUT ITM PANEL A Rank Sum Test p Value Option Expiration (in Days) Option Expiration (in Days) YEAR Min Max TOTAL YEAR Min Max TOTAL TOTAL TOTAL PANEL B Option Expiration (in Days) Option Expiration (in Days) D/E Min Max TOTAL D/E Min Max TOTAL TOTAL TOTAL TABLE 6 PUT OTM PANEL A Rank Sum Test p Value Option Expiration (in Days) Option Expiration (in Days) YEAR Min Max TOTAL YEAR Min Max TOTAL TOTAL TOTAL PANEL B Option Expiration (in Days) Option Expiration (in Days) D/E Min Max TOTAL D/E Min Max TOTAL TOTAL TOTAL

34 4d. Absolute and Relative Distance from the Market for BCC vs. G and BS The previous tables of results focused on BS biases and G s improvements relative to BS pricing errors. In this section we examine all matched pairs for distance differences to see which model is closest to the market price as measured by the absolute dollar and relative per cent distance from the market price. We now include BCC in comparisons with G and BS. Thus, for all matched pairs comparing BCC, G, and BS we determine which model is closest to the market price, and we also compute the relative percent pricing error. First we present these graphs for both time to expiration and moneyness. Time to Expiration In Figure 5A.(I) we present the average ITM absolute dollar pricing errors for matched pairs of index options with different times to expiration for BCC(SVJ), BS, and G. The average is across all strike prices for the same time to expiration and ATM is 5%. Fig5. A (I): Put ITM Absolute Dollar Pricing Errors Absolute Dollar Pricing Errors Option Expiration in Days Abs(CO-Market) Abs(BS-Market) Market Abs(SVJ-Market) 34

35 Here G is shown to be superior to both BCC and BS for ITM options with different times to expiration, and in this comparison BCC appears to be superior to BS except at short expirations. 33 Figure 5A.(II) presents the relative pricing errors for ITM options for different times to expiration. G s errors are almost always superior to BS and BCC, averaging about 3%, while BS and especially BCC errors have extremes and average more than 6%. Relative Pricing Errors Fig5. A (II): Put ITM Relative Pricing Errors Option Expiration in Days Abs(CO-Market)/Market Abs(BS-Market)/Market Market Abs(SVJ-Market)/Market Figure 5B.(I) presents similar average absolute dollar pricing errors for different times to expiration for BS, BCC (SVJ) and G computed for OTM matched pairs of index options. Absolute Dollar Pricing Errors Fig5. B (I): Put OTM Absolute Dollar Pricing Errors Option Expiration in Days Abs(CO-Market) Abs(BS-Market) Market Abs(SVJ-Market) 33 Here we present only the stochastic volatility with jumps version of BCC (SVJ) because it performs best. See Appendix for other comparisons. 35

36 This graph again demonstrates that G is superior to both BS and BCC for OTM options with different times to expiration. However, in this OTM comparison it is not clear that BCC is superior to BS since the vast majority of index put options traded have less than 150 days to expiration. In this expiration interval G is superior to BS and BS is superior to BCC. Figure 5B.(II) adds some clarification to the G-BS-BCC (SVJ) comparisons by presenting the relative pricing errors for OTM options for different times to expiration. G s errors are again always superior to both BCC and BS. However, in this graph it appears BS and G are closer 34, and that BS is superior to BCC. In Section 4f we show that G is closer to the market relative to BS (BCC) on 99% (65-90%, SVJ, SV, SVSI) of all OTM matched pair comparisons, resulting in considerable relative economic advantage. Furthermore, this section confirms that BS is sometimes relatively superior to BCC. Fig5. B (II): Put OTM Relative Pricing Errors Relative Pricing Errors Option Expiration in Days Abs(CO-Market)/Market Abs(BS-Market)/Market Market Abs(SVJ-Market)/Market 34 Previously in section 4d comparing only BS and G, Tables 3 and 4 for OTM index options demonstrated that G was much superior to BS. Here the scale for OTM relative pricing errors must be large to show BCC s extreme errors which makes the differences between G and BS appear smaller. 36

37 Moneyness Figure 6A(I) presents the average ITM absolute dollar pricing errors versus moneyness, where moneyness ranges from 1.05 to 1.25 when ATM is defined as 5%. The average is taken across different expirations for matched pairs of index put options with the same moneyness. Absolute Dollar Pricing Errors Fig6. A (I): Put ITM Absolute Dollar Pricing Errors K/S Abs(CO-Market) Abs(BS-Market) Market Abs(SVJ-Market) G is again shown superior to both BCC (SVJ) and BS for ITM options for all the different ITM amounts. In this comparison BCC and BS are somewhat similar, with BCC (BS) sometimes being superior to BS (BCC). Figure 6A(II) presents the average relative pricing errors for ITM options for different moneyness amounts. The figure shows that G s errors are always superior to BCC (SVJ) and BS, and average about 3%, while BCC and BS errors average more than 5%. 37

38 Fig6. A (II): Put ITM Relative Pricing Errors Relative Pricing Errors K/S Abs(CO-Market)/Market Abs(BS-Market)/Market Market Abs(SVJ-Market)/Market Figure 6B(I) presents the average absolute pricing errors for OTM index options for different moneyness amounts when ATM is defined as 5%. Fig6. B (I): Put OTM Absolute Dollar Pricing Errors Absolute Dollar Pricing Errors K/S Abs(CO-Market) Abs(BS-Market) Market Abs(SVJ-Market)/Market>1.0 38

39 This graph again demonstrates that G is superior to both BCC (SVJ) and BS for OTM options as index option moneyness varies. Once again, in this comparison BCC s errors fluctuate relative to BS, with BCC (BS) sometimes being superior to BS (BCC). Figure 6B(II) presents the relative pricing errors for put options for different OTM amounts with ATM again at 5%. The figure shows that G s errors always superior to both BCC (SVJ) and BS. 35 However, here BS appears to dominate BCC because BCC has many options with extreme pricing errors which increase BCC s relative pricing error computed per option in each moneyness category. Fig6. B (II): Put OTM Relative Pricing Errors Relative Pricing Errors K/S Abs(CO-Market)/Market Abs(BS-Market)/Market Market Abs(SVJ-Market)/Market The above graphs offer a visual image of G, BCC (SVJ), and BS absolute and relative pricing errors in terms of distance from the market price for both different times to expiration and different moneyness amounts. Each graph is constructed from tens of thousands of matched pairs of options (options with the same strike K and time to expiration T). The superiority of G relative to BCC and BS in terms of 35 As mentioned in the previous footnote, in section 4d comparing only BS and G, Tables 3 and 4 for OTM index options demonstrated that G was much superior to BS, and the closeness here is because of scale necessary to include BCC s pricing errors. 39

40 closeness to the market price is highly statistically significant in all comparisons. The next section presents evidence on the economic significance of the differences between BCC (SVJ), BS, and G. 4e. Economic Significance of Geske Improvements Compared to BS and BCC To complement all previous graphs and tables comparing the pricing errors of BCC (SVJ), BS, and G, here we report the economic significance of G s relative improvements for ITM options in Tables 7 and OTM options in Tables 8. Specifically, Tables 7 and 8 show results when G s model is compared to BCC and BS on three dimensions: i) by the number of matched pairs that G is a closer absolute distance to the market price, ii) by the dollar value of this G s improvement, and iii) by the basis points (bp) that G s improvement implies for an option portfolio. These comparisons are categorized by both calendar year and by leverage. 36 First, consider Table 7a comparing G and BS for ITM options when ATM is defined as either 5% (0%). In Panel A the columns left to right represent the year, the present value of all ITM put index option matched pairs for that year, the total number of the matched pairs that year, the number of those matched pairs where BS is closer to the market price in absolute distance, the number of matched pairs where G is closer to the market price, the dollar value of the BS improvement, the dollar value of G improvement, and the net basis point advantage or disadvantage of G s model for that year. 36 Economic improvement (bp) herein is based on a portfolio of 1of each option per day when actual daily volume experienced by market makers (or dealers) in each option is much greater. G s improvements are relative to the alternative model, not the market, and in no way do the improvements imply option market inefficiency or arbitrage opportunities. 40

41 TABLE 7a PUT ITM: CO vs. BS PANEL A BASIS POINT IMPROVEMENTS YEAR PV NUMBER NUMBER NUMBER YEAR PV NUMBER NUMBER NUMBER TOTAL BS CO BS CO BP TOTAL BS CO BS CO BP , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,485 1, , , ,359 1,369 5,990 1, , , , , , , , ,179 1, , , , TOTAL 2,851, ,853 3,016 19,837 2, , TOTAL 4,108, ,452 5,640 44,812 4, , PANEL B D/E PV NUMBER NUMBER NUMBER D/E PV NUMBER NUMBER NUMBER TOTAL BS CO BS CO BP TOTAL BS CO BS CO BP , , , , , , , , , , , , ,317, ,706 1,323 14, , , , , , , ,149 1,135 10, , , , , , , , , , , , , , , , , , , , , ,619 1,022 3,597 1, , , , , , , , ,312 1, , TOTAL 2,851, ,853 3,016 19,837 2, , TOTAL 4,108, ,452 5,640 44,812 4, , Panel B of Table 7a presents the same information categorized by the D/E ratio instead of by year, where D/E ranges from %. The totals for each column and each row are also presented. The total number of ITM matched pairs of options compared in Table 7a, Panels A and B for the two ATM definitions of 5% (0%) is 22,853 (50,452). G s model is closer to the market price than the BS model for 19,837 (44,812) of these ITM matched pairs (about 90%) and BS is closer on 3,016 (5,640) pairs. Table 7a shows that by G being closer to the market price than BS on 90% of the ITM option matched pairs results in a basis point (bp) net improvement on average of 167 bp (134) for ITM options in a one of each option portfolio of options when ATM is defined as 5% (0%), respectively. These numbers are 41

42 calculated by constructing a one of each option portfolio containing one option for each strike price and time to expiration for each day and finding the market value of that one of each option portfolio each day for all days in a year. These numbers present a lower bound on G s economic advantage relative to BS. In the following we explain in more detail the computation of the dollar and basis point improvement. More specifically, dollar improvement for each model is measured by considering all those matched pairs where a specific model is closer to the market price than the alternative model in absolute distance measured in dollars. The basis point advantage of Geske s model is then computed by dividing the net dollar improvement for that year or leverage category by the total value of options in that category. For example, in Table 7a, Panel A, across the sample years the G s compound option model has a total dollar improvement of $50, and BS has a dollar improvement of $2, Thus, the net dollar improvement of Geske s model is $47,754.85, and that divided by the total value of each option in this ITM portfolio, $2,851,273.74, produces the 167 net basis point improvement. When the 0% ATM definition of a single most at the money option is used this ITM portfolio value increases to $4,108, because of the inclusion of previously excluded 5% ATM options. This larger number of near the money options reduces the average errors, and the basis point improvement of G relative to BS drops to 134 bp. While the percent pricing error of G s improvement relative to BS is monotonic in leverage, basis point improvement need not be since this depends on the dollar value of the options. Next, consider Table 7d comparing BCC s SVJ model and G for ITM options when ATM is defined as 5% (0%). 37 Table 7d, Panels A and B, incorporates the same format as Table 7a. The number of ITM 37 Again, for brevity we only present BCC s best model, SVJ (Table 7d and 8d), which is somewhat better than their SV model (Table 7b and 8b) and much better than their SVSI (Table 7c and 8c) model. Readers interested in more detail can see the Appendix. For example, in the Appendix the matched pair comparison for G versus BCC s SVSI model shows G is closer to the market on 90% of all matched pairs. 42

43 matched pairs of options compared in Table 7d, Panels A and B, for the two ATM definitions of 5(0)% is 23,792 (59,569), respectively. G s model is closer to the market price than the BCC (SVJ) model for 15,123 (43,733) of these 23,792 (59,569) ITM matched pairs (about 75%) and BCC is closer on 8,669 (15,836) pairs. Here, Table 7d shows the basis points net improvements from using the G model and being closer than BCC s SVJ model to the market price for the two definitions of ATM are on average 202 bp (298 bp), respectively, for ITM options. Thus, we see that when comparing the basis points improvements of G relative to BS (167 bp or 134 bp) to the basis point improvements of G relative to BCC s SVJ (202 or 298) for ITM index put options, G s pricing improvements are greater relative to BCC, not BS. This is true even though G is closer to the market price for a much greater percentage (90% BS versus 75% BCC) of matched pair comparisons. This is because BCC produces more extreme values than BS. Again, these basis point numbers are calculated as discussed above for Table 7a as if constructing a one of each option portfolio containing one option for each strike price and time to expiration for each day and finding the market value of that one of each option portfolio each day for all days in a year. Once again, the portfolio basis point and dollar value improvements would generally be much larger for professionals who hold more than one of each option in their portfolio, but instead hold all options in multiple amounts based on each dealer s share of total trades The ATM options have the greatest trading volume and eliminating the 5% defined as ATM in order to find the pricing errors of the ITM s and OTM s excludes a large number of options as these Tables show. 43

44 TABLE 7d PUT ITM: CO vs. SVJ PANEL A BASIS POINT IMPROVEMENTS YEAR PV NUMBER NUMBER NUMBER YEAR PV NUMBER NUMBER NUMBER TOTAL SVJ CO SVJ CO BP TOTAL SVJ CO SVJ CO BP , , , , , ,397 1,166 5, , , , , , , ,095 1,647 6,448 1, , , ,002 1,581 1,421 6, , , ,683 2,773 4,910 8, , , ,479 1,273 1,206 4, , , ,212 2,674 4,538 7, , , ,226 1,331 1,895 4, , , ,578 2,327 5,251 6, , , ,780 1,388 2,392 3, , , ,292 2,008 5,284 4, , , ,566 1,668 2,898 4, , , ,503 2,387 6,116 4, , , , , , , , , , , , , TOTAL 3,014, ,792 8,669 15,123 25, , TOTAL 4,643, ,569 15,836 43,733 33, , PANEL B D/E PV NUMBER NUMBER NUMBER D/E PV NUMBER NUMBER NUMBER TOTAL SVJ CO SVJ CO BP TOTAL SVJ CO SVJ CO BP , ,031 1,392 1,639 4, , , ,117 3,009 6,108 7, , , ,294 2,997 4,297 10, , ,533, ,778 5,724 13,054 14, , , ,901 1,680 3,221 4, , , ,810 3,117 9,693 5, , , , ,002 1, , , ,168 1,335 5,833 1, , , , , , , , , , , ,921 1, , , ,295 1,113 4,182 2, , , , ,228 2, , , ,486 1,046 2,440 2, , TOTAL 3,014, ,792 8,669 15,123 25, , TOTAL 4,643, ,569 15,836 43,733 33, , Table 8a presents the same number, dollar improvement, and basis point improvement analysis for OTM put index options during this sample period when ATM is again defined either as 5% or 0% and is depicted both yearly and by D/E ratio. 44

45 TABLE 8a PUT OTM: CO vs. BS PANEL A BASIS POINT IMPROVEMENTS YEAR PV NUMBER NUMBER NUMBER YEAR PV NUMBER NUMBER NUMBER TOTAL BS CO BS CO BP TOTAL BS CO BS CO BP , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , TOTAL 962, , , , TOTAL 1,714, , , , PANEL B D/E PV NUMBER NUMBER NUMBER D/E PV NUMBER NUMBER NUMBER TOTAL BS CO BS CO BP TOTAL BS CO BS CO BP , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , TOTAL 962, , , , TOTAL 1,714, , , , Like Table 7a, Table 8a again shows the total dollar value of options by year or D/E category, the total number of options in that category, the number where BS is closer to the market and the number where the G s model is closer to the market, the dollar value of both BS and G s improvements, and the basis point (bp) advantage of G s compound option model relative to BS. Here, in Table 8a, for the number of OTM matched pairs of options compared with the two ATM definitions, 75,300 (109,301), G s model is closer to the market price than the BS model for 75,052 (108,437) of these matched pairs. Remarkably, G is closer than BS on 99% of all matched pairs of OTM index put options. 45

46 Thus for the two ATM definitions of 5% (0%) this advantage results in G s model now having a larger dollar improvement for OTM options of $55, ($65,009.57) while BS has a smaller dollar improvement for OTM options of $ ($377.56). Thus, the net dollar improvement of G s model is $55, ($64,632.01), and that divided by the total value of each option in this smaller valued OTM portfolio, $962, ($1,714,112.97), produces the 574 (377) basis point improvement. Both the dollar amount and basis point improvements for G compared to BS are larger for the less valuable OTM relative to ITM options. Now, consider Table 8d comparing G and BCC s best SVJ model for OTM options when ATM is defined as 5% (0%). Table 8d, Panels A and B, incorporates the same format as Table 7d. The number of OTM matched pairs of options compared in Table 8d, Panels A and B, for the two ATM definitions of 5% (0%) is 95,186 (139,016), respectively. G s model is closer to the market price than the BCC s best SVJ model for 58,466 (90,230) of these matched pairs (about 65%). So, by comparison we see that G is closer than BS to the OTM put market prices for 99% of the matched pairs, but G is only closer than BCC (SVJ) on about 65% of the matched pairs. However, because BCC produces many extreme values, we next learn that the economic improvement of G relative to BCC is greater than G s basis point improvement relative to BS. This economic improvement of G relative to BCC s SVJ results in a much larger dollar improvement for OTM put options relative to ITM options for the two definitions of ATM of $411, ($524,297.02), respectively, while BCC has a dollar improvement for OTM index puts of $105, ($120,231.61). 46

47 TABLE 8d PUT OTM: CO vs. SVJ PANEL A BASIS POINT IMPROVEMENTS YEAR PV NUMBER NUMBER NUMBER YEAR PV NUMBER NUMBER NUMBER TOTAL SVJ CO SVJ CO BP TOTAL SVJ CO SVJ CO BP , ,833 3,782 6,051 4, , , ,094 6,011 10,083 6, , , ,248 4,893 9,355 8, , , ,396 6,924 14,472 10, , , ,806 6,460 8,346 25, , , ,044 8,514 12,530 29, , , ,608 5,897 7,711 27, , , ,597 7,584 12,013 31, , , ,846 3,143 6,703 9, , , ,325 4,175 10,150 11, , , ,477 3,629 5,848 9, , , ,433 4,538 8,895 10, , , ,424 4,359 6,065 10, , , ,309 5,269 9,040 11, , , ,108 4,156 6,952 8, , , ,945 5,152 10,793 9, , , , , , , , , , TOTAL 970, ,186 36,720 58, , , TOTAL 2,030, ,016 48,786 90, , , PANEL B D/E PV NUMBER NUMBER NUMBER D/E PV NUMBER NUMBER NUMBER TOTAL SVJ CO SVJ CO BP TOTAL SVJ CO SVJ CO BP , ,073 6,597 10,476 27, , , ,904 8,765 16,139 31, , , ,832 12,352 19,480 41, , , ,652 16,269 29,383 47, , , ,199 6,945 11,254 13, , , ,175 10,110 18,065 16, , , ,126 3,703 6,423 7, , , ,210 5,026 10,184 8, , , ,562 1,464 3,098 2, , , ,968 2,017 4,951 3, , , ,071 3,760 5,311 8, , , ,331 4,422 7,909 8, , , ,323 1,899 2,424 3, , , ,776 2,177 3,599 4, , TOTAL 970, ,186 36,720 58, , , TOTAL 2,030, ,016 48,786 90, , , The net dollar improvement of G s compound option model is $306, ($404,065), and that divided by the total value of each option in the OTM portfolios, $970, ($2,030,645.23) produces a 3160 (1990) basis point improvement. Thus, we again see that when comparing the basis points improvements of G relative to BS (574 bp or 377 bp) to the basis point improvements of G relative to BCC s SVJ (3160 bp or 1990 bp) for OTM index put options for the two ATM comparisons, G s pricing improvements are greater relative to BCC, not BS. This is true even though we show G is closer to the market price for a much greater percentage of matched pair comparisons (99% of BS versus 65% BCC comparisons). Again, this occurs because BCC produces more extreme values than BS. 47

48 In this section we have demonstrated the considerable economic improvement of G s model relative to the BCC (SVJ) or BS models for pricing the world s most widely traded equity index options on the S&P 500. We have shown that the data necessary to implement G s model for valuing index options are readily available. This research can be further extended to other contracts involving leverage, such as mortgages and cross currency swaps, and to credit derivatives, such as credit default swaps, credit spread options, risk neutral default probabilities, and new price based measurement of the total market credit risk. Also, G s model has been extended in a variety of ways which make it appropriate for other options which are different than the S&P 500 index options, such as individual stock options, American options, and options with payouts. In this paper we have shown that existing implied market value of aggregate market leverage is both statistically and economically important to pricing the S&P 500 put options. Herein, we stress the importance of separating the economic effects of stochastic leverage and its induced stochastic volatility from any other assumed stochastic effects. Leverage is always present in the market and the implied market value of aggregate leverage has now been shown to be important to pricing equity index options. Thus, if leverage is not properly treated prior to modeling other assumed stochastic effects, then the resultant parameters will be misestimated because of the relevant omitted variable. The next section summarizes and presents our conclusions. 48

49 5. Conclusions This paper demonstrates the Geske no arbitrage, partial equilibrium compound option model which incorporates the implied market value of aggregate debt can be used to price the world s most widely traded equity index options on the S&P 500 using only contemporaneous market price data. The Geske option model characterizes how the market value of aggregate leverage causes the market equity index risk to change stochastically and inversely with the implied market value of aggregate leverage in the index. We believe we are the first to measure and show the importance of the implied market value of debt and leverage on asset prices. We have shown empirically that both the implied market value of the aggregate debt and the time series variations in this implied market value of aggregate leverage is sufficient to produce very significant improvements by Geske s model over models which omit leverage, such as the more complex models of Bakshi, Cao, Chen (SV, SVSI, & SVJ) and the seminal model of Black-Scholes. These improvements are shown to be conclusively and directly related to the implied market value of aggregate debt and leverage. Furthermore, we show why the improvements are greater for options with longer time to expiration because these options are effected by leverage for a longer period. We demonstrate the relative improvement attributable to leverage is both statistically and economically significant for all strikes and all times to expiration. We also demonstrate for comparison that these conclusions are independent of the implementation methodologies required for the more complex models discussed. However, we believe the new implied methodology presented here is superior to the compared alternatives. We reason that if leverage is a relevant variable effecting the underlying equity distribution that is not properly included prior to assuming stochastic characteristics for other variables which also effect this distribution, such as volatility, interest rates, or jumps, then the additional necessary parameters will be estimated with error because leverage is an important omitted variable. 49

50 APPENDIX I In this appendix we compare BS and G to BCC s other versions which do not include jumps (SVJ) and which incorporate stochastic volatility (SV) and stochastic volatility and stochastic interest rates (SVSI). We also use the alternate volatility estimation methodology of finding the volatility that minimizes the sum of squared errors for pricing equity index options on any day. This comparison allows us to show that the G model dominates BCC and BS when the models are implemented with identical methodologies. Furthermore, we lag the volatility estimate by one day in order for the estimate to be out of sample, as in BCC. As mentioned above, this methodology is necessary to implement models such as BCC which assume many other stochastic complexities and require many more option prices in order to estimate their required parameters. In Tables 11a and 11b we compare BS and BCC with these methodological changes and show that the conclusions of the paper do not change. BCC (SVJ) is slightly better than BCC (SV) and much better that BCC (SVSI). BS is slightly inferior to BCC (SVJ) in the number of matched pairs of options that BS is closer to the market price than BCC (SVJ). However, BS is superior to BCC (SVJ) in basis point improvements. This is because BCC has many large errors in pricing some index option matched pairs. BCC (SVJ) is slightly superior to BCC (SV) in basis points of errors. In Tables 12a and 12b we show that G s model is superior to all versions of BCC (SV, SVSI, and SVJ), even when volatility is estimated by minimizing the sum of squared pricing errors lagged one day. Furthermore, these tables also confirm that G is superior to BS using this alternate volatility estimation methodology. In all comparisons, BCC (SVSI) performs the least well of the three BCC versions. 50

51 Tables 11a and 11b BS vs BCC: Using Sigma Min SSE Lagged One Day BS: USING THE LAGGED SIGMAS_BS PER T_EXP TABLE 11a PUT ITM: BS vs. BCC PANEL A: BS vs. SV BASIS POINT IMPROVEMENTS YEAR PV NUMBER NUMBER NUMBER YEAR PV NUMBER NUMBER NUMBER TOTAL SV BS SV BS BP TOTAL SV BS SV BS BP , , , , , ,397 2,414 3,983 2, , , ,639 1,252 1,387 2, , , ,095 3,709 4,386 5, , , ,002 1,972 1,030 11, , , ,683 4,260 3,423 17, , , ,479 1, , , , ,212 4,055 3,157 17, , , ,226 1,492 1,734 5, , , ,578 3,100 4,478 9, , , ,780 1,853 1,927 6, , , ,292 3,232 4,060 9, , , ,566 2,319 2,247 6, , , ,503 3,888 4,615 9, , , ,372 1,012 1,360 2, , , ,971 1,964 4,007 3, , , , , TOTAL 3,014, ,792 12,371 11,421 46, , TOTAL 4,643, ,569 26,728 32,841 75, , PANEL B: BS vs. SVSI YEAR PV NUMBER NUMBER NUMBER YEAR PV NUMBER NUMBER NUMBER TOTAL SVSI BS SVSI BS BP TOTAL SVSI BS SVSI BS BP , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,080 1, , , , , , , , ,640 1, , , , , , , , ,814 1, , , , , , , , , , , , , TOTAL 3,014, ,792 1,074 22,718 1, , TOTAL 4,643, ,569 3,894 55,675 6, ,128, PANEL C: BS vs. SVJ YEAR PV NUMBER NUMBER NUMBER YEAR PV NUMBER NUMBER NUMBER TOTAL SVJ BS SVJ BS BP TOTAL SVJ BS SVJ BS BP , , , , , ,397 2,608 3,789 2, , , ,639 1,195 1,444 2, , , ,095 3,719 4,376 5, , , ,002 1,966 1,036 11, , , ,683 4,331 3,352 17, , , ,479 1, , , , ,212 4,124 3,088 17, , , ,226 1,507 1,719 5, , , ,578 3,207 4,371 9, , , ,780 1,875 1,905 6, , , ,292 3,342 3,950 9, , , ,566 2,360 2,206 6, , , ,503 4,002 4,501 9, , , ,372 1,025 1,347 2, , , ,971 2,006 3,965 3, , , , , TOTAL 3,014, ,792 12,436 11,356 46, , TOTAL 4,643, ,569 27,442 32,127 76, ,

52 TABLE 11b PUT OTM: BS vs. BCC PANEL A: BS vs. SV BASIS POINT IMPROVEMENTS YEAR PV NUMBER NUMBER NUMBER YEAR PV NUMBER NUMBER NUMBER TOTAL SV BS SV BS BP TOTAL SV BS SV BS BP , ,833 3,743 6,090 5, , , ,094 6,012 10,082 7, , , ,248 4,716 9,532 9, , , ,396 6,991 14,405 11, , , ,806 5,966 8,840 27, , , ,044 8,110 12,934 31, , , ,608 5,383 8,225 26, , , ,597 7,334 12,263 31, , , ,846 3,072 6,774 13, , , ,325 4,465 9,860 16, , , ,477 3,415 6,062 11, , , ,433 4,583 8,850 13, , , ,424 4,335 6,089 13, , , ,309 5,499 8,810 15, , , ,108 3,965 7,143 9, , , ,945 4,947 10,998 10, , , , , , , , , , TOTAL 970, ,186 34,961 60, , , TOTAL 2,030, ,016 48,511 90, , , PANEL B: BS vs. SVSI YEAR PV NUMBER NUMBER NUMBER YEAR PV NUMBER NUMBER NUMBER TOTAL SVSI BS SVSI BS BP TOTAL SVSI BS SVSI BS BP , ,833 1,232 8,601 1, , , ,094 1,809 14,285 1, , , ,248 1,780 12,468 3, , , ,396 2,228 19,168 4, , , ,806 3,294 11,512 15, , , ,044 3,835 17,209 16, , , ,608 2,599 11,009 12, , , ,597 3,119 16,478 13, , , ,846 1,242 8,604 4, , , ,325 1,706 12,619 6, , , ,477 1,968 7,509 6, , , ,433 2,464 10,969 7, , , ,424 3,208 7,216 9, , , ,309 3,743 10,566 10, , , ,108 3,328 7,780 7, , , ,945 4,052 11,893 8, , , , , , , , , , TOTAL 970, ,186 19,042 76,144 61, ,915, TOTAL 2,030, ,016 23, ,486 68, ,547, PANEL C: BS vs. SVJ YEAR PV NUMBER NUMBER NUMBER YEAR PV NUMBER NUMBER NUMBER TOTAL SVJ BS SVJ BS BP TOTAL SVJ BS SVJ BS BP , ,833 3,876 5,957 5, , , ,094 6,316 9,778 7, , , ,248 4,946 9,302 9, , , ,396 7,336 14,060 12, , , ,806 6,109 8,697 27, , , ,044 8,400 12,644 32, , , ,608 5,384 8,224 26, , , ,597 7,388 12,209 31, , , ,846 3,186 6,660 13, , , ,325 4,697 9,628 17, , , ,477 3,548 5,929 11, , , ,433 4,833 8,600 14, , , ,424 4,437 5,987 13, , , ,309 5,679 8,630 15, , , ,108 4,023 7,085 9, , , ,945 5,012 10,933 10, , , , , , , , , , TOTAL 970, ,186 35,883 59, , , TOTAL 2,030, ,016 50,229 88, , ,

53 Tables 12a and 12b G vs BCC: Using Sigma Min SSE Lagged One Day G versus BCC: ITM USING THE LAGGED SIGMAV PER T_EXP TABLE 12a PUT ITM: CO vs. BCC PANEL A: CO vs. SV BASIS POINT IMPROVEMENTS YEAR PV NUMBER NUMBER NUMBER YEAR PV NUMBER NUMBER NUMBER TOTAL SV CO SV CO BP TOTAL SV CO SV CO BP , , , , ,397 2,264 4,133 2, , , ,639 1,060 1,579 1, , , ,095 3,402 4,693 4, , , ,002 1,390 1,612 5, , , ,683 3,573 4,110 9, , , ,479 1,014 1,465 3, , , ,212 3,307 3,905 10, , , ,226 1,209 2,017 3, , , ,578 2,734 4,844 6, , , ,780 1,641 2,139 4, , , ,292 2,899 4,393 6, , , ,566 1,760 2,806 3, , , ,503 3,101 5,402 5, , , , ,732 1, , , ,971 1,431 4,540 2, , , , , TOTAL 3,014, ,792 9,447 14,345 23, , TOTAL 4,643, ,569 22,810 36,759 47, , PANEL B: CO vs. SVSI YEAR PV NUMBER NUMBER NUMBER YEAR PV NUMBER NUMBER NUMBER TOTAL SVSI CO SVSI CO BP TOTAL SVSI CO SVSI CO BP , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,690 1, , , , , , , , , , , , , , , , , , , , , TOTAL 3,014, , ,841 1, , TOTAL 4,643, ,569 3,555 56,014 5, ,193, PANEL C: CO vs. SVJ YEAR PV NUMBER NUMBER NUMBER YEAR PV NUMBER NUMBER NUMBER TOTAL SVJ CO SVJ CO BP TOTAL SVJ CO SVJ CO BP , , , , , ,397 2,487 3,910 2, , , ,639 1,030 1,609 1, , , ,095 3,451 4,644 4, , , ,002 1,385 1,617 5, , , ,683 3,663 4,020 10, , , ,479 1,025 1,454 3, , , ,212 3,388 3,824 10, , , ,226 1,191 2,035 3, , , ,578 2,826 4,752 6, , , ,780 1,655 2,125 4, , , ,292 3,019 4,273 6, , , ,566 1,794 2,772 3, , , ,503 3,174 5,329 5, , , , ,700 1, , , ,971 1,495 4,476 2, , , , , TOTAL 3,014, ,792 9,543 14,249 24, , TOTAL 4,643, ,569 23,599 35,970 49, ,

54 G versus BCC: ITM USING THE LAGGED SIGMAV PER T_EXP TABLE 12b PUT OTM: CO vs. BCC PANEL A: CO vs. SV BASIS POINT IMPROVEMENTS YEAR PV NUMBER NUMBER NUMBER YEAR PV NUMBER NUMBER NUMBER TOTAL SV CO SV CO BP TOTAL SV CO SV CO BP , ,833 3,506 6,327 4, , , ,094 5,607 10,487 6, , , ,248 4,177 10,071 6, , , ,396 6,221 15,175 8, , , ,806 5,022 9,784 16, , , ,044 7,011 14,033 19, , , ,608 4,502 9,106 16, , , ,597 6,254 13,343 20, , , ,846 2,604 7,242 8, , , ,325 3,828 10,497 11, , , ,477 2,893 6,584 7, , , ,433 3,916 9,517 9, , , ,424 3,402 7,022 7, , , ,309 4,458 9,851 9, , , ,108 3,455 7,653 6, , , ,945 4,336 11,609 7, , , , , , , , , , TOTAL 970, ,186 29,915 65,271 75, , TOTAL 2,030, ,016 42,179 96,837 92, , PANEL B: CO vs. SVSI YEAR PV NUMBER NUMBER NUMBER YEAR PV NUMBER NUMBER NUMBER TOTAL SVSI CO SVSI CO BP TOTAL SVSI CO SVSI CO BP , ,833 1,122 8,711 1, , , ,094 1,662 14,432 1, , , ,248 1,506 12,742 2, , , ,396 1,917 19,479 3, , , ,806 2,489 12,317 7, , , ,044 2,971 18,073 8, , , ,608 1,916 11,692 6, , , ,597 2,405 17,192 7, , , , ,880 2, , , ,325 1,393 12,932 3, , , ,477 1,593 7,884 3, , , ,433 2,039 11,394 4, , , ,424 2,418 8,006 5, , , ,309 2,897 11,412 6, , , ,108 2,806 8,302 4, , , ,945 3,474 12,471 5, , , , , , , , , , TOTAL 970, ,186 15,196 79,990 34, ,959, TOTAL 2,030, ,016 19, ,704 40, ,600, PANEL C: CO vs. SVJ YEAR PV NUMBER NUMBER NUMBER YEAR PV NUMBER NUMBER NUMBER TOTAL SVJ CO SVJ CO BP TOTAL SVJ CO SVJ CO BP , ,833 3,675 6,158 4, , , ,094 5,918 10,176 6, , , ,248 4,438 9,810 7, , , ,396 6,604 14,792 9, , , ,806 5,175 9,631 16, , , ,044 7,266 13,778 20, , , ,608 4,526 9,082 16, , , ,597 6,343 13,254 20, , , ,846 2,736 7,110 8, , , ,325 4,042 10,283 11, , , ,477 3,033 6,444 8, , , ,433 4,161 9,272 9, , , ,424 3,502 6,922 8, , , ,309 4,638 9,671 9, , , ,108 3,533 7,575 6, , , ,945 4,430 11,515 7, , , , , , , , , , TOTAL 970, ,186 30,978 64,208 77, , TOTAL 2,030, ,016 43,951 95,065 94, ,

55 APPENDIX II The equation below from BCC, p. 210, # 7, describes the dynamics for all three BCC embedded models SV, SVSI, and SVJ subject to the relevant parameters and boundary condition for a put or call option. The table below from BCC, p. 218, Table 3, shows a parameterization for all three BCC models. 55