A Volatility Smirk that Defaults: The Case of the S&P 500 Index Options

Transcription

1 A Volatility Smirk that Defaults: The Case of the S&P 500 Index Options Andreou C. Panayiotis (Lecturer in Finance) Durham University Durham Business School, Mill Hill Lane, DH1 3LB, Durham, UK, First Version: June 19 th, 2009 Abstract Modern financial engineering has dedicated significant effort in developing sophisticated option pricing models to replicate the implied volatility smirk anomaly. Nonetheless, there is limited empirical evidence to examine the causes of this anomaly implied by market options data. The primary purpose of this study is to investigate the time-series economic determinants that affect the shape of the S&P 500 index Implied Volatility Function (IVF). The analysis is carried out on a daily basis and covers the period from January 1998 to December One of the most important contributions of this study is to investigate how the market default risk affects the shape of the risk-neutral density function implied by the S&P 500 index options. In order to create the proxy for the market default risk, I compute the daily probability-to-default measure for all individual, non-financial, firms included in the S&P 500 index. The daily probability-to-default is calculated with the distance-to-default measure, which is based on s (1974) option pricing model. Part of my analysis includes discussions of the different versions of the default risk that I compute and compare with some key results reported in Bharath and Shumway (2008). My analysis shows that the market default risk has a dual role to play, since it can potentially capture both, the market leverage effect, as well as, the market s perceptions about the future growth/state of the economy. As such, market default risk has been found to affect the shape of the S&P 500 index IVF in numerous ways. The results suggest that, besides options pricing models that admit stochastic volatility and random jumps, it is also worthwhile to exploit models that take into account market leverage such as the ones of Geske (1979) and Toft and Prucyk (1997). More importantly, in a regression analysis where I disentangle the role of market leverage by separating it from the asset return, I show that the contemporaneous index return is still important in explaining the shape of the S&P 500 index IVF. The results of this study illuminate a set of economic determinants that are found to affect the risk-neutral density function of the index. These factors are related to characteristics of the underlying asset and micro-structure variables characterizing the option market itself. Financial engineers can exploit the role and importance of these factors in the future, in order to improve the forecasting accuracy, as well as, the hedging and risk management performance of option pricing models. JEL classification: G12, G13, G14 Keywords: default risk, implied volatility smirk, deterministic volatility functions, determinants, trading volume. Electronic copy available at:

2 1. Introduction Fischer Black and Myron Scholes introduced in 1973 their milestone parametric option pricing model that nowadays is known as the Black-Scholes (BS) formula. This model was a breakthrough in the pricing of options and still has a tremendous influence on the way that practitioners price various derivative securities. Despite its long lasting endurance in time, the BS formula is built on a set of simplistic assumptions. As such, it assumes that the asset price follows a geometric Brownian motion with constant volatility, thus, all options on the same asset should provide the same implied volatility. Many researchers have documented systematic biases (i.e. Rubinstein, 1994, Bakshi et al., 1997, Bates, 1996 and 2000) whereas, the BS implied volatilities for the S&P 500 index options tend to vary across option moneyness and time-to-maturity. This repudiates the principle that, under the assumptions of the BS model of a lognormal density, the index s option Implied Volatility Function (IVF) should be flat and constant through time. The S&P 500 index IVF forms a smile pattern prior to the October 1987 market crash, where out-of-the-money and in-the-money implied volatilities are higher compared to at-the-money ones. According to Rubinstein (1994, see also Dumas et al., 1998), after the crash a smirk pattern is most prominent where implied volatilities decrease monotonically as the option strike price rises relative to the index level (this stylized fact is also apparent in the dataset that I use herein). Moreover, implied volatilities (especially those of at-the-money options) also present a term structure pattern that again cannot be rationalized by the BS model. The modern financial engineering research has invented more elaborated option pricing models to mitigate the market biases associated with the BS model. For example, recent developments in this area relate to option pricing models that admit stochastic volatility (SV) or stochastic volatility and jump (SVJ) risk factors (e.g. Heston, 1993, Bates, 1991, 1996 and 2000, Bakshi et al., 1997). SV models are not capable of generating high levels of skewness and kurtosis at short maturities under reasonable parameterizations. The addition of a jump component enhances the distributional flexibility of the model, since it can now internalize additional levels of negative skewness and excess kurtosis. This allows the SVJ model to explain better the heavy-tailedness of the asset return distribution and to generate more realistic implied volatility smirk patterns, especially, at short maturities. Nevertheless, 1 Electronic copy available at:

3 Bakshi et al. (1997) point out, that SV and SVJ models are still misspecified, since their implied parameters (after being transformed from the risk-neutral to the real measure) are statistically inconsistent with those implicit by the time-series of the S&P 500 returns. In addition, for a given time-to-maturity, the modern theoretic parametric models cannot always generate implied volatility smirks, as sharp, as those that are typically observed in practise (see Das and Sundaram, 1999, Dumas et a., 1998, Pan, 2002, Backus et al., 2004). Based on Das and Sundaram (1999), it is even less obvious whether the theoretical predictions of such option pricing models are or can be made consistent with the observed term structure of the IVF. Another way to mitigate the implied volatility smirk anomaly is to employ interpolative techniques to model the local structure of the IVF; this is equivalent as of backing out the implied risk neutral density function from market option prices. As such, the implied binomial tree method of Dupire (1994), Derman and Kani (1994) and Rubinstein (1994) is so flexible that, in-sample, is capable of achieving a perfect match when fitted to option market prices across moneyness and timeto-maturity. Dumas et al. (1998) estimate regression-based Deterministic Volatility Function (DVF) specifications of quadratic forms that provide unique per-contract volatility estimates (in contrast to the overall average volatility estimates of Whaley, 1982), and examine how well they predict option prices out-of-sample. They find that this regression-based approach 1 produces significantly more accurate option prices and hedging parameters compared to the implied volatility trees. Bollen and Whaley (2004) find that the slope of the daily S&P 500 index IVF is erratic across time, which is able to explain the poor performance of the implied volatility trees approach when used for pricing and hedging options out-of-sample. Andreou et al. (2008a) extend the regression-based DVF models of Dumas et al. (1998) by developing a semi-parametric option pricing model able to produce Generalized Parameter Functions for the S&P 500 index IVF. Their semi-parametric method provides an enhancement of parameters beyond volatility (like skewness and kurtosis) without specifying a-priori a deterministic parametric functional form like in the case of DVF. This method results in excellent option pricing performance, 1 Option pricing models that require asset return volatility to be a deterministic function of the underlying asset price and time are appealing because they offer a practical approach to option pricing under stochastic volatility (see for instance Rosenberg, 2000). 2 Electronic copy available at:

4 which outperforms the regression-based DVF parametric counterparts in pricing and hedging the S&P 500 index options out-of-sample. It is also competitive in performance to models that admit stochastic volatility and random jumps. The superiority of this method comes from the fact that it captures the salient features embedded in the options data, since it does not impose any restrictive shapes for the higher moments across time. Andreou et al. (2008a) report that their semi-parametric models predict a smirk effect for short and medium term options, and also, skewness that is increasing in moneyness and decreasing in maturity, and kurtosis that is hump shaped in moneyness. Such results hint to serious misspecification of the stochastic processes that govern the dynamics of the parametric option pricing models 2. The former indicates that the discrepancy we observe between risk-adjusted option implied parameters and the estimated parameters from the time-series (see Bakshi et al., 1997, Pan, 2000) can be also attributed to micro-structure characteristics of the option markets, which are not related to the distributional characteristics of the underlying asset returns. From the previous discussions, it is obvious that the shape of the IVF 3 is significantly affected by economic variables not included in the existing and widely used theoretical (parametric) options pricing models. For instance, Peña et al. (1999) report evidence that variables that are related to liquidity and trading costs can explain the pronounced shape of the Spanish IBEX-35 index IVF. Bollen and Whaley (2004) report that changes in the shape of the S&P 500 index IVF are directly related to net buying pressure (computed from options trading volume) coming from public order flow. 2 Moreover, Das and Sundaram (1999) examine whether option pricing models that admit stochastic volatility or jumps can fit the data well by capturing the level of skewness and kurtosis implied by the data for all maturities. They state that (compared with empirical observations) jump models allow too rapid decay in skewness and kurtosis, and stochastic volatility models exhibit a hump shape overly pronounced for intermediate maturities. 3 It is well established in the literature that the shape of the IVF is directly linked to the shape of the risk-neutral density implied by option market prices. Shimko (1993) was the first one to show how to recover the riskneutral probability density function by fitting a smooth curve to the implied volatility smirk and by exploiting the idea of Breeden and Litzenberger (1978), that the second partial derivative of the European option with respect to strike price is proportional to the risk-neutral density of the asset s returns. Moreover, Bakshi et al. (2003) and Backus et al. (2004) have shown that the IVF is directly linked to the risk-neutral skewness and kurtosis of the implied distribution. Zhang and Xiang (2008) derive analytical formulas that allow them to recover the risk-neutral probability distribution from a smirk implied volatility function. Dennis and Mayhew (2002) find that there is one-to-one mapping between the risk neutral density function and the implied volatility curve, whereas negatively sloped volatility curves correspond to negative skewness in the risk-neutral density. Bakshi et al. (2003) verify a high correlation between their measure of the risk-neutral skewness and the slope of the implied volatility curve. Thus, it is obvious that investigating the shape of the daily S&P 500 index IVF is tantamount to investigating the shape of its option implied risk-neutral probability density function. 3

5 I model the S&P 500 index IVF using the regression-based DVF approach of Dumas et al. (1998). This method allows me to simultaneously model the slope and convexity of the IVF across option moneyness and time-to-maturity. Such treatment of the IVF is more elaborated compared with studies that ignore the time dimension and investigate the slope and/or the convexity of the IVF across moneyness only (e.g. see Peña et al., 1999, Dennis and Maynew, 2002, Bollen and Whaley, 2004, Chang et al., 2009, etc). Moreover, many studies in this area model the IVF using OLS estimation (e.g. Ncube, 1996, Peña et al., 1999, Chang et al., 2009). Yet, Christoffersen and Jacobs (2004) demonstrate that OLS estimates yield biased option pricing results and that a Nonlinear Least Squares pricing loss function should be used instead. For this reason, I estimate the regression-based DVF model using Nonlinear Least Squares - where the loss function is the price difference between the BS and the option market-price. It is well established in the literature that leverage, or equivalently default risk, affects the market prices of options (for instance, see Geske, 1979). There are also other theoretical proposals to support that the presence of leverage can give birth to a monotonically downward IVF, whereas the steepness of the smile depends on the level of leverage (see Toft and Prucyk, 1997). Thus, part of the empirical anomaly related to the implied volatility smirk can be attributed to default risk as well. For example, Geske and Zhou (2007) show that the leverage effect causes the index volatility to be both stochastic and inversely related to the level of the index. As a result, the index s implied returns distribution will have a fatter left tail and a thinner right tail than the BS assumption of a normal return distribution. Effectively, the former translates to a downward sloping volatility smile like the one we empirically observe for the S&P 500 index options. Figlewski and Wang (2000) find that the size of the leverage effect is distinctly greater for the S&P 100 index than for single stock options. Despite, the extant empirical evidence looking into the intimate relationship of default risk and implied volatility smirk is extremely limited and concentrated mainly on equity options 4. Moreover, such type of research for the S&P 500 index options is rather unexploited and merits further analysis. This study addresses this gap by investigating the time-series economic determinants that affect the shape of the S&P 500 index IVF. The analysis is carried out on a daily basis and covers the period from January 4 For instance, Walter (2007) manages to locate only a handful of studies in this area, with most of them focusing on the relationship between equity default risk and the (stock option) slope at the at-the-money point. 4

6 1998 to December More importantly, it assesses how one of the economics determinants considered, namely the market default risk, affects the shape of the risk-neutral density function implied by the S&P 500 index options 5. In order to create the proxy for the market default risk, I compute the daily probability-todefault for all individual, non-financial, firms included in the S&P 500 index. I employ the distance-to-default (thereinafter DD) measure as employed by Bharath and Shumway (2008) 6 to forecast corporate default values. As Bharath and Shumway (2008) assert, this is an innovative default probability forecasting model, which has been widely applied in both academic research and practice. The DD model combines the framework of s (1974) bond pricing model and, in conjunction with the contingent claims methodology of Black and Scholes (1973), it treats the equity as a call option on the firm s total assets with a strike price equal to the face value of the firm s debt. Given the value of the volatility of the firm s equity (a known quantity, readily available from the market), a procedure is applied to solve a system of two nonlinear equations in order to retrieve the value of the firm s total assets and volatility (both are unknown quantities). The distance-todefault measure captures the difference between the market-value of the firm s assets and the face value of its debt, scaled by the volatility of the firm s assets value that is measured to reflect the time horizon of the required forecast. Subsequently, the DD probability-to-default (i.e. default risk) for a particular firm (namely, π ) is calculated by plugging the total value of the firm s assets and the firm s volatility along with other observables into a z-score functional form 7. All in all, the DD model allows me to measure the value of leverage in market terms, and by using it, I gain a significant advantage compared to previous studies that measure leverage using book-values of debt retrieved from Compustat (e.g. Toft and Prucyk, 1997, Figlewski and Wang, 2000, Dennis and Mayhey, 2002, etc). 5 The S&P 500 index options are European styled and allow me to examine the effect of market default risk without having to control for any early exercise premium. 6 In the later sections, for compatibility purposes, I follow most of the nomenclature used by these authors. 7 This particular application of the (1974) model was developed by the proprietors of the KMV Corporation which was acquired by the Moody s in The model as applied in Bharath and Shumway (2008) differs in several potentially important aspects than that of Moddy s KMV, albeit they have found high correlations between the default probabilities produced by the DD model and the ones produced by the Moody s KMV model. The differences between the two alternative models are discussed in section 2.2 of Bharath and Shumway (2008). 5

7 I estimate the individual firm s default risk ( π Specifically, I consider the case when the expected return on the firm s asset ( ) using three different expected returns. ) is equal to: i) the firm s stock returns over the previous year ( r ), ii) the prevailing risk free rate ( r ), and iii) a return r G E ( ) which is given implicitly by the hedge parameters (i.e. Greek letters) that can be calculated as part of the solution of the DD model. Bharath and Shumway (2008) have considered only the μ V first two cases ( r and E r ), while to the best of my knowledge there is lack of empirical evidence regarding the third case ( ). Moreover, for comparison purposes with the results of Bharath and Shumway (2008), I also estimate their naïve alternative default probability measure ( default risk measure, mimics the functional form of ( r G π π naive ) but is much simpler to calculate. ). This Overall, I find that the market default risk is a primary economic determinant of the S&P 500 index IVF. Figlewski and Wang (2000) by using monthly returns data and quarterly book-values of debt, find a strong leverage effect for the S&P 100 index options but only in down markets. In contrast to them, I find a strong leverage effect even when the S&P 500 index is rising. Moreover, I find that the market default risk affects the shape of the IVF in other ways as well. There can be two explanations for this to happen. First, market default risk obtained via the DD model is a much better and a forward looking proxy of the leverage effect compared with the face value of the debt that Figlewski and Wang (2000) use 8. Moreover, the use of market default risk allows me to capture the leverage effect on a daily basis, while Figlewski and Wang (2000) investigate the leverage effect by measuring the change in volatility that results from the quarterly change in book-values of leverage. Second, my results suggest the existence of another important role between the contemporaneous index return and the shape of IVF. Specifically, the contemporaneous index return is still highly 8 Figlewski and Wang (2000) recognise this drawback by saying that: one problem with this specification, shared with previous articles on this subject, is that leverage should be measured using market-values for firm securities, but only the book-value of debt is available from Compustat. As a partial remedy to this, they construct a monthly series of leverage by assigning changes in outstanding bonds to the final month of the quarter in which they occur. Other studies that investigate the leverage effect rely also on the crucial assumption that book-value of debt is an adequate proxy for market-values (e.g. Toft and Prucyk, 1997, Dennis and Mayhey, 2002). 6

8 significant in explaining the S&P 500 index IVF even after I remove from it the effect of leverage 9. Thus, some of the inconsistencies that Figlewski and Wang (2000) find for the role of leverage might be explained by the fact, that the contemporaneous index return might not be the best proxy for market leverage in all time instances. Consistent with Figlewski and Wang (2000) though, my regression results reveal that the leverage effect cannot be the sole explanation for the implied volatility skew anomaly. Specifically, variables that are related to uncertainty, trading activity and direction of the underlying asset s future return, seem to be key determinants of the shape of the implied volatility function as well. In the following, I briefly discuss previous literature that is related to this topic. Subsequently, I review the DD model, I explain how I compute the market default risk and I also compare and contrast the different versions I consider with some key results reported in Bharath and Shumway (2008). Then, I discuss the methodology used in order to model the daily S&P 500 index IVF. Following that, I review the different datasets and I discuss the results. Finally, I conclude. 2. Relation to previous studies There is only a limited number of studies to investigate the link between the equity volatility smile and the default risk, like the ones of Toft and Prucyk (1997) and Dennis and Meyhew (2002). For instance, Dennis and Meyhew (2002) test whether leverage, firm size beta, trading volume and/or the put/call volume ratio can explain the cross-section variation in the equity risk-neutral skew. To the best of my knowledge, this is the first study to investigate whether market default risk acts as an economic determinant of the S&P 500 index IVF (or equivalently of the shape of the riskneutral index return density). Geske and Zhou (2007) is the only other study to elaborate on pertinent issues 10. Specifically, Geske and Zhou (2007) examine whether existing market leverage has an effect 9 The index return net of the leverage effect, continues to be negatively related to the level of the at-the-money implied volatility regardless of the period, but its relationship with the slope of the IVF swaps sign conditional on the market momentum (i.e. bullish or bearish). Moreover, the relation between the sign and magnitude of the market default risk with the slope of the IVF is conditional on the market momentum as well. 10 Bollen and Whaley (2004) focus their attention on how changes in the implied volatilities of the S&P 500 index options are related to the net buying pressure (an option volume related variable) that arises from public order flow. In the regressions, they also include as control variables, the contemporaneous index return, the trading volume of the index and one-day lagged change in the implied volatility. I also control for such variables in the regression analysis I consider later. 7

9 on the pricing of S&P 500 index put options. They separate the extent of the existing leverage effect in S&P 500 index options by measuring and using market leverage of the index that is retrieved using the Geske s (1979) model. My paper differs significantly from theirs with distinct contributions to the literature. First, I compute daily, the market-value of assets and the market-value of the leverage using a firm-by-firm approach; thus, depending on the year, I estimate daily around 400 firm default risk measures, which I aggregate in order to proxy the market default risk level. On the contrary, Geske and Zhou (2007) estimate only one market value of their variables per day, by summing the observations for all of the firms included in the S&P 500 index. I believe that my approach is better since I exclude all financial firms from my calculations because the debt of such firms has a totally different meaning 11. As a result, my proxy for the market default risk should be measured with higher accuracy. In addition, my approach enables me to observe daily the probability-to-default distribution across all firms included in my sample. On the contrary, Geske and Zhou (2007) have only a single point estimate of market-value of leverage. Nonetheless, by investigating the characteristics of the market default risk distribution we might reveal other important dimensions on the way that agents price and trade options in the market. Second, Geske and Zhou (2007), in order to estimate the market-values of their variables, use concurrently data from the underlying and the option market. In such cases, the validity of the empirical results is based on the premise that the option market is fully integrated with the spot market, sharing the same price dynamics and the same market prices of risk (see Pan, 2002) 12. One example that invalidates this premise happens when the option s market is somehow segmented from the spot market, because of some option-specific factors, such as liquidity. Bollen and Whaley (2004) find that the volatility smirk can be attributed to the buying pressure (excess demand) of specific option series and the limited ability of arbitrageurs to bring prices back to alignment (see also discussions in Peña et al., 1999, and Chang et al., 2009). I also find that options trading activity is a 11 It is usual to exclude financial firms because high leverage that is normal for these firms probably does not have the same meaning as for non-financial firms, where high leverage more likely indicates financial distress (see Fama and French, 1992). 12 As noted by Das and Sundaram (1999), Chernov and Ghysels (2000) and Christoffersen et al. (2009) for the purpose of option valuation, parameters estimated from option prices are preferable to parameters estimated from the underlying returns. 8

10 significant economic determinant of the S&P 500 index IVF. Unlike Geske and Zhou (2007), my purpose in this study is to see whether the market default risk is an economic determinant of the S&P 500 index IVF. In my case though, I do not have to bear the restricting assumption that the option market is fully integrated with the spot market because the individual firm s probability-to-default measures are estimated using data of the underlying market only. Third, following Campbell et al. (2008), I align each firm s fiscal year appropriately with the calendar year, and then I lag accounting data by two months. This ensures that all quarterly accounting data needed for the DD model are publicly available before each estimation period, and limits significantly any look-ahead bias in the construction of the market default risk measure. On the contrary, Geske and Zhou (2007) do not allow for such lag period when they use quarterly balance sheet information collected from the S&P s Compustat database. Finally, the main purpose of Geske and Zhou (2007) is to investigate the out-of-sample pricing performance of the Geske s leverage-based option pricing model. In other words, they want to see how stochastic characteristics for volatility, jumps and interest rates are estimated better after accounting for leverage. In contrast, my primary purpose is to identify the set of economic determinants that affect the shape of the S&P 500 index IVF. At the end, the findings of my study offer an empirical justification for the results reported by Geske and Zhou (2007). Moreover, my results offer an empirical motivation that, besides options pricing models that admit stochastic volatility and random jumps, it is also worthwhile to exploit models that take into account market leverage such as the ones of Geske (1979) and Toft and Prucyk (1997). 3. The Default Risk Model and Methodology for Modeling the S&P 500 Index IVF Below, I first describe the DD model and the different default risk measures that I consider. Following that, I rationalize the use of DD model for meeting the needs of this study, and I describe the methodology used to model daily the S&P 500 index IVF. 9

11 3.1 The DD Model In corporate finance, equity holders are considered to be residual claimants on the firm s assets after all other obligations have been met. Therefore, equity can be considered to be a call option on the firm s total assets, whereas equity holders will exercise only when the underlying value of the firm s assets is greater than the book-value (i.e. face value) of the firm s liabilities. The market-value of the firm is the sum of the market-values of the firm s debt and equity. The market-value of equity can be easily computed by multiplying the firm s market-value of its stock times the number of outstanding shares, whilst only the book-value of the debt is readily available 13. As explained by Bharath and Shumway (2008), in such a case, the DD model estimates the market-value of the debt by applying the classic (1974) bond pricing model. This model posits that the dynamics of the equity value of the company can be described by a geometric Brownian motion where E is the value of the firm s equity, de = μ E Edt + σ EdW, (1) E μ E is the expected continuously compounded return on E (i.e. the instantaneous drift), σ is the instantaneous volatility of firm s equity values, and dw is a standard Gauss-Wiener process. (1974) shows that the dynamics of the total value of a firm can also be described by a geometric Brownian motion E dv = μ V Vdt + σ VdW, (2) V where V is the total value of the firm s assets, μ V is the expected continuously compounded return on V (i.e. the instantaneous drift), and σ is the instantaneous volatility of firm value. V The model assumes that the capital structure of the firm includes: i) the residual claim, equity, and ii) a single, homogeneous class of debt (i.e. discount bond) that pays of the total amount to the bondholders on a specified calendar date, T, in the future. Under the classic (1974) model, the market-value of equity, E, can be viewed as a call option on the total value of the assets, V (i.e. the underlying asset), with exercise price equal to the face value of the debt, F, and time-to-maturity 13 In addition, the firm s total value volatility is not observable and should be estimated somehow from other quantities. 10

12 T. Therefore, the market-value of the equity can be written at any point in time as a function of the value of the firm and time (i.e. E = f ( V, T ) ), thus, it can be described by the Black and Scholes (1973) formula for call options, as follows, where Fe rt E = VΝ(d) Ν( d σ T ), (3) ln( V / F) + ( r + 0.5σV ) T d = σ T V 2, (4) with r to be the instantaneous risk-free rate, σ to be the volatility of the firm value, and Ν (.) to be V the cumulative standard normal distribution. The volatility of the firm s total asset is not readily available from the market, but it can be retrieved by an explicit functional relationship that links it with other observables. To achieve this, the model explores the fact that E = f ( V, T ), and uses Ito s Lemma to derive another expression for the dynamics of the equity, given by: E E E E de = 0.5σVV + μ V dt V dw 2 V + + σv V V t. (5) V By comparing the diffusion terms of Eq. (1) and Eq. (5), it follows directly that the volatility of the equity returns,, equals: σ E V E σ E = σv. (6) E V By noticing also that E / V = Ν( d) is the delta hedge parameter that is computed by Eq. (3), the resulting relationship becomes: V σ E = Ν( d) σv, (7) E where d is given by Eq. (4). Eq. (7) reflects the fact that because of leverage, the volatility of an option (i.e. the equity in this case) is always greater than or equal to the volatility of the underlying asset (i.e. the market-value of the total assets). Moreover, Eq. (7) can offer an explanation to the stylized negative correlation between the index price and its volatility; that is, when the equity index level drops, the equity index volatility rises and vice versa. 11

13 It is obvious that E, σ E, and r can be observed from the financial markets, F and T can be observed from the financial statements of the firm, whilst V and σ V should be inferred using the DD model, since neither of them is directly observable. Nonetheless, it is easy to construct measures of these variables by solving Eqs. (3) and (7) simultaneously. Once this numerical solution is obtained for V and σ V, the distance-to-default value at each time instance is: DD t ln( V / F) + ( μ = σ V V 0.5σ T 2 V ) T. (8) The resulting probability-to-default value is computed using the normal cumulative distribution, which is the distribution implied by the (1974) model 14, and is given as follows: π = Ν DD ). (9) ( t Vassalou and Xing (2004) and Bharath and Shumway (2008) follow a complicated iterative procedure that uses historical returns data in order to find the implicit values 15 of V and σ V. In this paper, I solve Eqs. (3) and (7) simultaneously via a numerical nonlinear root finding algorithm. I do this for three reasons. First, this estimation scheme is straightforward and it is considered sufficient approach for the given problem; for example, it has been employed recently by Campbell et al. (2008), Agarwal and Taffler (2008), and Hillegeist et al. (2004) among others. Second, Vassalou and Xing (2004) and Bharath and Shumway (2008) apply their approach on a monthly basis, which is much less timeconsuming from applying it on a daily basis like in my case. Third, and more importantly, the empirical results of Bharath and Shumway (2008) support that the simultaneously solved π has actually better out-of-sample predicting performance than the one that is solved iteratively. 14 As Vassalou and Xing (2004, pg. 837) explain, strictly speaking, π is not a default probability because it does not correspond to the true probability-of-default in large samples. In contrast, the default probabilities calculated by Moody s KMV are indeed default probabilities because they are calculated using the empirical distribution of distance-to-default values. As pointed out by Bharath and Shumway (2008), albeit the empirical distribution of distances-to-default is an important input to default probabilities, it is not required for ranking firms by their relative probability. 15 First, they use daily data from the past 12 months to obtain an estimate of the equity volatility σ E, which is then used to initialize σ V and they use this value in Eq. (3) to infer the market-value of each firm s assets every day. They then calculate the implied log-return on assets each day and use that returns series to generate new estimates for σ and using a numerical nonlinear root finding algorithm. They iterate on in this manner V μ V σv until it converges, so the absolute difference in adjacent values is less than a small threshold. 12

14 3.2. The Naïve DD Model Bharath and Shumway (2008) have proposed a naïve estimator of the distance-to-default measure that requires the same basic inputs and resembles the functional form and structure of the original DD model. Its intriguing feature is that is simple enough to implement, since it does not require solving any equations or estimating any difficult quantities. Bharath and Shumway (2008) approximate the total volatility of the firm for their naïve estimator as follows: E F naive σv = σ E + naive σ D, (10) E + F E + F where naive σ D is their approximation for the volatility of each firm s debt, given by: naive σ = (11) D σ E They include the five percentage points to represent the term structure of volatility, and they include 25% times equity volatility to allow for volatility associated with default risk. Moreover, they set the expected return on the firm s assets equal to the firm s stock return over the previous year, naive μ =, (12) V r E which allows them to capture part of the same information captured by the DD model when estimated with the iterative procedure conditional on an entire year of equity return data. They define the naïve distance-to-default value as: naive DD t ln[( E + F) / F] + ( re 0.5 naiveσ = naiveσ T V 2 V ) T, (13) and the naïve probability-to-default as: π = Ν(-naive DD ). (14) naive Bharath and Shumway (2008) show empirically that the naïve estimator captures approximately the same quantity of information as the DD probability that is computed with the complicated iterative approach using historical equity returns (like the one used in Vassalou and Xing, 2004). t 3.3. Alternative market-based probability-to-default estimators Before moving into the main analysis, I would like to gauge the relation between the different versions of the probability-to-default estimators. It is very interesting to see whether the empirical 13

15 relationships of the estimators reported before in Bharath and Shumway (2008) hold in my data as well. I consider three alternative cases where Eqs. (3) and (7) are solved simultaneously 16, but every time a different expected return on the firm s total assets ( μ ) is used to compute the default risk value given by Eq. (9). The first two have been considered by Bharath and Shumway (2008). The first V predictor is r as μv = E π, where the expected return on the firm s sets is equal to the firm s stock return μ V = r over the previous year ( r ). The second estimator is π, where the prevailing risk-free rate ( r ) E μ = V is used as the expected return on the firm s assets. The third esti mator denoted by r G π, explores the fact that the expected return on the firm s assets can be expressed as a function of hedge parameters (i.e. Greek letters) computed via Eq. (3). Specifically, by comparing the drift terms of Eqs. (1) and (5), we have that and by rearranging terms, μ E E E E E = 0.5σVV + μvv + V 2, V t r G 2 2 E E E μv = ( μ E E 0.5σVV ) / V 2 V t V 2, (15) where the hedge parameters are given as follows (see Hull, 2008): delta : E V = Ν( d), (16) 2 E N' ( d) gamma : = 2 V V T σ V, (17) E theta : t V Ν' ( d) σ = 2 T V rfe rt Ν( d σ V T ), (18) with Ν '( d) to denote the density function for the standard normal distribution. This estimator has not been considered by prior studies such as the ones of Bharath and Shumway (2008), Campbell et al. (2008) and Vassalou and Xing (2004). 16 For the simultaneous estimation, the starting value for V is set to E + F, whilst for is set to σ E ( E / E + F). 14 σ V

16 After estimating daily the probability-to-default measures for each one of the firms in my sample, I create the alternative measures of market default risk by aggregating the individual firmspecific default risks. For example, the market default risk on trading day t is computed by aggregating the firm-specific default risks μv = re π as follows: Market default risk: n t 1 V re V re Π μ = ( t) = π μ = ( i, t), (19) n t i= 1 where V r n denotes the number of firms in trading day t, and π E ( i, t) t μ = represents the default risk computed for firm i on trading day t using the DD model. The market default risk measures using the rest probability-to-default estimators are estimated accordingly in a similar manner The DD model as a proxy for default risk Bharath and Shumway (2008) conclude that the DD model is useful method for modeling the firm s default risk, albeit not a totally sufficient statistic for accurately forecasting the (exact) probability-to-default value. In a similar view, Campbell et al. (2008) assess that an econometric approach that allows volatility and leverage to enter into the model with free coefficients is a better alternative compared to the DD model. Nevertheless, both studies attest that the DD model captures important aspects of the process that determines the corporate failure and that it can provide useful guidance for building default forecast models. My goal in this paper is not to predict corporate failures as in the studies of Campbell et al. (2008), Bharath and Shumway (2008) and Hillegeist et al. (2004). I rather seek to investigate whether the market default risk is an economic determinant of option related variables and more specifically, whether it can explain the daily shape of the S&P 500 index IVF. In this context, I presume that the values computed via the DD model is an adequate proxy for the market default risk, for several reasons. 15

17 First, the DD model is estimated using market variables, which by virtue are forwardlooking and reflect investor s expectations about future. This is most relevant for my analysis, since information embedded in option data is forwarding looking as well. Second, because the model employs market-based variables, it is able to capture timely information about default risk faster than traditional agency rating models 17 and econometric approaches that rely on accounting ratio-based data 18 (see Bharath and Shumway, 2008, and Agarwal and Taffler, 2008). Evidently, the DD model gives me a significant advantage over previous studies that measure leverage using book-values of debt retrieved from Compustat (e.g. Toft and Prucyk, 1997, Figlewski and Wang, 2000, Dennis and Mayhey, 2002, etc). Third, Agarwal and Taffler (2008) conclude that in terms of predicting accuracy, there is little difference between the market-based and accounting-based models. Despite this, the DD model preserves two significant features over traditional econometric models that are developed using accounting ratio-based variables (such the ones of Altman, 1968, and Ohlson, 1980), or more recent leading econometric alternatives that make use of market-based variables as well. One significant feature is that the DD model can produce the default risk values on a daily basis, for every firm in my sample and at any given point in time. Such capability coincides with the needs of my analysis. In contrast, recent leading alternative estimators, such as the dynamic logit model of Campbell et al. (2008) that makes use of both accounting and market data, is shown to outperform the DD when the models are estimated at a monthly frequency. The other significant feature is that the DD model is neither a time nor a sample specific estimator, since it can be estimated independently for any firm. In contrast, the default risk forecasting performance of econometric approaches is likely to be sample specific. As argued by Hillegeist et al. (2004) and Agarwal and Taffler (2008), econometric estimators are typically developed by searching through a large sample of 17 Most of the times, bond-ratings changes take place with a significant delay. Thus, the rating change can be a poor estimator of the probability-to-default measure (see Vassalou and Xing, 2004). 18 Vassalou and Xing (2004) conjecture that there are several concerns about the use of accounting models in estimating the default risk of equities, since they use information derived from financial statements. As they say, such information is inherently backward looking, since financial statements aim to report a firm s past performance, rather than its future prospects. Agarwal and Taffler (2008) criticize the validity of accountingbased models because accounting numbers are subject to manipulation by the management and in addition, conservatism and historical cost accounting imply that the true asset values may be very different from the bookvalues. Hillegeist et al. (2004) argue that because financial statements are formulated under the going-concern principle, their ability of assessing the individual s firm default risk will be limited by design. 16

18 accounting and/or market ratios with the ratio weightings estimated on a large sample of failed and non-failed firms (traded in many different stock exchanges like NYSE, AMEX, and NASDAQ). In this study, I focus my analysis on the companies included in the S&P 500 index only, making it a very specific group of companies. These are leading firms in leading industries of the U.S. economy, and by nature are very large in market capitalization, have high financial viability and high prices per share. Campbell et al. (2008) report that financially distressed firms tend to be relatively small, have severely low financial viability, and tend to trade at very low prices per share. Under this, it is high unlikely that the superiority of the econometric models over the DD model will hold in my sample too Option pricing approach: The Black and Scholes model The dividend adjusted Black and Scholes (1973) formula (see also, 1973) for the value of the European call option is: c BS = Se d T y o N( δ) Ke r T o o N( d δ T ), (20.1) o and for the value of the European put option is: with BS r T d T o o y o p = Ke N( δ + σ T ) Se N( δ), (20.2) o ln( S / K) + ( ro - d y ) To + ( σ δ = σ T o T o ) 2 / 2. (20.3) In the above expressions, S is the spot price of the underlying asset; K is the strike price of the r o option; is the continuousl y compounded risk free interest rate; d is the continuous dividend yield T o paid by the underlying asset; is the time left until the option expiration date; σ is the yearly variance of the rate of re turn for the underlying asset and N(.) stands for the standard normal cumulative distribution. y 2 19 In addition, since the majority of the bankrupt firms do not match the characteristics of the ones included in the S&P 500 index, it would be very difficult to develop an econometric model to provide an accurate default risk measure for such firms. 17

19 The abundant empirical evidence regarding the smirk behavior of the BS implied volatility is indicative of implied return distributions that are negatively skewed and heavier tailed than those im plied by the BS log-normal distribution (see Bakshi et al., 1997 and Bates, 2000). Thus, the regression-based DVF approach of Dumas et al. (1998) is a way to mitigate the volatility smirk anomaly, by allowing the implied volatilities to be deterministic functions of the option s strike price ( K ) and maturity ( T o ) 20. According to Dumas et al. (1998), the approach of smoothing the BS implied volatilities across strike prices and maturities exhibits superior in- and out-of-sample performance for pricing European options. Christoffersen and Jacobs (2004) have found that this approach outperforms the Heston (1993) SV model. The DVF parameters can be estimated using either OLS - where the loss function is the difference between the estimated and the actual contractspecific implied volatility - or Nonlinear Least Squares - where the loss function is the difference between the theoretical/model-based and the actual/market-based option prices. In fact, many studies in this area concentrate only on OLS estimation (e.g. Ncube, 1996, Peña et al., 1999, Chang et al., 2009). Yet, Christoffersen and Jacobs (2004) demonstrate that OLS estimates of such regression based models yield biased option pricing results and that a Nonlinear Least Squares pricing loss function should be used instead. Akin to Andreou et al. (2008b), I model the S&P 500 index IVF using Nonlinear Least Squares based on the following DVF specification: o 4 o 5 o σ = max(0.01, a + a LnX + a ( LnX ) + a T + a ( LnX ) T + a T ) (21) where X = S / K (i.e. the moneyness ratio). Following Dumas et al. (1998), a minimum value of 0.01 is imposed to prevent negative values of volatility. An intriguing advantage of this specification is that, for a given maturity, the at-the-money implied volatility value is explicitly defined by the intercept a 0. This happens because the natural logarithm of an option that is exactly at-the money (X =1) equals zero. The coefficient a 1 ( a 2 ) captures the slope (curvature) of the implied volatility smirk with respect to moneyness. For instance, Zhang and Xiang (2008) derive analytical formulas that relate the level ( a 0 ), the slope ( a1 ) and the convexity ( a 2 ) of the IVF, to standard deviation, 20 This is equivalent saying that the DVF approach is able to capture the salient features of negative skewness and excess kurtosis of the risk-neutral distribution that is implied by the market option data at different maturities. 18

20 skewness and excess kurtosis of the risk-neutral distribution of the underlying asset. The coefficient a 3 ( a 5 ) captures the slope (curvature) of the term structure/time-dimension. The use of a quadratic time-term is motivated by the empirical presence of curvature (even humps) in the term structure of implied volatility 21. The coefficient a 4 reflects the importance of the cross product ( LnX ) T that allows the DVF slope with respect to moneyness to be time dependent, and the slope with respect to time to depend on the moneyness level, thus capturing changes in the shape of the volatili ty smirk over different maturities. The regression based-dvf coefficients in Eq. (21) are obtained via Nonlinear Least Squares by using the Whaley s (1982) simultaneous equation procedure 22. According to this method, option mrk market prices ( o ) are assumed to be the corresponding BS-DVF estimates ( o ) plus a BS DVF BS DVF random additive disturbance term ( ε ): mrk o = o BS DVF + ε BS DVF (22) To find the implied DVF coefficient values daily, I estimate an optimization problem that has the following form: P t BS DVF j 2 SSE ( t) = min ( ε ) (23) ξ DVF j= 1 where P t refers to the number of different option transaction datapoints available in day t, and ξ DVF to the unknown DVF coefficients in Eq. (21). The SSE in Eq. (23) is minimized using the Levenberg- Marquardt method with a line search based on a mixed quadratic and cubic polynomial interpolation and extrapolation method offered by Matlab. Daily recalibration of the implied coefficients is also adopted by Christoffersen and Jacobs (2004) (see also discussions in Ncube, 1996, Peña et al., 1999, Rosenberg, 2000, Hull and Suo, 2002). 21 Empirical evidence of such curvature in the time structure of implied volatility can be found in Table II of Bakhsi et al. (1997), in Table 1 of Christoffersen et al. (2009). I have confirm that such curvature is present in my option data as well. 22 The methodology employed here for estimating daily the DVF coefficients via Nonlinear Least Squares is similar to that in previous studies that adopt the Whaley s (1982) simultaneous equation procedure to minimize a price deviation function with respect to the unobserved parameters (see for instance Bates, 1991, Bakshi et al., 1997, Dumas et al., 1998, Christoffersen and Jacobs, 2004, Christoffersen et al., 2009). See also Andreou et al. (2008b) for more details. 19