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Abstract:
We find that variance risk premia, defined as the spread between the option-implied and expected variances, have a prominent predictability for the credit default swap spreads at individual firm level. Such a predictability cannot be crowded out by that of the market and firm level credit risk factors identified in previous research. We demonstrate that the strong predictability of implied variance for credit spreads is mostly explained by either variance premium or expected variance. Our findings suggest that variance risk premium is a relatively clean measure of a firm’s exposure to macroeconomic uncertainty or systematic variance risk, while option-implied variance may be contaminated by idiosyncratic variance risk. Such a result is consistent with the market level evidence that variance risk premium predicts credit spread variation.

variance risk premia, credit default swap spreads, option-implied variance, expected variance, realized variance

Abstract:
This paper introduces a new methodology for measuring and analyzing capital structure effects on option prices of individual firms in the economy. By focusing on individual firms we examine the cross sectional effects of leverage on option prices. Our methodology allows the market value of each firm's debt to be implied directly from two contemporaneous, liquid, at-the-money option prices without the use of any historical price data. We compare Geske's parsimonious model to the alternative models of Black Scholes (BS) (1973), Bakshi, Cao, and Chen (BCC, 1997) (stochastic volatility (SV), stochastic volatility and stochastic interest rates (SVSI), and stochastic volatility and jumps (SVJ)), and Pan (2002) (no-risk premia (SV0), volatility-risk premia(SV), jump-risk premia (SVJ0), volatility and jump risk premia (SVJ)) which allows state-dependent jump intensity and adopts implied state-GMM econometrics. These alternative models do not directly incorporate leverage effects into option pricing, and except for Black-Scholes these model calibrations require the use of historical prices, and many more parameters which require complex estimation procedures. The comparison demonstrates that firm leverage has significant statistical and economic cross sectional effects on the prices of individual stock options. The paper confirms that by incorporating capital structure effects using our methodology to imply the market value of each firm's debt, Geske's model reduces the errors pricing options on individual firms by 60% on average, relative to the models compared herein (BS, BCC, Pan) which omit leverage as a variable. However, we would be remiss in not noting that after including leverage there is still room for improvement, and perhaps by also incorporating jumps or stochastic volatility at the firm level would result in an even better model.

Derivatives, Options, Leverage, Stochastic Volatility

Abstract:
The primary purpose of this paper is to examine whether leverage has a significant statistical and economic effect on the pricing of S&P 500 index put options. The secondary purpose is to present information regarding the shape and persistent smile rather than skew of the implied volatility function. This is the first paper to directly test for leverage effects in stock index put options. To analyze these effects we use the Geske (1979) compound option model. The Geske model is closed form, implies stochastic equity volatility, is consistent with Modigliani and Miller, incorporates debt refinancing, and includes possibly differential default and bankruptcy. Black-Scholes (1973) is a special case of the Geske model. In this paper we show that during the years 1996-2004 the aggregate market based debt to equity (D/E) ratio of the firms comprising the S&P 500 equity index varies from about 40-120 percent. We believe that we are the first to present a market D/E ratio derived from option theory. We also present evidence that on an average of about 200,000 options during this 8 year period the implied volatility most often exhibits a smile not a smirk or skew. Next and more importantly we are the first to report the details of the statistically significant economic effects that market leverage has on pricing S&P 500 index put options. We measure that the Geske model improves the net option valuation of listed in the money (or out of the money) S&P 500 index put options on average by about 37% (19%) compared to Black-Scholes values. We demonstrate that the improvement is directly (and monotonically) related to both the time to expiration of the option and the amount of leverage in this market index. For options with longer expirations and/or periods of higher market leverage the improvement is greater, ranging from about 50% to 85%. We also demonstrate economic significance in basis points by showing that dealers making a book in index options can expect benefits of at least several 100 basis points using Geske instead of Black-Scholes. Finally we show that the per cent pricing errors compare very favorably with Heston-Nandi (2000).

Derivatives, Stochastic Stock Volatility, Leverage

Abstract:
This paper presents empirical tests of seven alternative estimators for the volatility of the S&P 500 equity index. Two of the estimators are the implied volatilities derived from the option models of Geske (1979) and Black-Scholes (1973) and using market prices for options. The Geske volatility estimator is stochastic, and this paper presents its first empirical tests. The Black-Scholes estimator in theory is deterministic, but herein when computed daily it is ad hoc allowed to change randomly with changes in the index level. The other five estimators are empirical, based on sample data. Four of the empirical estimators are GARCH approaches (GARCH11, EGARCH, TGARCH, and Heston-Nandi GARCH) which accommodate stochastic volatilities, and the fifth empirical estimator is simply the sample variance viewed as an historical estimate. All seven estimators are used to predict the actual realized volatility over the life of each option. If the market processes information efficiently, when the actual realized volatility over the option life is regressed on each estimator, the slope coefficient and intercept should be 1.0 and 0.0, respectively. In the case of the implied volatility estimators, this can be considered a test of these models. Our results show that in all cases the slope coefficients and intercepts of the implied volatilities are much closer to 1.0, more highly significant, have intercepts closer to 0.0 which are insignificant, and exhibit higher R2 than the empirical estimators. Furthermore, the Geske estimator appears better in these respects than this ad hoc Black-Scholes estimator. Encompassing regressions are run paring the Geske estimator separately with each of the five empirical estimators. In every encompassing regression test the slope coefficients and intercepts of the Geske estimator remain much closer to 1.0 and 0.0, respectively, while the competing estimators slope coefficients are much reduced from 1.0 toward 0.0, and are often insignificant. Thus, the Geske stochastic volatility estimator appears to capture a very significant portion but not all of the information relevant for predicting the volatility of the S&P 500 equity index.

Derivatives, Volatility Forecast, Stochastic Stock Volatility, Leverage

Abstract:
The purpose of this paper is to examine whether leverage has a significant statistical and economic effect on the pricing of S&P 500 index options. This is the first paper to directly test for leverage effects in stock index options. To analyze these effects we use the Geske (1979) compound option model. The Geske model is closed form, implies stochastic equity volatility, is consistent with Modigliani and Miller, incorporates debt refinancing, and includes possibly differential default and bankruptcy. Black-Scholes (1973) is a special case of the Geske model. In this paper we show that during the years 1996-2004 the aggregate market based debt to equity (D/E) ratio of the firms comprising the S&P 500 equity index varies from about 40-120 percent. We believe this is the first presentation of a market D/E ratio derived from option theory. Next and more importantly we are the first to report the details of the statistically significant economic effects that market leverage has on pricing S&P 500 index call options. We measure that the Geske model improves the net option valuation of listed in the money (or out of the money) S&P 500 index call options on average by about 35% (28%) compared to Black-Scholes values. We demonstrate that the improvement is directly (and monotonically) related to both the time to expiration of the option and the amount of leverage in this market index. For options with longer expirations and/or periods of higher market leverage the improvement is greater, ranging from about 40% to 80%. We also demonstrate economic significance in basis points by showing that dealers making a book in index options can expect benefits of at least several 100 basis points using Geske instead of Black-Scholes.

Derivatives, Stochastic Stock Volatility, Leverage