The Merton model imposes a lot of structure on the pricing relation. I therefore also …t a linear model using the same inputs as the Merton model. Following Campbell and Taksler (2003) who show that historical volatility helps price bonds in a linear setting, I regress observed spreads on explanatory variables and di¤erent measures of volatility.
I include leverage, time to maturity and average rating spread (the same as the model inputs).
Table 6 Panel B reports the results. I report results from three di¤erent spec- i…cations: including historical volatility and option implied volatility separately and including both together. I do not use predicted volatility because this would impose a restriction on the relative importance of the two measures in the regression. When including the measures separately, both historical and option implied volatility come in with the expected sign and are signi…cant. The regression including option im- plied volatility has a slightly higher …t than the regression including historical volatility.
When both measures are included together, both are signi…cant. Relative to includ- ing only historical volatility, the R2 improves from 24.5% to 27.4%. I next focus on the time-series variation and include an issue …xed e¤ect. Both measures of volatility are signi…cant, both when included separately and when included together. The R2 improves from 17.8% to 22.6% when including both measures of volatility relative to using only historical volatility.28 These results provide evidence that it is better to use a forward looking measure of volatility when pricing bonds.29
28These results are consistent with independent work by Cremers, Driessen, Maenhout, Weinbaum (2006) who also use single stock option implied volatility and skewness to price bonds in a linear speci…cation similar to that reported in Table 6 Panel B.
29Nevertheless, the resultingR2of 27.4% (of overall variation) and 22.6% (of time series variation) are far from perfect. These results are therefore consistent with the evidence in Collin-Dufresne, Goldstein, and Martin (2001) that changes in fundamentals cannot explain most of the variation in observed credit spread changes.
6 Conclusion
This paper contributes to the existing empirical corporate bond pricing literature by demonstrating in two ways that the corporate bond market is forward looking with respect to volatility. First, I …nd that implied volatility, calculated from yield spreads, contains substantial information about future volatility. Added to a regression of future on historical volatility, implied volatility is a statistically and economically signi…cant predictor of future volatility and provides noticeable incremental explanatory power.
This is true mainly in the time-series. Implied volatility retains explanatory power when included together with stock option implied volatility. These results are consistent with Hotchkiss and Ronen (2002) who …nd a high level of informational e¢ ciency in the bond market. The results also suggest that the bond market, though in parts not very transparent (Goldstein, Hotchkiss, and Sirri 2006), nevertheless re‡ects important information about future market conditions.
Second, I …nd that using predicted volatility is better at pricing bonds than historical volatility. Calculating model implied spreads using the information from the options market results in better explanatory power of the time-series variation in yield spreads.
In a linear regression of spreads on explanatory variables, option implied volatility comes in signi…cantly and improves the …t when included together with historical volatility.
This evidence is consistent with Cremers, Maenhout, Driessen and Weinbaum (2006) who price bonds in a linear setting.
In addition, the results have implications for the usefulness of structural bond pricing models. Schaefer and Strebulaev (2004) argue that structural models are useful since they give accurate predictions of how bond prices respond to changes in the …rm’s equity value (the hedge ratio or delta). In this paper, I use a structural model to explore the relation between changes in expected future volatility and current bond prices. The results provide insight about the sensitivity of bond spreads to volatility (option vega) and suggest that the theoretical and empirical sensitivities are quite close. This evidence further underscores that structural models can help to explain patterns in bond prices.
The results also have broader implications for prices in di¤erent markets. The evidence that the bond market re‡ects information available in the equity and option markets may shed light on the possibility of implementing pro…table capital structure arbitrage strategies: If a …rm’s outstanding equity and bonds are priced e¢ ciently it is less likely that such a strategy will return positive economic pro…ts. In related work, Carr and Linetsky (2006) and Carr and Wu (2006) model joint pricing of credit and equity derivatives, speci…cally considering the e¤ect of credit events on option valuation.
The results in this paper suggest that, more generally, credit, equity and option markets share the same information.
A The Merton model
In the Merton (1974) model, risky debt is priced as safe debt plus a short put option.
Using the Black Scholes (1973) option pricing formula for a put option and collecting terms, the value of risky debt is given by
Bt=VtN( d1) +F exp ( rT)N(d2): Since st = T1 log BFt r and de…ning w = BVt
t = exp ( (r+s)T)FVt
t this expression can be rewritten as
N( d1)
w + exp (st T)N(d2) = 1 where
d1 = log (wt) + s 12 2A T
A
pT d2 = d1 Ap
T :
In order to calculate equity volatility from asset volatility, it is necessary to know the equity delta, the sensitivity of the equity value with respect to changes in asset value.
Since the value of equity at any point in time is given by
St=Vt Bt =Vt(1 N( d1)) exp ( rT)FtN(d2);
the sensitivity of equity to asset value is
@S
@V = 1 N( d1):
The level of implied equity volatility can then be calculated as
implied
t = AVt
St
@S
@V = AVt
St(1 N( d1)) = A1 N( d1)
1 w :
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Panel A: Baseline regression sample
Mean 107 9.2 0.26 0.40 0.32 0.33
Median 92 6.9 0.20 0.38 0.30 0.31
St. Dev. 65 7.7 0.20 0.14 0.12 0.12
Min 10 0.1 0.001 0.13 0.13 0.10
Max 1355 30.0 0.97 0.95 0.88 0.92
Observations: 20,716
Panel B: Correlations of log volatility measures
Historical volatility 0.69 1
Implied volatility 0.04 0.04
Observations: 20,716
Historical volatility 0.47 1
Implied volatility 0.48 0.54
Observations: 14,599
Table 1: Summary statistics
Panel A reports summary statistics for bond spreads, maturity, leverage, and measures of volatility for the baseline sample.
Spread is the yield spread in basis points over the closest U.S. Treasury on fixed-rate U.S. dollar, non-callable, non-puttable, non-sinking fund, non-convertible bond price observations from 1995-1999 with available accounting and rating data.
Leverage is total debt to capitalization, maturity is the remaining time to maturity of the bond. Implied volatility is the level of volatility needed to match the bond spread. Historical and future volatility are calculated for 180 days before and after the bond price observation. In Panel B time-series correlation is the bond issue de-meaned correlation of volatility measures.
Correlations are computed for the sub-sample with at least 8 observations for each bond issue.
Future volatility
Future volatility
Historical volatility
Implied volatility
Spread Maturity Leverage Historical
volatility
correlation
Future volatility
Historical volatility correlation
(time-series)
Regression of future volatility on historical and implied volatility
(1) (2) (3) (4) (5) (6)
0.708 0.707 0.494 0.308
(139.04)** (138.84)** (61.92)** (34.01)**
0.044 0.016 0.733 0.492
(6.22)** (3.08)** (64.47)** (37.81)**
Constant -0.307 -1.116 -0.292
(48.08)** (152.36)** (36.80)**
Observations 20716 20716 20716 14599 14599 14599
# of firms 606 606 606 355 355 355
# of bonds 3015 3015 3015 1074 1074 1074
Bond fixed effect X X X
min # of obs./bond 1 1 1 8 8 8
R-squared 0.483 0.002 0.483
within R-squared 0.221 0.235 0.295
Absolute value of t-statistics in parentheses
* significant at 5%; ** significant at 1%
Table 2: Predictive regressions
Historical volatility Implied volatility
This table reports results from regressing log future volatility on historical volatility and implied volatility. Historical and future volatility are measured as the sample standard deviation of daily returns over a 180 day period before and after the bond transaction respectively. The sample corresponds to the baseline sample discussed in Table 1. For the regressions (4) - (6), the sample is restricted to observations for issues with at least 8 transactions for each bond issue. Constants and fixed effects are not reported for the fixed effect regressions.
Regression including maturity and leverage interactions
(1) (2) (3)
0.308 0.237 0.286
(34.01)** (25.57)** (31.09)**
0.492 0.160 0.612
(37.81)** (7.83)** (30.97)**
0.149 (12.89)**
0.272 (19.12)**
0.391 (23.85)**
0.462 (20.49)**
-0.014 (1.23) -0.071 (5.25)**
-0.141 (9.47)**
-0.202 (9.45)**
Observations 14599 14599 14599
Bond fixed effect X X X
Number of bonds 1074 1074 1074
within R-squared 0.2954 0.329 0.305
Absolute value of t-statistics in parentheses
* significant at 5%; ** significant at 1%
Implied volatility
5<maturity<=8*implied volatility
Table 3: Maturity and leverage effects
Historical volatility
This Table reports results from regressing log future volatility on historical volatility and implied volatility. The sample corresponds to that used in Table 2, specifications (4)-(6). Constants and fixed effects are not reported.
Specification (1) is the same as specification (6) in Table 2.
8<maturity<=12*implied volatility 12<maturity<=30*implied volatility 3<maturity<=5*implied volatility
0.5<leverage<=1*implied volatility 0.1<leverage<=0.2*implied volatility 0.2<leverage<=0.3*implied volatility 0.3<leverage<=0.5*implied volatility
Future volatility on historical, bond, and option implied volatility
(1) (2) (3) (4) (5) (6)
0.466 0.257 0.164 0.057
(46.23)** (22.24)** (11.66)** (4.01)**
0.752 0.540 0.444
(52.46)** (31.92)** (26.23)**
0.637 0.495 0.389
(55.11)** (29.57)** (23.42)**
Observations 9766 9766 9766 9766 9766 9766
Bond fixed effect X X X X X X
within R-squared 0.192 0.234 0.253 0.274 0.264 0.316
Absolute value of t-statistics in parentheses
* significant at 5%; ** significant at 1%
Historical volatility Bond implied volatility Option implied volatility
Table 4: Predictive regressions including option implied volatility
This table reports results from regressing log future volatility on historical, bond implied and option implied volatility. The sample is restricted to observations for bond issues with at least 8 transactions for each issue. To compare R-squareds, results are reported for the same sample throughout.
Realized and model implied spreads (in basis points)
Mean 109 152 144 141
Median 95 53 53 64
St. Dev. 65 220 202 181
Min 10 0.001 0.001 0.002
5th percentile 38 0.04 0.07 0.26
95th percentile 223 630 594 543
Max 1355 1392 1192 1002
Observations: 13,710
Table 5: Predicted yield spreads using different measures of volatility
Spread (option implied
volatility)
This table reports summary statistics for observed and model implied spreads using historical, option implied and predicted volatility as inputs. Predicted volatility is measured as the fitted value from a regression of future volatility on historical and option implied volatility. The sample includes the set of observations for which there are predicted spreads for all measures of volatility.
Spread Spread
(historical volatility)
Spread (predicted
volatility)
(1) (2) (3) (4) (5) (6)
0.092 0.144
(30.21)** (31.48)**
0.108 0.201
(32.52)** (36.71)**
0.109 0.259
(29.93)** (37.00)**
Constant 91.071 89.851 89.545
(121.51)** (119.81)** (114.53)**
Observations 8603 8603 8603 8603 8603 8603
Bond fixed effect X X X
R-squared 0.096 0.110 0.094
within R-squared 0.111 0.145 0.147
Panel B: Regression of spread on explanatory variables
(1) (2) (3) (4) (5) (6)
Historical volatility 192.728 70.649 243.733 109.161
(39.87)** (9.25)** (42.36)** (13.97)**
Option implied volatility 249.083 183.714 325.950 233.241
(44.35)** (20.38)** (47.65)** (24.60)**
Leverage -2.252 -7.987 -9.261 132.505 99.202 103.852
(0.83) (2.97)** (3.46)** (10.84)** (8.25)** (8.72)**
Rating spread 0.772 0.690 0.699 0.313 0.322 0.268
(29.36)** (26.43)** (26.83)** (4.03)** (4.23)** (3.55)**
Maturity 1.620 1.560 1.592
(23.11)** (22.62)** (23.15)**
Constant -58.115 -65.251 -68.115
(18.90)** (21.38)** (22.29)**
Observations 10422 10422 10422 10422 10422 10422
Bond fixed effect X X X
R-squared 0.245 0.268 0.274
within R-squared 0.178 0.211 0.226
Absolute value of t-statistics in parentheses
* significant at 5%; ** significant at 1%
Spread (predicted volatility)
Table 6: Pricing bonds with different measures of volatility
Panel A reports results from regressing realized spreads on predicted spreads. Predicted spreads are calculated using historical, option implied and predicted volatility. Panel B reports results from regressions of spreads on characteristics, historical and option implied volatility. The sample is restricted to observations for bond issues with at least 8 transactions for each issue.
Spread (historical volatility) Spread (option implied volatility)
Panel A: Regression of spreads on predicted spreads
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