Assessing the Impact of Credit Risk on Equity Options via Information Contents and Compound Options
Abstract
:1. Introduction
2. Firm’s Claims as Compound Options
Call and Put Equity Options as a $(n+1)$–Fold Compound Options on Asset Value
3. Information Content Ratios as Measure of Impact of Credit Risk
4. Data and Estimation Methodology
4.1. Dataset
4.2. Estimation of the Model Parameters
5. Empirical Tests
5.1. Cross Sectional Differences in Calls and Puts
5.2. SubSamples Analysis Based on Sectors
5.3. Explaining the Skew
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Sample Availability
Notes
1  https://www.risk.net/derivatives/1505339/jpmorganchaselaunchesequitydefaultswaps (accessed on 11 October 2023). 
2  
3  Shorting bonds is even more difficult in the cash market as the repo market for corporate bond is often illiquid, and the tenor of the agreement is usually very short. 
4  These are the assumptions in Merton (1974) and Geske (1977). Specifically: (A.1) there are no transactions costs, taxes, or problems with indivisibilities of assets; (A.2) there are a sufficient number of investors with comparable wealth levels so that each investor believes that he can buy and sell as much of an asset as he wants at the market price; (A.3) there exists an exchange market for borrowing and lending at the same rate of interest; (A.4) shortsales of all assets, with full use of the proceeds, is allowed; (A.5) trading in assets takes place continuously in time. 
5  That is the value of equity before paying the bond. E.g., if the continuation value of the equity ${S}^{\u2605}$ is 20 and the face value of debt is 30, then equity is worthless ($S=0$). 
6  
7  To be precise, the option’s payoff consistent with the compound option model of equity should be
$${P}_{T}^{\left(\xi \right)}=\xi \left({S}_{T}\left(V\right){\mathbb{1}}_{\left\{\tau >T\right\}}+{S}_{i}^{\u2605}{\mathbb{1}}_{\left\{\tau \le T\right\}}K\right){\mathbb{1}}_{\left\{\xi {S}_{T}\left(V\right)\ge \xi K\right\}},$$

8  In Information Theory, information content (or surprisal) of a signal is the amount of information gained when it is sampled. It is defined as minus the logprobability of the event: the less likely the event, the greater is the “surprise” associated if it happens. See Cover and Thomas (2006) for further details. 
9  Usually, the $\mathbb{P}$entropy of a discrete random variable X is defined as ${H}_{b}^{\mathbb{P}}\left(X\right)={\sum}_{i}\mathbb{P}\left(X={x}_{i}\right){log}_{b}\mathbb{P}\left(X={x}_{i}\right)$ as the chosen base is usually $b=\{2,e,10\}$. Here, instead, having a base $b\in (0,1)$, the minus is not necessary as the ${log}_{b}$ function is already positive. 
10  Unreported empirical tests show that ${b}_{i}$ are indeed approximately constant for the sample. By construction, ${b}_{i}$ is already bounded in $(0,1)$; moreover it is the probability of the intersection of the option expiring ITM and the firm surviving up to ${t}_{i}$. Therefore, as the probability of the intersection is smaller of the probability of the single events, it should not surprise that ${b}_{i}$ is quite small and stable. 
11  This estimation technique is based on Brigo and Mercurio (2006) and is further discussed in Maglione (2022), Section 3.3. We refer to these references for further details. 
12  Other values of loss given default have been investigated as a robustness and results are available upon request. 
13  More specifically, after estimating the asset volatility surface from options, the average value ${\overline{\sigma}}_{V}$ is used to compute the asset value such that (4) holds. Subsequently, the model implied market value of debt is obtained. 
14  The compound option model of default used here is able to model default only by the mean of financial leverage: if at reimbursement dates the equity of the firm is not large enough to repay the face value of the liability due, then the firm defaults. It should be clear that realworld default may occur not only in case of excessive financial leverage. Indeed, other sources of default are investigated in Carr and Wu (2017). What we refer as ‘apparent’ leverage effect is the possibility of observing a sizeable and similar skew both in the Black–Scholes and compound option implied volatilities when a firm is highly levered: since the compound options accounts for financial leverage, observing a large skew after having accounted for the latter may suggest that put option price the possibility of a large fall in asset prices for reasons other than leverage. 
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Ticker  SIC  Division 

AAPL  3663  Manufacturing 
ABT  2834  Manufacturing 
ACN  8742  Services 
ALL  6331  Finance, Insurance and Real Estate 
AMGN  2836  Manufacturing 
AMZN  5961  Wholesale Trade 
BA  3721  Manufacturing 
BAC  6020  Finance, Insurance and Real Estate 
BMY  2834  Manufacturing 
C  6199  Finance, Insurance and Real Estate 
CAT  3531  Manufacturing 
CL  2844  Manufacturing 
CMCSA  4841  Transportation, Communications, Electric, Gas and Sanitary service 
COF  6141  Finance, Insurance and Real Estate 
COP  1311  Mining 
COST  5399  Wholesale Trade 
CSCO  3576  Manufacturing 
CVS  5912  Retail Trade 
CVX  2911  Manufacturing 
DD  2821  Manufacturing 
DIS  4888  Transportation, Communications, Electric, Gas and Sanitary service 
EMR  3823  Manufacturing 
EXC  4911  Transportation, Communications, Electric, Gas and Sanitary service 
F  3711  Manufacturing 
FDX  4513  Transportation, Communications, Electric, Gas and Sanitary service 
GD  3721  Manufacturing 
GE  4911  Transportation, Communications, Electric, Gas and Sanitary service 
HAL  1389  Mining 
HD  5211  Wholesale Trade 
IBM  7370  Services 
INTC  3674  Manufacturing 
JNJ  2834  Manufacturing 
JPM  6020  Finance, Insurance and Real Estate 
KO  2086  Manufacturing 
LLY  2834  Manufacturing 
LOW  5211  Wholesale Trade 
MCD  5812  Retail Trade 
MDT  3845  Manufacturing 
MMM  2670  Manufacturing 
MO  2111  Manufacturing 
MON  5169  Retail Trade 
MRK  2834  Manufacturing 
MS  6211  Finance, Insurance and Real Estate 
MSFT  7372  Services 
ORCL  7370  Services 
OXY  1311  Mining 
PEP  2080  Manufacturing 
PFE  2834  Manufacturing 
PG  2840  Manufacturing 
PM  2111  Manufacturing 
RTN  3812  Manufacturing 
SLB  1389  Mining 
SO  4911  Transportation, Communications, Electric, Gas and Sanitary service 
SPG  6798  Finance, Insurance and Real Estate 
T  4812  Transportation, Communications, Electric, Gas and Sanitary service 
TGT  5331  Wholesale Trade 
TWX  8748  Services 
TXN  3674  Manufacturing 
UNH  6324  Finance, Insurance and Real Estate 
UNP  4011  Transportation, Communications, Electric, Gas and Sanitary service 
USB  6020  Finance, Insurance and Real Estate 
UTX  3724  Manufacturing 
VZ  4812  Transportation, Communications, Electric, Gas and Sanitary service 
WFC  6020  Finance, Insurance and Real Estate 
WMT  5331  Retail Trade 
XOM  1311  Mining 
Calls  Puts  

$\mathit{K}$  $\mathit{T}$  $\mathit{K}$  $\mathit{T}$  
Correlation (w/o CDS)  −0.0307  −0.0053  −0.0897  0.1291 
pvalue  0.000 ***  0.155  0.000 ***  0.000 *** 
Correlation (with CDS)  −0.0723  −0.0857  −0.0751  0.1250 
pvalue  0.000 ***  0.000 ***  0.000 ***  0.000 *** 
(a): ${\overline{\mathit{A}\mathit{I}\mathit{C}\mathit{R}}}^{\prime}$regressed onto$\overline{\mathit{A}\mathit{I}\mathit{C}\mathit{R}}$(both constructed on calls).  
Regressand  Adj${R}^{2}$:  0.7416  
${\overline{AICR}}_{1}^{\prime}$  
Regressors  Coefficient  Robust Standard Error  tstat  pvalue  
${\overline{AICR}}_{1}$  0.7936  0.0050  160.07  0.000  *** 
${\alpha}_{1}$  0.0018  0.0001  29.42  0.000  *** 
(b): ${\overline{AICR}}^{\prime}$ regressed onto $\overline{AICR}$ (both constructed on puts).  
Regressand  Adj${R}^{2}$:  0.9054  
${\overline{AICR}}_{1}^{\prime}$  
Regressors  Coefficient  Robust Standard Error  tstat  pvalue  
${\overline{AICR}}_{1}$  0.8357  0.0145  57.63  0.000  *** 
${\alpha}_{1}$  −0.0002  0.00004  −4.21  0.000  *** 
(a): $\overline{\mathit{A}\mathit{I}\mathit{C}\mathit{R}}$ (constructed on calls) regressed onto $\mathit{L}\mathit{E}\mathit{V}$.  
Regressand  Adj${R}^{2}$:  0.0309  
${\overline{AICR}}_{1}$  
Regressors  Coefficient  Robust Standard Error  tstat  pvalue  
$LEV$  0.0002  0.0004  0.53  0.630  
${\alpha}_{1}$  0.0017  0.0006  2.74  0.071  * 
Industry−FE  ✓  
Year−FE  ✓  
(b): ${\overline{AICR}}^{\prime}$ (constructed on calls and CDSs) regressed onto $LEV$.  
Regressand  Adj${R}^{2}$:  0.1120  
${\overline{AICR}}_{1}^{\prime}$  
Regressors  Coefficient  Robust Standard Error  tstat  pvalue  
$LEV$  0.0066  0.0013  5.16  0.014  ** 
${\alpha}_{1}$  0.0029  0.0011  2.59  0.081  * 
Industry−FE  ✓  
Year−FE  ✓  
(c): $\overline{AICR}$ (constructed on puts) regressed onto $LEV$.  
Regressand  Adj${R}^{2}$:  0.4967  
${\overline{AICR}}_{1}$  
Regressors  Coefficient  Robust Standard Error  tstat  pvalue  
$LEV$  0.0253  0.0015  16.72  0.000  *** 
${\alpha}_{1}$  −0.0060  0.0020  −3.07  0.054  * 
Industry−FE  ✓  
Year−FE  ✓  
(d): ${\overline{AICR}}^{\prime}$ (constructed on puts and CDSs) regressed onto $LEV$.  
Regressand  Adj${R}^{2}$:  0.5326  
${\overline{AICR}}_{1}^{\prime}$  
Regressors  Coefficient  Robust Standard Error  tstat  pvalue  
$LEV$  0.0236  0.0019  12.41  0.001  *** 
${\alpha}_{1}$  −0.0070  0.0023  −2.98  0.059  * 
Industry−FE  ✓  
Year−FE  ✓ 
(a): Extrainformation provided by CDSs when calls are used to infer credit risk.  
Regressand  Adj${R}^{2}$:  0.3948  
${\eta}_{1}$  
Regressors  Coefficient  Robust Standard Error  tstat  pvalue  
$LEV$  0.0064  0.0014  4.74  0.018  ** 
${\alpha}_{1}$  −0.0002  0.0122  −0.17  0.875  
Industry−FE  ✓  
Year−FE  ✓  
(b): Extrainformation provided by CDSs when puts are used to infer credit risk.  
Regressand  Adj${R}^{2}$:  0.0557  
${\eta}_{1}$  
Regressors  Coefficient  Robust Standard Error  tstat  pvalue  
$LEV$  0.0024  0.0008  2.98  0.058  * 
${\alpha}_{1}$  −0.0018  0.0009  −1.90  0.153  
Industry−FE  ✓  
Year−FE  ✓ 
(a): Financials  
Regressand  Adj${R}^{2}$:  0.5205  
${\overline{AICR}}_{1}^{\prime}$  
Regressors  Coefficient  Robust Standard Error  tstat  pvalue  
$LEV$  0.0255  0.0008  32.20  0.000  *** 
${\alpha}_{1}$  −0.0092  0.0012  −7.50  0.000  *** 
Year−FE  ✓  
(b): Mining, Energy and Utilities  
Regressand  Adj${R}^{2}$:  0.3916  
${\overline{AICR}}_{1}^{\prime}$  
Regressors  Coefficient  Robust Standard Error  tstat  pvalue  
$LEV$  0.0200  0.0013  15.72  0.000  *** 
${\alpha}_{1}$  −0.0013  0.0005  −2.69  0.007  *** 
Year−FE  ✓  
(c): Manufacturing  
Regressand  Adj${R}^{2}$:  0.3541  
${\overline{AICR}}_{1}^{\prime}$  
Regressors  Coefficient  Robust Standard Error  tstat  pvalue  
$LEV$  0.0249  0.0031  8.08  0.000  *** 
${\alpha}_{1}$  −0.0040  0.0006  −6.87  0.000  *** 
Year−FE  ✓  
(d): Retail, Wholesale and Services  
Regressand  Adj${R}^{2}$:  0.1081  
${\overline{AICR}}_{1}^{\prime}$  
Regressors  Coefficient  Robust Standard Error  tstat  pvalue  
$LEV$  0.0009  0.0001  14.38  0.000  *** 
${\alpha}_{1}$  0.0001  0.00002  4.91  0.000  *** 
Year−FE  ✓ 
Financials  Energy and Utilities  Manufacturing  Sales and Services  

$\overline{LEV}$  1.3436  0.4478  0.2237  0.2286 
${\overline{\overline{AICR}}}_{1}^{\prime}$  0.0265  0.0072  0.0008  0.0033 
(a): Predictive regression for shortterm skew.  
Regressand  ${R}^{2}$ (between):  0.0036  
$\Delta {\mathrm{Skew}}_{T<1}$  ${R}^{2}$ (within):  0.0004  
Regressors  Coefficient  Robust Standard Error  tstat  pvalue  
${\overline{AICR}}_{1,T<1}^{\prime}$  −0.4308  0.9730  −0.44  0.659  
${\alpha}_{T<1}$  −0.0035  0.0009  −3.81  0.000  *** 
firm−FE  ✓  
year−FE  ✓  
(b): Predictive regression for longterm skew.  
Regressand  ${R}^{2}$ (between):  0.3096  
$\Delta {\mathrm{Skew}}_{T>1}$  ${R}^{2}$ (within):  0.0004  
Regressors  Coefficient  Robust Standard Error  tstat  pvalue  
${\overline{AICR}}_{1,T>1}^{\prime}$  0.3360  0.0956  3.51  0.001  *** 
${\alpha}_{T>1}$  −0.0108  0.0017  −6.36  0.000  *** 
firm−FE  ✓  
year−FE  ✓ 
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Maglione, F.; Mancino, M.E. Assessing the Impact of Credit Risk on Equity Options via Information Contents and Compound Options. Risks 2023, 11, 183. https://doi.org/10.3390/risks11100183
Maglione F, Mancino ME. Assessing the Impact of Credit Risk on Equity Options via Information Contents and Compound Options. Risks. 2023; 11(10):183. https://doi.org/10.3390/risks11100183
Chicago/Turabian StyleMaglione, Federico, and Maria Elvira Mancino. 2023. "Assessing the Impact of Credit Risk on Equity Options via Information Contents and Compound Options" Risks 11, no. 10: 183. https://doi.org/10.3390/risks11100183