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 –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. Sub-Samples Analysis Based on Sectors
5.3. Explaining the Skew
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Sample Availability
1 | https://www.risk.net/derivatives/1505339/jp-morgan-chase-launches-equity-default-swaps (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) short-sales 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 is 20 and the face value of debt is 30, then equity is worthless (). |
6 | |
7 | To be precise, the option’s payoff consistent with the compound option model of equity should be
|
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 log-probability 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 -entropy of a discrete random variable X is defined as as the chosen base is usually . Here, instead, having a base , the minus is not necessary as the function is already positive. |
10 | Unreported empirical tests show that are indeed approximately constant for the sample. By construction, is already bounded in ; moreover it is the probability of the intersection of the option expiring ITM and the firm surviving up to . Therefore, as the probability of the intersection is smaller of the probability of the single events, it should not surprise that 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 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 real-world 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 | |||
---|---|---|---|---|
Correlation (w/o CDS) | −0.0307 | −0.0053 | −0.0897 | 0.1291 |
p-value | 0.000 *** | 0.155 | 0.000 *** | 0.000 *** |
Correlation (with CDS) | −0.0723 | −0.0857 | −0.0751 | 0.1250 |
p-value | 0.000 *** | 0.000 *** | 0.000 *** | 0.000 *** |
(a): regressed onto(both constructed on calls). | |||||
Regressand | Adj-: | 0.7416 | |||
Regressors | Coefficient | Robust Standard Error | t-stat | p-value | |
0.7936 | 0.0050 | 160.07 | 0.000 | *** | |
0.0018 | 0.0001 | 29.42 | 0.000 | *** | |
(b): regressed onto (both constructed on puts). | |||||
Regressand | Adj-: | 0.9054 | |||
Regressors | Coefficient | Robust Standard Error | t-stat | p-value | |
0.8357 | 0.0145 | 57.63 | 0.000 | *** | |
−0.0002 | 0.00004 | −4.21 | 0.000 | *** |
(a): (constructed on calls) regressed onto . | |||||
Regressand | Adj-: | 0.0309 | |||
Regressors | Coefficient | Robust Standard Error | t-stat | p-value | |
0.0002 | 0.0004 | 0.53 | 0.630 | ||
0.0017 | 0.0006 | 2.74 | 0.071 | * | |
Industry−FE | ✓ | ||||
Year−FE | ✓ | ||||
(b): (constructed on calls and CDSs) regressed onto . | |||||
Regressand | Adj-: | 0.1120 | |||
Regressors | Coefficient | Robust Standard Error | t-stat | p-value | |
0.0066 | 0.0013 | 5.16 | 0.014 | ** | |
0.0029 | 0.0011 | 2.59 | 0.081 | * | |
Industry−FE | ✓ | ||||
Year−FE | ✓ | ||||
(c): (constructed on puts) regressed onto . | |||||
Regressand | Adj-: | 0.4967 | |||
Regressors | Coefficient | Robust Standard Error | t-stat | p-value | |
0.0253 | 0.0015 | 16.72 | 0.000 | *** | |
−0.0060 | 0.0020 | −3.07 | 0.054 | * | |
Industry−FE | ✓ | ||||
Year−FE | ✓ | ||||
(d): (constructed on puts and CDSs) regressed onto . | |||||
Regressand | Adj-: | 0.5326 | |||
Regressors | Coefficient | Robust Standard Error | t-stat | p-value | |
0.0236 | 0.0019 | 12.41 | 0.001 | *** | |
−0.0070 | 0.0023 | −2.98 | 0.059 | * | |
Industry−FE | ✓ | ||||
Year−FE | ✓ |
(a): Extra-information provided by CDSs when calls are used to infer credit risk. | |||||
Regressand | Adj-: | 0.3948 | |||
Regressors | Coefficient | Robust Standard Error | t-stat | p-value | |
0.0064 | 0.0014 | 4.74 | 0.018 | ** | |
−0.0002 | 0.0122 | −0.17 | 0.875 | ||
Industry−FE | ✓ | ||||
Year−FE | ✓ | ||||
(b): Extra-information provided by CDSs when puts are used to infer credit risk. | |||||
Regressand | Adj-: | 0.0557 | |||
Regressors | Coefficient | Robust Standard Error | t-stat | p-value | |
0.0024 | 0.0008 | 2.98 | 0.058 | * | |
−0.0018 | 0.0009 | −1.90 | 0.153 | ||
Industry−FE | ✓ | ||||
Year−FE | ✓ |
(a): Financials | |||||
Regressand | Adj-: | 0.5205 | |||
Regressors | Coefficient | Robust Standard Error | t-stat | p-value | |
0.0255 | 0.0008 | 32.20 | 0.000 | *** | |
−0.0092 | 0.0012 | −7.50 | 0.000 | *** | |
Year−FE | ✓ | ||||
(b): Mining, Energy and Utilities | |||||
Regressand | Adj-: | 0.3916 | |||
Regressors | Coefficient | Robust Standard Error | t-stat | p-value | |
0.0200 | 0.0013 | 15.72 | 0.000 | *** | |
−0.0013 | 0.0005 | −2.69 | 0.007 | *** | |
Year−FE | ✓ | ||||
(c): Manufacturing | |||||
Regressand | Adj-: | 0.3541 | |||
Regressors | Coefficient | Robust Standard Error | t-stat | p-value | |
0.0249 | 0.0031 | 8.08 | 0.000 | *** | |
−0.0040 | 0.0006 | −6.87 | 0.000 | *** | |
Year−FE | ✓ | ||||
(d): Retail, Wholesale and Services | |||||
Regressand | Adj-: | 0.1081 | |||
Regressors | Coefficient | Robust Standard Error | t-stat | p-value | |
0.0009 | 0.0001 | 14.38 | 0.000 | *** | |
0.0001 | 0.00002 | 4.91 | 0.000 | *** | |
Year−FE | ✓ |
Financials | Energy and Utilities | Manufacturing | Sales and Services | |
---|---|---|---|---|
1.3436 | 0.4478 | 0.2237 | 0.2286 | |
0.0265 | 0.0072 | 0.0008 | 0.0033 |
(a): Predictive regression for short-term skew. | |||||
Regressand | (between): | 0.0036 | |||
(within): | 0.0004 | ||||
Regressors | Coefficient | Robust Standard Error | t-stat | p-value | |
−0.4308 | 0.9730 | −0.44 | 0.659 | ||
−0.0035 | 0.0009 | −3.81 | 0.000 | *** | |
firm−FE | ✓ | ||||
year−FE | ✓ | ||||
(b): Predictive regression for long-term skew. | |||||
Regressand | (between): | 0.3096 | |||
(within): | 0.0004 | ||||
Regressors | Coefficient | Robust Standard Error | t-stat | p-value | |
0.3360 | 0.0956 | 3.51 | 0.001 | *** | |
−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