# Risk Management and Agency Theory: Role of the Put Option in Corporate Bonds

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## Abstract

**:**

## 1. Introduction

#### 1.1. The Greeks

#### 1.2. Relation between the Greeks and Risk/Agency Issues

#### 1.3. Hypotheses

#### 1.4. Results

## 2. Literature Survey

#### 2.1. Agency Issues and the Choice for Gamma and Vega

#### 2.2. Information Asymmetry and the Choice for Gamma and Vega

#### 2.3. Credit Risk and the Choice for Delta

#### 2.4. Interest-Rate Risk and the Choice for Rho

## 3. Data

#### 3.1. Firm and Issue Characteristics

#### 3.2. Call and Put Profiles

#### 3.3. Issuances in 1998

## 4. Methodology

#### 4.1. Model

#### 4.2. Indenture Provisions and Bankruptcy Reorganization

#### 4.3. Optimal Exercise of Call and Put Provisions

#### 4.4. Numerical Method

#### 4.5. Model Parameters

- For the lower bound, denoted by ${\sigma}_{l}$, V is calculated using the market-value of equity plus the book-value of debt and the book-value of preferreds. The book values are assumed to be constant. This assumption presumes debt and preferred price movements are inconsequential, relative to equity price movements, when estimating the firm’s volatility.2
- For the upper bound, denoted by ${\sigma}_{u}$, V is defined, simply, as the market value of equity. This assumption presumes debt and preferred price movements are perfectly positively correlated with equity.3

#### 4.6. Estimators for the Greeks

## 5. Results

#### 5.1. Relative Pricing Errors (RPE)

#### 5.2. Estimates for the Greeks

- (0,0) indicates neither the call nor the put is incorporated in the valuation model, i.e., the issue is Straight;
- (1,0) indicates the call is incorporated but not the put, i.e., the issue is Callable only;
- (0,1) indicates the put is incorporated but not the call, i.e., the issue is Putable only;
- (1,1) indicates that both provisions are incorporated in the valuation model, i.e., the issue is Callable and Putable.

#### 5.2.1. Greeks at the Upper Bound

- $Delt{a}_{\left(0,0\right)}$ = 0.070851 (Straight)
- $Delt{a}_{\left(1,0\right)}$ = 0.062580 (Callable)
- $Delt{a}_{\left(0,1\right)}=$ 0.021306 (Putable)
- $Delt{a}_{\left(1,1\right)}=$ 0.010445 (Callable and Putable)

- $Gamm{a}_{\left(0,0\right)}$ = −0.000245 $Veg{a}_{\left(0,0\right)}$ = −0.150381 (Straight)
- $Gamm{a}_{\left(1,0\right)}$ = −0.000227 $Veg{a}_{\left(1,0\right)}$ = −0.131519 (Callable)
- $Gamm{a}_{\left(0,1\right)}=$ −0.000081 $Veg{a}_{\left(0,1\right)}=$ −0.047370 (Putable)
- $Gamm{a}_{\left(1,1\right)}=$ −0.000060 $Veg{a}_{\left(1,1\right)}=$ −0.024703 (Callable and Putable)

- $Rh{o}_{\left(0,0\right)}$ = 8.781216 (Straight)
- $Rh{o}_{\left(1,0\right)}$ = 5.676433 (Callable)
- $Rh{o}_{\left(0,1\right)}=$ 8.053633 (Putable)
- $Rh{o}_{\left(1,1\right)}=$ 4.645235 (Callable and Putable)

- % Reduction in Delta by (call, put, call and put) = (−11.67%, −69,93%, −85.26%)
- % Reduction in Gamma by (call, put, call and put) = (−7.35%, −66.94%, −75.51%)
- % Reduction in Vega by (call, put, call and put) = (−12.54%, −68.50%, −83.57%)
- % Reduction in Rho by (call, put, call and put) = (−35.36%, −8.29%, −47.10%)

- % Reduction in Delta (by put relative to call) = −65.95%
- % Reduction in Gamma (by put relative to call) = −64.76%
- % Reduction in Vega (by put relative to call) = −63.98%
- % Reduction in Rho (by call relative to put) = −29.52%

#### 5.2.2. Greeks at the Lower Bound

#### 5.2.3. Individual-Security Sensitivity Analysis of the Greeks

#### 5.2.4. Results on the Greeks

#### 5.3. Cross-Sectional Analysis of RPE

#### 5.3.1. Explanatory Variables

#### 5.3.2. Multicollinearity

- Group 1:
- (Delta, Gamma, and Vega)
- Group 2:
- (Rho, Call-Def, Put-Def, and Maturity)
- Group 3:
- (Coupon, Call-Idx, and Put-Idx)

#### 5.3.3. Reduced Set of Explanatory Variables

#### 5.3.4. Cross-Sectional Analysis

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Notes

1 | Alternatively, when the put provision is inactive, bankruptcy alters the stated lower-boundary condition. If the recovery rate for debt is γ and equity is wiped out in bankruptcy, then the domain is [$\gamma B,\infty $) and the corresponding boundary conditions are $f\left(\gamma B,\tau \right)=\gamma B$ and ${f}_{v}\left(\infty ,\tau \right)=0$. This raises the valuation function, i.e., the model’s prediction of the value of debt rises. If however equity retains the value ε in bankruptcy, then the domain is [$\gamma B+\epsilon ,\infty $) and the corresponding boundary conditions are $f\left(\gamma B+\epsilon ,\tau \right)=\gamma B$ and ${f}_{v}\left(\infty ,\tau \right)=0$. This may lower the valuation function if the size of ε is sufficiently large relative to $\gamma B$. But this is unlikely to hold in practice given the low recovery of equity in bankruptcy (Bhandari and Weiss 1996, p. 249). Furthermore, whatever the choice for $\left(\gamma ,\epsilon \right)$, its impact on the valuation function is reduced significantly owing to the impeding put, which lowers the investor’s horizon from about 19 years (average maturity) to 4.2. years (average time to first put). |

2 | Denote Market Value of Equity by MVE, Book Value of Debt by BVD and Book Value of Preferred Stock by BVP. The firm’s market value is then approximated by V = MVE + BVD + BVP. Assuming the book values of debt and preferred are constant, the change in the firm’s market value can be written at dV = dMVE and the instantaneous return in the firm’s value may be written as $dV/V=dMVE/\left(MVE+BVD+BVP\right)$. The standard deviation of equity value (${\sigma}_{E}$) is related to the standard deviation of the firm’s value (${\sigma}_{V}$) using Ito’s Lemma:${\sigma}_{E}={\sigma}_{V}\left(\partial E/\partial V\right)\left(V/E\right).$ In our notation, this expression from Ito’s Lemma can be written as ${\sigma}_{E}={\sigma}_{V}\left(\partial MVE/\partial V\right)\left(MVE+BVD+BVP/MVE\right)={\sigma}_{V}\left(MVE+BVD+BVP/MVE\right)$ since $\partial MVE/\partial V=1.$ Solving for ${\sigma}_{V}$ yields ${\sigma}_{V}={\sigma}_{E}(MVE/(MVE+BVD+BVP\left)\right)$ for the lower bound of the firm’s volatility. Thus, the firm’s volatility ${\sigma}_{V}$ is less than the volatility of equity ${\sigma}_{E}$ under this assumption of price movements, and data for MVE, BVD and BVP are available to calculate the firm’s volatility. |

3 | Denote Market Value of Equity by MVE, Market Value of Debt by MVD and Market Value of Preferred Stock by MVP. If MVD and MVP are perfectly correlated with $MVE$, then $MVD={k}_{D}MVE$ and $MVP={k}_{p}MVE$, where ${k}_{D}$ and ${k}_{p}$ are constants (assuming all three types of securities approach zero concurrently, which holds in contingent-claims valuation theory). The firm’s market value, $V=MVE+MVD+MVP$, may then be written as $V=MVE\left(1+{k}_{D}+{k}_{p}\right)$, and the instantaneous return in the firm’s value may be written as $dV/V=dMVE/MVE$. Thus, the firm’s volatility ${\sigma}_{V}$ equals the volatility of equity ${\sigma}_{E}$ under this assumption of price co-movements, and data for MVE are available to calculate the firm’s volatility. |

4 | For comparison, the cross-sectional analysis of other studies in the literature that evaluate pricing at issuance is listed. Datta et al. (1997) examine initial-day and aftermarket performance of straight-debt. |

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**Figure 1.**Schematic representation of call and put profiles of sample issues. The call profile is stepwise and continuous; the put profile is discontinuous and active either once (solid dot) or at a periodic interval (open dots).

**Figure 2.**Schematic representation of the optimal exercise of the call and put options. $f\left(V,\tau \right)$ represents the bond’s value; $V$ represents the firm’s value; $\tau $ represents the time-to-maturity. The valuation function is represented by the solid lines labeled 1 and 2. Line 1 depicts the valuation function when both the call provision ($\kappa \left(\tau \right)$ and the put provision ($p\left(\tau \right)$ are inactive. Line 2 depicts the valuation function when both provisions are active (two characteristic forms are shown, corresponding to low interest rates and/or volatility—the upper line; high interest rates and/or volatility—the lower line). When the put provision is active, early exercise of the put at ${V}_{min}$ (i.e., before the firm’s value drops to $V=p\left(\tau \right)$) may be optimal for sufficiently high interest rates and/or volatility. Conversely, when the call provision is active, early exercise of the call at ${V}_{max}$ (prior to maturity) may be optimal for sufficiently low interest rates and/or volatility.

**Figure 3.**Relative pricing error (RPE) at the lower- and upper-bound of the firm’s risk. RPE is calculated as (Offer Price-Model Price)/Model Price. Negative values of RPE indicate that the model overprices the market or that the security is underpriced at issuance. The mean RPE is −5.56% (stdev. = 3.59%) at the lower-bound of the firm’s risk and −4.97% (stdev. = 3.77%) at the upper-bound of the firm’s risk.

Industry | Number of Issues (Full Sample) | Number of Issues (In 1998) |
---|---|---|

Agriculture | 1 | 0 |

Communication | 3 | 3 |

Finance | 27 | 10 |

Insurance | 6 | 2 |

Manufacturing | 21 | 15 |

Mineral | 6 | 2 |

Real Estate | 7 | 7 |

Retail | 11 | 10 |

Transportation | 3 | 2 |

Utilities | 63 | 16 |

Total | 159 | 69 |

**Table 2.**Greeks (Delta, Gamma, Vega, and Rho) evaluated at the Upper Bound of the firm’s volatility.

Contract Provision | Index | Estimate | Delta | Gamma | Vega | Rho |
---|---|---|---|---|---|---|

Panel A: | ||||||

Straight | (0, 0) | Mean | 0.070851 | −0.000245 | −0.150381 | 8.781216 |

(9.75) *** | (−2.02) ** | (−8.41) *** | (42.58) *** | |||

Callable | (1, 0) | Mean | 0.062580 | −0.000227 | −0.131519 | 5.676433 |

(9.41) *** | (−1.96) ** | (−7.69) *** | (27.60) *** | |||

Putable | (0, 1) | Mean | 0.021306 | −0.000081 | −0.047370 | 8.053633 |

(7.88) *** | (−2.91) *** | (−6.97) *** | (34.68) *** | |||

Callable and Putable | (1, 1) | Mean | 0.010445 | −0.000060 | −0.024703 | 4.645235 |

(5.37) *** | (−2.95) *** | (−4.11) *** | (29.76) *** | |||

Panel B: | ||||||

Marginal Impact of Call | (1, 0)–(0, 0) | Mean | −0.008271 | 0.000018 | 0.018861 | −3.104787 |

(−6.54) *** | (1.90) ** | (7.50) *** | (−16.06) *** | |||

% Change | −11.67% | −7.35% | −12.54% | −35.36% | ||

Marginal Impact of Put | (0, 1)–(0, 0) | Mean | −0.049545 | 0.000164 | 0.103011 | −0.727584 |

(−8.32) *** | (1.68) ** | (7.51) *** | (−5.76) *** | |||

% Change | −69.93% | −66.94% | −68.50% | −8.29% | ||

Impact of Call and Put | (1, 1)–(0, 0) | Mean | −0.060406 | 0.000185 | 0.125678 | −4.135981 |

(−8.91) *** | (1.75) ** | (8.63) *** | (−20.18) *** | |||

% Change | −85.26% | −75.51% | −83.57% | −47.10% | ||

Panel C: | ||||||

Put relative to Call | (0, 1)–(1, 0) | Mean | −0.041274 | 0.000147 | 0.084149 | |

(−7.35) *** | (1.58) * | (6.33) *** | ||||

% Change | −65.95% | −64.76% | −63.98% | |||

Call Relative to Put | (1, 0)–(0, 1) | Mean | −2.377199 | |||

(−8.87) *** | ||||||

% Change | −29.52% |

**Table 3.**Greeks (Delta, Gamma, Vega, and Rho) evaluated at the Lower Bound of the firm’s volatility.

Contract Provision | Index | Estimate | Delta | Gamma | Vega | Rho |
---|---|---|---|---|---|---|

Panel A: | ||||||

Straight | (0, 0) | Mean | 0.032502 | −0.000211 | −0.045297 | 9.435541 |

(5.71) *** | (−1.73) ** | (−5.41) *** | (43.89) *** | |||

Callable | (1, 0) | Mean | 0.025772 | −0.000192 | −0.036631 | 5.785801 |

(5.49) *** | (−1.67) ** | (−4.87) *** | (25.68) *** | |||

Putable | (0, 1) | Mean | 0.012044 | −0.000061 | −0.014993 | 8.805308 |

(4.16) *** | (−1.98) ** | (−5.22) *** | (36.91) *** | |||

Callable and Putable | (1, 1) | Mean | 0.002882 | −0.000040 | −0.003548 | 4.806963 |

(2.61) *** | (−1.98) ** | (−3.10) *** | (30.46) *** | |||

Panel B: | ||||||

Marginal Impact of Call | (1, 0)–(0, 0) | Mean | −0.006730 | 0.000019 | 0.008666 | −3.649740 |

(−4.15) *** | (2.04) ** | (4.76) *** | (−17.26) *** | |||

% Change | −20.71% | −9.00% | −19.13% | −38.68% | ||

Marginal Impact of Put | (0, 1)–(0, 0) | Mean | −0.020457 | 0.000150 | 0.030304 | −0.630233 |

(−5.03) *** | (1.53) * | (4.47) *** | (−4.77) *** | |||

% Change | −62.94% | −71.09% | −66.90% | −6.68% | ||

Impact of Call and Put | (1, 1)–(0, 0) | Mean | −0.029620 | 0.000171 | 0.041749 | −4.628578 |

(−5.64) *** | (1.62) * | (5.34) *** | (−21.55) *** | |||

% Change | −91.13% | −81.04% | −92.17% | −49.05% | ||

Panel C: | ||||||

Put relative to Call | (0, 1)–(1, 0) | Mean | −0.013727 | 0.000131 | 0.021638 | |

(−3.53) *** | (1.41) * | (3.36) *** | ||||

% Change | −53.26% | −68.23% | −59.07% | |||

Call Relative to Put | (1, 0)–(0, 1) | Mean | −3.019507 | |||

(−10.31) *** | ||||||

% Change | −34.29% |

Impact of Contract Provision * | Estimate | Delta | Gamma | Vega | Rho |
---|---|---|---|---|---|

Marginal Impact of Call: | |||||

$\frac{Gree{k}_{\left(1,0\right)}^{i}-Gree{k}_{\left(0,0\right)}^{i}}{Gree{k}_{\left(0,0\right)}^{i}}$ | Mean | −32.43% | −27.32% | −33.26% | −34.07% |

−11.88 *** | −10.36 *** | −12.18 *** | −17.70 *** | ||

Marginal Impact of Put: | |||||

$\frac{Gree{k}_{\left(0.1\right)}^{i}-Gree{k}_{\left(0,0\right)}^{i}}{Gree{k}_{\left(0,0\right)}^{i}}$ | Mean | −45.38% | −49.24% | −45.48% | −8.86% |

−15.25 *** | −15.04 *** | −15.33 *** | −5.73 *** | ||

Total Impact of Call and Put: | |||||

$\frac{Gree{k}_{\left(1,0\right)}^{i}-Gree{k}_{\left(0,0\right)}^{i}}{Gree{k}_{\left(0,0\right)}^{i}}$ | Mean | −81.18% | −79.00% | −82.67% | −45.12% |

−35.47 *** | −28.70 *** | −37.16 *** | −25.09 *** |

Panel A: Correlation matrix | ||||||||||

Groups: | Delta | Gamma | Vega | Rho | Call-Def | Put-Def | Maturity | Coupon | Call-Idx | Put-Idx |

Group 1:Delta | 1.000 | |||||||||

Gamma | −0.628 | 1.000 | ||||||||

Vega | −0.863 | 0.337 | 1.000 | |||||||

Group 2:Rho | −0.019 | 0.071 | 0.069 | 1.000 | ||||||

Call-Def | 0.097 | 0.048 | −0.115 | 0.771 | 1.000 | |||||

Put-Def | −0.089 | 0.184 | 0.043 | 0.585 | 0.774 | 1.000 | ||||

Maturity | −0.058 | 0.116 | 0.076 | 0.490 | 0.499 | 0.504 | 1.000 | |||

Group 3:Coupon | 0.207 | −0.180 | −0.125 | 0.063 | 0.064 | −0.272 | −0.259 | 1.000 | ||

Call-Idx | 0.277 | −0.266 | −0.141 | 0.145 | −0.041 | −0.410 | −0.125 | 0.730 | 1.000 | |

Put-Idx | 0.216 | −0.154 | −0.168 | 0.114 | −0.098 | −0.265 | 0.024 | 0.117 | 0.436 | 1.000 |

Panel B: Regression estimates for variables in Groups 1, 2 and 3 | ||||||||||

Group 1 | Constant | Delta | Gamma | Vega | Adj R^{2} | |||||

Regression | −0.0502 | −0.9505 | 1.0326 | −0.4204 | 0.181 | |||||

(−16.09) *** | (−2.71) *** | (−0.07) | (−6.38) *** | |||||||

Reduced | −0.0501 | −0.9649 | −0.4233 | 0.186 | ||||||

(−17.07) *** | (−4.43) *** | (−6.00) *** | ||||||||

Group 2 | Constant | Rho | Call-Def | Put-Def | Maturity | Adj R^{2} | ||||

Regression | −0.0258 | −0.0081 | −0.0005 | 0.0023 | 0.0004 | 0.099 | ||||

(−3.20) *** | (−3.48) *** | (−0.27) | (1.50) | (0.90) | ||||||

Reduced | −0.0226 | −0.0059 | 0.088 | |||||||

(−3.08) *** | (−4.03) *** | |||||||||

Group 3 | Constant | Coupon | Call-Idx | Put-Idx | Adj R^{2} | |||||

Regression | −0.0243 | −0.0019 | −0.0170 | −0.0150 | 0.156 | |||||

(−2.26) ** | (−1.26) | (−1.79) * | (−1.70) * | |||||||

Reduced | −0.0128 | −0.0039 | −0.0233 | 0.144 | ||||||

(−1.47) | (−4.00) *** | (−3.04) *** |

Variables | Exogenous | Greeks | Provisions | Combined |
---|---|---|---|---|

1 | 2 | 3 | 4 | |

Intercept | −0.0666 | −0.0264 | −0.0128 | −0.0239 |

(−5.32) *** | (−3.87) *** | (−1.47) | (−1.81) * | |

Equity β | 0.0061 | 0.0024 | ||

(1.57) | (0.75) | |||

Issue Size | −0.1163 | −0.0722 | ||

(−2.22) ** | (−1.40) * | |||

Issue Rating | 0.0023 | 0.0024 | ||

(2.19) ** | (2.73) *** | |||

Delta | −0.9018 | −0.3705 | ||

(−4.30) *** | (−1.76) ** | |||

Vega | −0.3967 | −0.3005 | ||

(−5.83) *** | (−4.79) *** | |||

Rho | −0.0052 | −0.0051 | ||

(−3.79) *** | (−4.30) *** | |||

Coupon | −0.0039 | −0.0023 | ||

(−4.00) *** | (−2.60) *** | |||

Put−Idx | −0.0233 | −0.0193 | ||

(−3.04) *** | (−2.94) *** | |||

Adjusted R^{2} | 0.1303 | 0.2506 | 0.1436 | 0.4325 |

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**MDPI and ACS Style**

Tewari, M.; Ramanlal, P.
Risk Management and Agency Theory: Role of the Put Option in Corporate Bonds. *J. Risk Financial Manag.* **2022**, *15*, 61.
https://doi.org/10.3390/jrfm15020061

**AMA Style**

Tewari M, Ramanlal P.
Risk Management and Agency Theory: Role of the Put Option in Corporate Bonds. *Journal of Risk and Financial Management*. 2022; 15(2):61.
https://doi.org/10.3390/jrfm15020061

**Chicago/Turabian Style**

Tewari, Manish, and Pradipkumar Ramanlal.
2022. "Risk Management and Agency Theory: Role of the Put Option in Corporate Bonds" *Journal of Risk and Financial Management* 15, no. 2: 61.
https://doi.org/10.3390/jrfm15020061