Credit Risk, Capital Structure, and the Pricing of Equity Options: Hanke, Michael: 9783211005200: Book: Business & Economics: eCampus.com

Option pricing in firm-value-based ("structural") credit risk models is addressed in this book. Using modern techniques instead of directly solving partial differential equations (the main approach in the literature), closed-form pricing formulae for options on equity are derived for a range of well-known models. A common feature of these models is the assumption of an exogenously given firm value process, which leads to an endogenous stock price process that depends directly on the firm's capital structure. This allows for the study of credit risk effects in option prices and option price changes resulting from changes in a firm's capital structure. Numerical results illustrate the implications of these credit risk models.

List of Symbols

xiv

1 Option Pricing with an Exogenous Stock Price Process

(22)

1.1 A One-Period Pricing Model

(8)

1.1.1 Model Description

(1)

1.1.2 Replicating Portfolios

(1)

1.1.3 Absence of Arbitrage

(1)

1.1.4 Risk-Neutral Valuation and Equivalent Martingale Measures

(2)

1.1.5 Numerical Example

(2)

1.2 The Binomial Model

(4)

1.2.1 Model Description

(1)

1.2.2 Replicating Portfolios

(1)

1.2.3 Absence of Arbitrage

(1)

1.2.4 Martingale Measure

(1)

1.2.5 Numerical Example

(2)

1.3 The Black-Scholes-Merton Model

(10)

1.3.1 Model Description

(2)

1.3.2 Replicating Portfolios

(1)

1.3.3 Absence of Arbitrage and Risk-Neutral Pricing

(2)

1.3.4 "Measure-Independent" Derivation of Probabilities

(3)

1.3.5 Extension: Continuous Dividend Yield

(2)

2 Option Pricing with an Endogenous Stock Price Process

(16)

2.1 The Extended One-Period Option Pricing Model with Endogenous Stock Price Process

(3)

2.1.1 Model Description

(1)

2.1.2 Risk-Neutral Valuation of Corporate Securities

(1)

2.1.3 Risk-Neutral Valuation of Options

(1)

2.1.4 Numerical Examples

(1)

2.2 The Extended Binomial Option Pricing Model with Endogenous Stock Price Process

(5)

2.2.1 Model Description

(1)

2.2.2 Risk-Neutral Valuation of Corporate Securities

(1)

2.2.3 Risk-Neutral Valuation of Options

(1)

2.2.4 Numerical Examples

(3)

2.3 The Extended Black-Scholes Model with Endogenous Stock Price Process

(8)

2.3.1 Model Description

(1)

2.3.2 Risk-Neutral Valuation of Corporate Securities

(2)

2.3.2.1 Equity

(1)

2.3.2.2 Corporate Bond

(1)

2.3.3 Risk-Neutral Valuation of Options

(1)

2.3.4 "Measure-Independent" Derivation of Probabilities

(1)

2.3.5 Numerical Examples

(4)

3 Exotic Options

(20)

3.1 Non-Barrier Exotic Options

(1)

3.2 Barrier Options

(9)

3.2.1 Preliminaries

(1)

3.2.2 Option Types and Barrier-Dependent Probabilities

(7)

3.3 Applications: Barrier Heavisides, Calls and Puts

(5)

3.3.1 Barrier Heavisides

(1)

3.3.2 Barrier Calls

(1)

3.3.3 Barrier Puts

(1)

3.3.4 Relation to the Standard Approach

(2)

3.4 Numerical Examples

(5)

4 A Probabilistic, Firm Value Based Security Pricing Framework

(32)

4.1 Ericsson and Reneby (1998)

(10)

4.1.1 Assumptions

(1)

4.1.2 Valuation of the Building Blocks

(4)

4.1.3 Results

(4)

4.2 Ericsson and Reneby (2001)

(5)

4.3 Additional Building Blocks within the Probabilistic Framework

(17)

5 A Review of Firm Value Based Security Pricing Models from a Probabilistic Perspective

(24)

5.1 Finite-Maturity Discount Bonds, No Intermediate Default (Merton 1974)

(1)

5.2 Finite-Maturity, Continuous-Coupon Bonds, Intermediate Default (Black and Cox 1976)

(2)

5.3 Finite-Maturity, Discrete-Coupon Bonds, Intermediate Default (Geske 1977)

(1)

5.4 Finite-Maturity, Convertible Discount Bonds, No Intermediate Default (Ingersoll 1977a)

(3)

5.4.1 Convertible Discount Bonds

(1)

5.4.2 Callable Convertible Discount Bonds

(2)

5.5 Finite-Maturity, Discrete-Coupon Bonds, Intermediate Default, Discrete Dividends, Taxes, Stochastic Interest Rates (Brennan and Schwartz 1977,1978,1980)

(1)

5.6 Warrants (Galai and Schneller 1978)

(2)

5.7 Empirical Study of Firm Value Based Pricing of Corporate Bonds (Jones, Mason and Rosenfeld 1984)

101

(1)

5.8 Finite-Maturity, Continuous-Coupon Bonds, Intermediate Default, CIR Interest Rates (Kim, Ramaswamy and Sundaresan 1993)

102

(1)

5.9 Finite-Maturity, Continuous-Coupon Bonds, Intermediate Default, Vasicek Interest Rates (Longstaff and Schwartz 1995)

103

(1)

5.10 Infinite-Maturity, Continuous-Coupon Bonds, Taxes, Intermediate Default, Bankruptcy Costs (Leland 1994)

104

(3)

5.11 Finite-Maturity, Continuous-Coupon Bonds, Taxes, Intermediate Default, Bankruptcy Costs (Leland and Toft 1996)

107

(3)

5.12 Finite-Average-Maturity, Continuous-Coupon Bonds, Taxes, Intermediate Default, Bankruptcy Costs, Costly Debt Issuance (Leland 1998)

110

(1)

5.13 "Model A" : Finite-Maturity, Continuous-Coupon Bonds, Exponentially Increasing Debt, Intermediate Default, Bankruptcy Costs, Taxes, Deviations from Absolute Priority (Extended Leland and Toft)

111

(4)

6 Extension of the Probabilistic Security Pricing Framework to Derivative Securities

115

(24)

6.1 Ericsson and Reneby (1996)

115

(6)

6.1.1 Assumptions and Results

115

(3)

6.1.2 Correcting the Ericsson-Reneby (1996) Results

118

(3)

6.2 Reneby (1998)

121

(1)

6.3 Extending the Ericsson-Reneby (1996) Results

122

(17)

6.3.1 Lifting Assumptions

122

(1)

6.3.2 Down-and-Out Underlyings Other than Calls or Heavisides

123

(10)

6.3.3 Put Options on Down-and-Out Underlyings

133

(1)

6.3.4 Underlyings of the Up-Barrier Type

134

(5)

7 Review of Firm Value Based Pricing Models for Equity Derivatives from a Probabilistic Perspective

139

(8)

7.1 Option Pricing Extension of Merton (1974): Geske (1979)

139

(2)

7.2 Option Pricing Extension of Leland (1994): Toft and Prucyk (1997)

141

(2)

7.3 Option Pricing Extension of Galai and Schneller (1978): Hanke and P�tzelberger (2002)

143

(4)

8 Option Pricing Extensions for Several Classical Capital Structure Models

147

(8)

8.1 Model 1: Option Pricing Extension of Black and Cox (1976)

147

(2)

8.2 Option Pricing Extensions of Ingersoll (1977a)

149

(1)

8.2.1 Model 2: Convertible Discount Bonds

149

(1)

8.2.2 Model 3: Callable Convertible Discount Bonds

150

(1)

8.3 Model 4: Option Pricing Extension of Leland and Toft (1996)

150

(1)

8.4 Model 5: Option Pricing Extension of (a Restricted Version of) Leland (1998)

151

(1)

8.5 Model 6: Option Pricing Extension of Ericsson and Reneby (2001)

152

(1)

8.6 Model 7: Option Pricing Extension of Model A

153

(2)

9 Capital Structure Effects in Option Prices - The Static Case

155

(36)

9.1 Pricing Biases of the Black-Scholes Model - "Stylized Facts"

156

(2)

9.1.1 The Volatility Smile

156

(1)

9.1.2 The Term Structure of Volatilities

157

(1)

9.1.3 The Debt-Maturity Term Structure of Volatilities

158

(1)

9.2 Pure Debt-Equity Capital Structures

158

(28)

9.2.1 Merton (1974) /Geske (1979)

158

(6)

9.2.1.1 Volatility Smile

159

(3)

9.2.1.2 Term Structure of Volatilities

162

(2)

9.2.1.3 Debt-Maturity Term Structure of Volatilities

164

(1)

9.2.2 Black and Cox (1976)/Model 1

164

(8)

9.2.2.1 Positive Asset Payouts, No Barrier

165

(4)

9.2.2.2 Positive and Exponentially Increasing Barrier, No Asset Payouts

169

(3)

9.2.3 Leland (1994) / Toft and Prucyk (1997)

172

(1)

9.2.4 (Restricted) Leland (1998) / Model 5

173

(4)

9.2.5 Leland and Toft (1996) / Model 4

177

(3)

9.2.6 Ericsson and Reneby (2001) / Model 6

180

(2)

9.2.7 Model A / Model 7

182

(4)

9.2.8 Conclusions

186

(1)

9.3 Capital Structure Models with Convertibles

186

(5)

9.3.1 Ingersoll (1977a): Convertible Discount Bonds / Model 2

187

(1)

9.3.2 Ingersoll (1977a): Callable Convertible Discount Bonds / Model 3

187

(4)

10 Option Pricing Effects of Changes in a Firm's Capital Structure

191

(10)

10.1 Changes within Model 7

192

(5)

10.1.1 Changes in the Level of Debt

192

(2)

10.1.2 Changes in the Growth Rate of Debt

194

(1)

10.1.3 Changes in Debt Maturity

195

(1)

10.1.4 Changes in the Level of Debt Protection

196

(1)

10.2 Changes within Hanke and P�tzelberger (2002)

197

(4)

11 Conclusions and Directions for Further Research

201

(2)

Bibliography

203




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