Pricing Vulnerable European Options

Chaun-Ju Wang, November 1, 2007 1 / 35

Pricing Vulnerable European Options When the

Option’s Payoff Can Increase the Risk of Financial

Distress

Peter Klein, Michael InglisJournal of Banking & Finance

presenter: Chuan-Ju Wang

Outline

❖ Outline

Introduction

The model

Valuation equations

Valuation methods

Numerical examples

Conclusion


● Introduction

● The model

● Valuation equations

● Valuation methods

● Numerical examples

● Conclusion

Introduction

❖ Outline

Introduction

❖ Vulnerable options

❖ Related works❖ The idea of thispaper

The model

Valuation equations

Valuation methods

Numerical examples

Conclusion


Vulnerable options

❖ Outline

Introduction



The model

Valuation equations

Valuation methods

Numerical examples

Conclusion


● Many financial institutions actively trading derivativecontract with their corporate clients as well as with otherfinancial institutions in the over-the-counter (OTC)markets.

● No exchange or cleaning house to ensure that both partiesto a contract honor their obligations.

● The holder’s of these contracts are vulnerable tocounter-party credit risk.

Related works

❖ Outline

Introduction



The model

Valuation equations

Valuation methods

Numerical examples

Conclusion


● Most of the literature on vulnerable options assumes thatfinancial distress occurs when the value of writer’s assetsdrop below the value of its other liabilities.

● This assumption ignores the potential liability created bythe option itself.

Related works (cont.)

❖ Outline

Introduction



The model

Valuation equations

Valuation methods

Numerical examples

Conclusion


● Johnson and Stulz (1987)

✦ Allowing the occurrence of financial distress to dependon the value of the option that has been written.

✦ In the event of financial distress, they assume that theoption holder receives all the assets of the optionwriter.

Related works (cont.)

❖ Outline

Introduction



The model

Valuation equations

Valuation methods

Numerical examples

Conclusion


● Klein (1996)

✦ Default boundary does not depend on the value of theoption itself (fixed default boundary).

✦ Allowing for the presence of other liabilities in thecapital structure of the option writer.

● Rich (1996)

✦ Allowing the default boundary to be stochastic.

✦ But not explicitly connect to the stochastic boundaryto the value of the option that has been written.

The idea of this paper

❖ Outline

Introduction



The model

Valuation equations

Valuation methods

Numerical examples

Conclusion


● Allowing for the presence of other liabilities in the capitalstructure of the option writer while recognizing the growthin the value of the option itself may also cause financialdistress.

● Default barrier can be stochastic.

✦ A fixed component represents the other liabilities ofthe option writer.

✦ A stochastic component measures the potential payoffon the option itself.

The model

❖ Outline

Introduction

The model

❖ Assumption

Valuation equations

Valuation methods

Numerical examples

Conclusion


Assumption

❖ Outline

Introduction

The model

❖ Assumption

Valuation equations

Valuation methods

Numerical examples

Conclusion


● Summarizing the assumptions underlying the Klein (1996)model after appropriate adjustments to incorporate thevariable default boundary (VDB) condition.

Assumption (cont.)

❖ Outline

Introduction

The model

❖ Assumption

Valuation equations

Valuation methods

Numerical examples

Conclusion


Assumption (cont.)

❖ Outline

Introduction

The model

❖ Assumption

Valuation equations

Valuation methods

Numerical examples

Conclusion


Valuation equations

❖ Outline

Introduction

The model

Valuation equations

❖ Johnson and Stulz(1987)

❖ Klein (1996)

❖ Model of thispaper

Valuation methods

Numerical examples

Conclusion


Johnson and Stulz (1987)

❖ Outline

Introduction

The model

Valuation equations


❖ Klein (1996)


Valuation methods

Numerical examples

Conclusion


● Johnson and Stulz (1987) pricing equation of vulnerableEuropean calls can be written as

c = e−r(T−t)

[

E∗

{

ST −K ST ≥ K,VT ≥ ST −K

VT ST ≥ K,VT < ST −K

0 otherwise

}]

. (3)

Klein (1996)

❖ Outline

Introduction

The model

Valuation equations


❖ Klein (1996)


Valuation methods

Numerical examples

Conclusion


● Klein (1996) pricing equation of vulnerable European callscan be written as

c = e−r(T−t)

[

E∗

{

ST −K ST ≥ K,VT ≥ D∗

(1− α)VTST−K

D∗ST ≥ K,VT < D

∗

0 otherwise

}]

. (4)

Model of this paper

❖ Outline

Introduction

The model

Valuation equations


❖ Klein (1996)


Valuation methods

Numerical examples

Conclusion


● The pricing equation for vulnerable European calls in thispaper’s framework can be written as

c = e−r(T−t)

[

E∗

{

ST −K ST ≥ K,VT ≥ D∗ + ST −K

(1− α)VTST−K

D∗+ST−KST ≥ K,VT < D

∗ + ST −K

0 otherwise

}]

. (5)

Valuation methods

❖ Outline

Introduction

The model

Valuation equations

Valuation methods

❖ Numerical method❖ Approximateanalytical solution

Numerical examples

Conclusion


Numerical method

❖ Outline

Introduction

The model

Valuation equations

Valuation methods


Numerical examples

Conclusion


● Three-dimension binomial tree

● Orthogonal the two process to ensure zero correlationbetween the two state variables.

Approximate analytical solution

❖ Outline

Introduction

The model

Valuation equations

Valuation methods


Numerical examples

Conclusion


● Performing the standard log transformation and thenemploying a first order Taylor series approximation tolinearize the boundary conditions.

● The denominator in the second term of Eq.(5) must also belinearized through a first order Taylor series approximation.

● A standard rotation as outlined in Abramowitz and Stegun(1972) is used to eliminate S from the boundary conditionfor V , which enables us to rewrite the approximation interms of the cumulative bivariate normal distribution asfollows:

c=SN2(a1,b1,δ)−Ke−r(T−t)N2(a2,b2,δ)+

(1−α)SV exp

((

rσ2V

2 +(ρ−m)σSσV

)

(T−t)+m2

)

D∗−K+m1N2(a3,b3,−δ)−

(1−α)KV exp(m2)

D∗−K+m1N2(a4,b4,−δ). (6)

Approximate analytical solution (cont.)

❖ Outline

Introduction

The model

Valuation equations

Valuation methods


Numerical examples

Conclusion


● The approximation valuation equation depends on thepoint (p) around which the Taylor series is expanded.

✦ If D∗ = K, the valuation equation does not depend onthe point of expansion p.

■ The barrier depends only upon ln(ST ) which, afterlog transformation is already linear.


❖ Outline

Introduction

The model

Valuation equations

Valuation methods


Numerical examples

Conclusion



✦ If D∗ > K, the true default barrier is the convex lineshow in Fig. 1.

■ Since this line corresponds to the probability thatfinancial distress will occur.

■ An approximation will underestimate the effect ofcredit risk on the value of the vulnerable calloption.

■ The optimal value for the expansion point (p) willbe the value that minimizes the value ofvulnerable option.


❖ Outline

Introduction

The model

Valuation equations

Valuation methods


Numerical examples

Conclusion


● Fig. 1: Integration region for the vulnerable European callwhen D∗ > K.


❖ Outline

Introduction

The model

Valuation equations

Valuation methods


Numerical examples

Conclusion



✦ If D∗ < K, the correct default barrier is concave.

■ An approximation based on a tangent willunderestimate the value of the vulnerable calloption as shown in Fig. 2.

■ The optimal value for p will be the value thatmaximized the value of the vulnerable option.


❖ Outline

Introduction

The model

Valuation equations

Valuation methods


Numerical examples

Conclusion


● Fig. 2: Integration region for the vulnerable European callwhen D∗ < K.

Numerical examples

❖ Outline

Introduction

The model

Valuation equations

Valuation methods

Numerical examples

❖ Numericalexamples

Conclusion


Numerical examples

❖ Outline

Introduction

The model

Valuation equations

Valuation methods

Numerical examples


Conclusion


● Table 1: A comparison of FDB vs VDB

Numerical examples (cont.)

❖ Outline

Introduction

The model

Valuation equations

Valuation methods

Numerical examples


Conclusion


● Fig. 3: Vulnerable call values as a function of option’smoneyness: a comparison of the FDB and VDB models(base case)


❖ Outline

Introduction

The model

Valuation equations

Valuation methods

Numerical examples


Conclusion


● Fig. 4: Vulnerable call values as a function of option’smoneyness: a comparison of the FDB and VDB models(base case)


❖ Outline

Introduction

The model

Valuation equations

Valuation methods

Numerical examples


Conclusion


● Fig. 5: Vulnerable call values as a function of option’swriter’s assets: a comparison of the FDB and VDB models(base case)


❖ Outline

Introduction

The model

Valuation equations

Valuation methods

Numerical examples


Conclusion


● Fig. 6: Vulnerable call values as a function of option’swriter’s assets: a comparison of the FDB and VDB models(base case)


❖ Outline

Introduction

The model

Valuation equations

Valuation methods

Numerical examples


Conclusion


● Fig. 7: Vulnerable call values as a function of option’swriter’s assets: a comparison of the FDB and VDB models(out-of-the-money option)


❖ Outline

Introduction

The model

Valuation equations

Valuation methods

Numerical examples


Conclusion


● Fig. 8: Vulnerable call values as a function of option’swriter’s assets: a comparison of the FDB and VDB models(in-the-money option)


❖ Outline

Introduction

The model

Valuation equations

Valuation methods

Numerical examples


Conclusion


● Fig. 9: Vulnerable call values as a function of option’swriter’s assets: a comparison of the FDB and VDB models(ρ = 0.5)

Conclusion

❖ Outline

Introduction

The model

Valuation equations

Valuation methods

Numerical examples

Conclusion

❖ Conclusion


Conclusion

❖ Outline

Introduction

The model

Valuation equations

Valuation methods

Numerical examples

Conclusion

❖ Conclusion


● This paper extends the vulnerable European option pricingresults of Johnson and Stulz (1987) and Klein (1996).

✦ Allowing for other liabilities in the capital structure ofthe option writer.

✦ The default boundary depends on the payoff of theoption itself.

✦ Allowing the pay-out ratio to be linked to the value ofoption writer’s assets, and for correlation between theassets of the option writer and the asset underlyingthe option.

Economy & Finance

Pricing Vulnerable European Options