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Pricing vulnerable European options when the option’s payoff can increase the risk of financial distressPeter Klein, Michael InglisJournal of Banking & Finance
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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
Chaun-Ju Wang, November 1, 2007 2 / 35
● 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
Chaun-Ju Wang, November 1, 2007 3 / 35
Vulnerable options
❖ Outline
Introduction
❖ Vulnerable options
❖ Related works❖ The idea of thispaper
The model
Valuation equations
Valuation methods
Numerical examples
Conclusion
Chaun-Ju Wang, November 1, 2007 4 / 35
● 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
❖ Vulnerable options
❖ Related works❖ The idea of thispaper
The model
Valuation equations
Valuation methods
Numerical examples
Conclusion
Chaun-Ju Wang, November 1, 2007 5 / 35
● 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
❖ Vulnerable options
❖ Related works❖ The idea of thispaper
The model
Valuation equations
Valuation methods
Numerical examples
Conclusion
Chaun-Ju Wang, November 1, 2007 6 / 35
● 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
❖ Vulnerable options
❖ Related works❖ The idea of thispaper
The model
Valuation equations
Valuation methods
Numerical examples
Conclusion
Chaun-Ju Wang, November 1, 2007 7 / 35
● 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
❖ Vulnerable options
❖ Related works❖ The idea of thispaper
The model
Valuation equations
Valuation methods
Numerical examples
Conclusion
Chaun-Ju Wang, November 1, 2007 8 / 35
● 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
Chaun-Ju Wang, November 1, 2007 9 / 35
Assumption
❖ Outline
Introduction
The model
❖ Assumption
Valuation equations
Valuation methods
Numerical examples
Conclusion
Chaun-Ju Wang, November 1, 2007 10 / 35
● 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
Chaun-Ju Wang, November 1, 2007 11 / 35
Assumption (cont.)
❖ Outline
Introduction
The model
❖ Assumption
Valuation equations
Valuation methods
Numerical examples
Conclusion
Chaun-Ju Wang, November 1, 2007 12 / 35
Valuation equations
❖ Outline
Introduction
The model
Valuation equations
❖ Johnson and Stulz(1987)
❖ Klein (1996)
❖ Model of thispaper
Valuation methods
Numerical examples
Conclusion
Chaun-Ju Wang, November 1, 2007 13 / 35
Johnson and Stulz (1987)
❖ Outline
Introduction
The model
Valuation equations
❖ Johnson and Stulz(1987)
❖ Klein (1996)
❖ Model of thispaper
Valuation methods
Numerical examples
Conclusion
Chaun-Ju Wang, November 1, 2007 14 / 35
● 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
❖ Johnson and Stulz(1987)
❖ Klein (1996)
❖ Model of thispaper
Valuation methods
Numerical examples
Conclusion
Chaun-Ju Wang, November 1, 2007 15 / 35
● 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
❖ Johnson and Stulz(1987)
❖ Klein (1996)
❖ Model of thispaper
Valuation methods
Numerical examples
Conclusion
Chaun-Ju Wang, November 1, 2007 16 / 35
● 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
Chaun-Ju Wang, November 1, 2007 17 / 35
Numerical method
❖ Outline
Introduction
The model
Valuation equations
Valuation methods
❖ Numerical method❖ Approximateanalytical solution
Numerical examples
Conclusion
Chaun-Ju Wang, November 1, 2007 18 / 35
● 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 method❖ Approximateanalytical solution
Numerical examples
Conclusion
Chaun-Ju Wang, November 1, 2007 19 / 35
● 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 method❖ Approximateanalytical solution
Numerical examples
Conclusion
Chaun-Ju Wang, November 1, 2007 20 / 35
● 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.
Approximate analytical solution (cont.)
❖ Outline
Introduction
The model
Valuation equations
Valuation methods
❖ Numerical method❖ Approximateanalytical solution
Numerical examples
Conclusion
Chaun-Ju Wang, November 1, 2007 21 / 35
● The approximation valuation equation depends on thepoint (p) around which the Taylor series is expanded.
✦ 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.
Approximate analytical solution (cont.)
❖ Outline
Introduction
The model
Valuation equations
Valuation methods
❖ Numerical method❖ Approximateanalytical solution
Numerical examples
Conclusion
Chaun-Ju Wang, November 1, 2007 22 / 35
● Fig. 1: Integration region for the vulnerable European callwhen D∗ > K.
Approximate analytical solution (cont.)
❖ Outline
Introduction
The model
Valuation equations
Valuation methods
❖ Numerical method❖ Approximateanalytical solution
Numerical examples
Conclusion
Chaun-Ju Wang, November 1, 2007 23 / 35
● The approximation valuation equation depends on thepoint (p) around which the Taylor series is expanded.
✦ 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.
Approximate analytical solution (cont.)
❖ Outline
Introduction
The model
Valuation equations
Valuation methods
❖ Numerical method❖ Approximateanalytical solution
Numerical examples
Conclusion
Chaun-Ju Wang, November 1, 2007 24 / 35
● 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
Chaun-Ju Wang, November 1, 2007 25 / 35
Numerical examples
❖ Outline
Introduction
The model
Valuation equations
Valuation methods
Numerical examples
❖ Numericalexamples
Conclusion
Chaun-Ju Wang, November 1, 2007 26 / 35
● Table 1: A comparison of FDB vs VDB
Numerical examples (cont.)
❖ Outline
Introduction
The model
Valuation equations
Valuation methods
Numerical examples
❖ Numericalexamples
Conclusion
Chaun-Ju Wang, November 1, 2007 27 / 35
● Fig. 3: Vulnerable call values as a function of option’smoneyness: a comparison of the FDB and VDB models(base case)
Numerical examples (cont.)
❖ Outline
Introduction
The model
Valuation equations
Valuation methods
Numerical examples
❖ Numericalexamples
Conclusion
Chaun-Ju Wang, November 1, 2007 28 / 35
● Fig. 4: Vulnerable call values as a function of option’smoneyness: a comparison of the FDB and VDB models(base case)
Numerical examples (cont.)
❖ Outline
Introduction
The model
Valuation equations
Valuation methods
Numerical examples
❖ Numericalexamples
Conclusion
Chaun-Ju Wang, November 1, 2007 29 / 35
● Fig. 5: Vulnerable call values as a function of option’swriter’s assets: a comparison of the FDB and VDB models(base case)
Numerical examples (cont.)
❖ Outline
Introduction
The model
Valuation equations
Valuation methods
Numerical examples
❖ Numericalexamples
Conclusion
Chaun-Ju Wang, November 1, 2007 30 / 35
● Fig. 6: Vulnerable call values as a function of option’swriter’s assets: a comparison of the FDB and VDB models(base case)
Numerical examples (cont.)
❖ Outline
Introduction
The model
Valuation equations
Valuation methods
Numerical examples
❖ Numericalexamples
Conclusion
Chaun-Ju Wang, November 1, 2007 31 / 35
● 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)
Numerical examples (cont.)
❖ Outline
Introduction
The model
Valuation equations
Valuation methods
Numerical examples
❖ Numericalexamples
Conclusion
Chaun-Ju Wang, November 1, 2007 32 / 35
● 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)
Numerical examples (cont.)
❖ Outline
Introduction
The model
Valuation equations
Valuation methods
Numerical examples
❖ Numericalexamples
Conclusion
Chaun-Ju Wang, November 1, 2007 33 / 35
● 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
Chaun-Ju Wang, November 1, 2007 34 / 35
Conclusion
❖ Outline
Introduction
The model
Valuation equations
Valuation methods
Numerical examples
Conclusion
❖ Conclusion
Chaun-Ju Wang, November 1, 2007 35 / 35
● 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.
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