Pricing vulnerable European options when the option’s payoff can increase the risk of financial distressPeter Klein, Michael InglisJournal of Banking & Finance
Call Girls In Yusuf Sarai Women Seeking Men 9654467111
Pricing Vulnerable European Options When Payoff Can Increase Financial Distress
1. Pricing Vulnerable European Options When the
Option’s Payoff Can Increase the Risk of Financial
Distress
Peter Klein, Michael Inglis
Journal of Banking & Finance
presenter: Chuan-Ju Wang
Chaun-Ju Wang, November 1, 2007 1 / 35
2. Outline
y Outline
Introduction
q
Introduction
The model
The model
q
Valuation equations
Valuation methods
Valuation equations
q
Numerical examples
Valuation methods
q
Conclusion
Numerical examples
q
Conclusion
q
Chaun-Ju Wang, November 1, 2007 2 / 35
3. y Outline
Introduction
y Vulnerable options
y Related works
y The idea of this
paper
The model
Valuation equations
Introduction
Valuation methods
Numerical examples
Conclusion
Chaun-Ju Wang, November 1, 2007 3 / 35
4. Vulnerable options
y Outline
Many financial institutions actively trading derivative
q
Introduction
contract with their corporate clients as well as with other
y Vulnerable options
y Related works
financial institutions in the over-the-counter (OTC)
y The idea of this
markets.
paper
The model
No exchange or cleaning house to ensure that both parties
q
Valuation equations
to a contract honor their obligations.
Valuation methods
Numerical examples
The holder’s of these contracts are vulnerable to
q
Conclusion
counter-party credit risk.
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5. Related works
y Outline
Most of the literature on vulnerable options assumes that
q
Introduction
financial distress occurs when the value of writer’s assets
y Vulnerable options
y Related works
drop below the value of its other liabilities.
y The idea of this
paper
This assumption ignores the potential liability created by
q
The model
the option itself.
Valuation equations
Valuation methods
Numerical examples
Conclusion
Chaun-Ju Wang, November 1, 2007 5 / 35
6. Related works (cont.)
y Outline
Johnson and Stulz (1987)
q
Introduction
y Vulnerable options
3 Allowing the occurrence of financial distress to depend
y Related works
y The idea of this
on the value of the option that has been written.
paper
The model
3 In the event of financial distress, they assume that the
Valuation equations
option holder receives all the assets of the option
Valuation methods
writer.
Numerical examples
Conclusion
Chaun-Ju Wang, November 1, 2007 6 / 35
7. Related works (cont.)
y Outline
Klein (1996)
q
Introduction
y Vulnerable options
3 Default boundary does not depend on the value of the
y Related works
y The idea of this
option itself (fixed default boundary).
paper
The model
3 Allowing for the presence of other liabilities in the
Valuation equations
capital structure of the option writer.
Valuation methods
Numerical examples
Rich (1996)
q
Conclusion
3 Allowing the default boundary to be stochastic.
3 But not explicitly connect to the stochastic boundary
to the value of the option that has been written.
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8. The idea of this paper
y Outline
Allowing for the presence of other liabilities in the capital
q
Introduction
structure of the option writer while recognizing the growth
y Vulnerable options
y Related works
in the value of the option itself may also cause financial
y The idea of this
distress.
paper
The model
Default barrier can be stochastic.
q
Valuation equations
Valuation methods
3 A fixed component represents the other liabilities of
Numerical examples
the option writer.
Conclusion
3 A stochastic component measures the potential payoff
on the option itself.
Chaun-Ju Wang, November 1, 2007 8 / 35
9. y Outline
Introduction
The model
y Assumption
Valuation equations
Valuation methods
Numerical examples
The model
Conclusion
Chaun-Ju Wang, November 1, 2007 9 / 35
10. Assumption
y Outline
Summarizing the assumptions underlying the Klein (1996)
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Introduction
model after appropriate adjustments to incorporate the
The model
variable default boundary (VDB) condition.
y Assumption
Valuation equations
Valuation methods
Numerical examples
Conclusion
Chaun-Ju Wang, November 1, 2007 10 / 35
13. y Outline
Introduction
The model
Valuation equations
y Johnson and Stulz
(1987)
y Klein (1996)
y Model of this
Valuation equations
paper
Valuation methods
Numerical examples
Conclusion
Chaun-Ju Wang, November 1, 2007 13 / 35
14. Johnson and Stulz (1987)
y Outline
Johnson and Stulz (1987) pricing equation of vulnerable
q
Introduction
European calls can be written as
The model
Valuation equations
y Johnson and Stulz
(1987)
y Klein (1996)
y Model of this
ST − K ST ≥ K, VT ≥ ST − K
paper
c = e−r(T −t) E ∗ (3)
VT ST ≥ K, VT < ST − K .
Valuation methods
0 otherwise
Numerical examples
Conclusion
Chaun-Ju Wang, November 1, 2007 14 / 35
15. Klein (1996)
y Outline
Klein (1996) pricing equation of vulnerable European calls
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Introduction
can be written as
The model
Valuation equations
y Johnson and Stulz
(1987)
y Klein (1996)
y Model of this ST − K ST ≥ K, VT ≥ D∗
paper ST −K
c = e−r(T −t) E ∗ (4)
.
(1 − α)VT ST ≥ K, VT < D∗
D∗
Valuation methods
0 otherwise
Numerical examples
Conclusion
Chaun-Ju Wang, November 1, 2007 15 / 35
16. Model of this paper
y Outline
The pricing equation for vulnerable European calls in this
q
Introduction
paper’s framework can be written as
The model
Valuation equations
y Johnson and Stulz
(1987)
y Klein (1996)
y Model of this
ST ≥ K, VT ≥ D∗ + ST − K
ST − K
paper
ST −K
(1 − α)VT ST ≥ K, VT < D∗ + ST − K
c = e−r(T −t) E ∗ (5)
.
∗ +S −K
D
Valuation methods T
0 otherwise
Numerical examples
Conclusion
Chaun-Ju Wang, November 1, 2007 16 / 35
17. y Outline
Introduction
The model
Valuation equations
Valuation methods
y Numerical method
y Approximate
analytical solution
Valuation methods
Numerical examples
Conclusion
Chaun-Ju Wang, November 1, 2007 17 / 35
18. Numerical method
y Outline
Three-dimension binomial tree
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Introduction
The model
Orthogonal the two process to ensure zero correlation
q
Valuation equations
between the two state variables.
Valuation methods
y Numerical method
y Approximate
analytical solution
Numerical examples
Conclusion
Chaun-Ju Wang, November 1, 2007 18 / 35
19. Approximate analytical solution
y Outline
Performing the standard log transformation and then
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Introduction
employing a first order Taylor series approximation to
The model
linearize the boundary conditions.
Valuation equations
Valuation methods
The denominator in the second term of Eq.(5) must also be
q
y Numerical method
y Approximate
linearized through a first order Taylor series approximation.
analytical solution
Numerical examples
A standard rotation as outlined in Abramowitz and Stegun
q
Conclusion
(1972) is used to eliminate S from the boundary condition
for V , which enables us to rewrite the approximation in
terms of the cumulative bivariate normal distribution as
follows:
c=SN2 (a1 ,b1 ,δ)−Ke−r(T −t) N2 (a2 ,b2 ,δ)+
rσ 2
V
(1−α)SV exp 2 +(ρ−m)σS σV (T −t)+m2
N2 (a3 ,b3 ,−δ)−
D ∗ −K+m1
(1−α)KV exp(m2 )
N2 (a4 ,b4 ,−δ). (6)
D ∗ −K+m1
Chaun-Ju Wang, November 1, 2007 19 / 35
20. Approximate analytical solution (cont.)
y Outline
The approximation valuation equation depends on the
q
Introduction
point (p) around which the Taylor series is expanded.
The model
Valuation equations
3 If D ∗ = K, the valuation equation does not depend on
Valuation methods
the point of expansion p.
y Numerical method
y Approximate
The barrier depends only upon ln(ST ) which, after
analytical solution s
log transformation is already linear.
Numerical examples
Conclusion
Chaun-Ju Wang, November 1, 2007 20 / 35
21. Approximate analytical solution (cont.)
y Outline
The approximation valuation equation depends on the
q
Introduction
point (p) around which the Taylor series is expanded.
The model
Valuation equations
3 If D ∗ > K, the true default barrier is the convex line
Valuation methods
show in Fig. 1.
y Numerical method
y Approximate
Since this line corresponds to the probability that
analytical solution s
financial distress will occur.
Numerical examples
Conclusion
An approximation will underestimate the effect of
s
credit risk on the value of the vulnerable call
option.
The optimal value for the expansion point (p) will
s
be the value that minimizes the value of
vulnerable option.
Chaun-Ju Wang, November 1, 2007 21 / 35
22. Approximate analytical solution (cont.)
y Outline
Fig. 1: Integration region for the vulnerable European call
q
Introduction
when D∗ > K.
The model
Valuation equations
Valuation methods
y Numerical method
y Approximate
analytical solution
Numerical examples
Conclusion
Chaun-Ju Wang, November 1, 2007 22 / 35
23. Approximate analytical solution (cont.)
y Outline
The approximation valuation equation depends on the
q
Introduction
point (p) around which the Taylor series is expanded.
The model
Valuation equations
3 If D ∗ < K, the correct default barrier is concave.
Valuation methods
An approximation based on a tangent will
y Numerical method
s
y Approximate
underestimate the value of the vulnerable call
analytical solution
option as shown in Fig. 2.
Numerical examples
Conclusion
The optimal value for p will be the value that
s
maximized the value of the vulnerable option.
Chaun-Ju Wang, November 1, 2007 23 / 35
24. Approximate analytical solution (cont.)
y Outline
Fig. 2: Integration region for the vulnerable European call
q
Introduction
when D∗ < K.
The model
Valuation equations
Valuation methods
y Numerical method
y Approximate
analytical solution
Numerical examples
Conclusion
Chaun-Ju Wang, November 1, 2007 24 / 35
25. y Outline
Introduction
The model
Valuation equations
Valuation methods
Numerical examples
y Numerical
Numerical examples
examples
Conclusion
Chaun-Ju Wang, November 1, 2007 25 / 35
26. Numerical examples
y Outline
Table 1: A comparison of FDB vs VDB
q
Introduction
The model
Valuation equations
Valuation methods
Numerical examples
y Numerical
examples
Conclusion
Chaun-Ju Wang, November 1, 2007 26 / 35
27. Numerical examples (cont.)
y Outline
Fig. 3: Vulnerable call values as a function of option’s
q
Introduction
moneyness: a comparison of the FDB and VDB models
The model
(base case)
Valuation equations
Valuation methods
Numerical examples
y Numerical
examples
Conclusion
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28. Numerical examples (cont.)
y Outline
Fig. 4: Vulnerable call values as a function of option’s
q
Introduction
moneyness: a comparison of the FDB and VDB models
The model
(base case)
Valuation equations
Valuation methods
Numerical examples
y Numerical
examples
Conclusion
Chaun-Ju Wang, November 1, 2007 28 / 35
29. Numerical examples (cont.)
y Outline
Fig. 5: Vulnerable call values as a function of option’s
q
Introduction
writer’s assets: a comparison of the FDB and VDB models
The model
(base case)
Valuation equations
Valuation methods
Numerical examples
y Numerical
examples
Conclusion
Chaun-Ju Wang, November 1, 2007 29 / 35
30. Numerical examples (cont.)
y Outline
Fig. 6: Vulnerable call values as a function of option’s
q
Introduction
writer’s assets: a comparison of the FDB and VDB models
The model
(base case)
Valuation equations
Valuation methods
Numerical examples
y Numerical
examples
Conclusion
Chaun-Ju Wang, November 1, 2007 30 / 35
31. Numerical examples (cont.)
y Outline
Fig. 7: Vulnerable call values as a function of option’s
q
Introduction
writer’s assets: a comparison of the FDB and VDB models
The model
(out-of-the-money option)
Valuation equations
Valuation methods
Numerical examples
y Numerical
examples
Conclusion
Chaun-Ju Wang, November 1, 2007 31 / 35
32. Numerical examples (cont.)
y Outline
Fig. 8: Vulnerable call values as a function of option’s
q
Introduction
writer’s assets: a comparison of the FDB and VDB models
The model
(in-the-money option)
Valuation equations
Valuation methods
Numerical examples
y Numerical
examples
Conclusion
Chaun-Ju Wang, November 1, 2007 32 / 35
33. Numerical examples (cont.)
y Outline
Fig. 9: Vulnerable call values as a function of option’s
q
Introduction
writer’s assets: a comparison of the FDB and VDB models
The model
(ρ = 0.5)
Valuation equations
Valuation methods
Numerical examples
y Numerical
examples
Conclusion
Chaun-Ju Wang, November 1, 2007 33 / 35
35. Conclusion
y Outline
This paper extends the vulnerable European option pricing
q
Introduction
results of Johnson and Stulz (1987) and Klein (1996).
The model
Valuation equations
3 Allowing for other liabilities in the capital structure of
Valuation methods
the option writer.
Numerical examples
3 The default boundary depends on the payoff of the
Conclusion
y Conclusion
option itself.
3 Allowing the pay-out ratio to be linked to the value of
option writer’s assets, and for correlation between the
assets of the option writer and the asset underlying
the option.
Chaun-Ju Wang, November 1, 2007 35 / 35