Pricing vulnerable European options when the option’s payoff can increase the risk of financial distressPeter Klein, Michael InglisJournal of Banking & Finance

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.
Chaun-Ju Wang, November 1, 2007 4 / 35

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.
Chaun-Ju Wang, November 1, 2007 7 / 35

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)
q
Introduction
model after appropriate adjustments to incorporate the
The model
variable default boundary (VDB) condition.
y Assumption
Valuation equations
Valuation methods
Numerical examples
Conclusion
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11. Assumption (cont.)
y Outline
Introduction
The model
y Assumption
Valuation equations
Valuation methods
Numerical examples
Conclusion
Chaun-Ju Wang, November 1, 2007 11 / 35

12. Assumption (cont.)
y Outline
Introduction
The model
y Assumption
Valuation equations
Valuation methods
Numerical examples
Conclusion
Chaun-Ju Wang, November 1, 2007 12 / 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
q
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
q
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
q
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
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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
Chaun-Ju Wang, November 1, 2007 27 / 35

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
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34. y Outline
Introduction
The model
Valuation equations
Valuation methods
Numerical examples
Conclusion
Conclusion
y Conclusion
Chaun-Ju Wang, November 1, 2007 34 / 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