Pricing Vulnerable European Options When the
      Option’s Payoff Can Increase the Risk of Financial
                     ...
Outline

y Outline
                                Introduction
                            q
Introduction

The model
    ...
y Outline

Introduction
y Vulnerable options
y Related works
y The idea of this
paper

The model

Valuation equations

   ...
Vulnerable options

y Outline
                                Many financial institutions actively trading derivative
     ...
Related works

y Outline
                                Most of the literature on vulnerable options assumes that
       ...
Related works (cont.)

y Outline
                                Johnson and Stulz (1987)
                            q
In...
Related works (cont.)

y Outline
                                Klein (1996)
                            q
Introduction
y...
The idea of this paper

y Outline
                                Allowing for the presence of other liabilities in the ca...
y Outline

Introduction

The model
y Assumption

Valuation equations

Valuation methods

Numerical examples

             ...
Assumption

y Outline
                                Summarizing the assumptions underlying the Klein (1996)
            ...
Assumption (cont.)

y Outline

Introduction

The model
y Assumption

Valuation equations

Valuation methods

Numerical exa...
Assumption (cont.)

y Outline

Introduction

The model
y Assumption

Valuation equations

Valuation methods

Numerical exa...
y Outline

Introduction

The model

Valuation equations
y Johnson and Stulz
(1987)
y Klein (1996)
y Model of this

       ...
Johnson and Stulz (1987)

y Outline
                                Johnson and Stulz (1987) pricing equation of vulnerabl...
Klein (1996)

y Outline
                                Klein (1996) pricing equation of vulnerable European calls
       ...
Model of this paper

y Outline
                                The pricing equation for vulnerable European calls in this
...
y Outline

Introduction

The model

Valuation equations

Valuation methods
y Numerical method
y Approximate
analytical sol...
Numerical method

y Outline
                                Three-dimension binomial tree
                            q
In...
Approximate analytical solution

y Outline
                                Performing the standard log transformation and ...
Approximate analytical solution (cont.)

y Outline
                                The approximation valuation equation de...
Approximate analytical solution (cont.)

y Outline
                                The approximation valuation equation de...
Approximate analytical solution (cont.)

y Outline
                                Fig. 1: Integration region for the vuln...
Approximate analytical solution (cont.)

y Outline
                                The approximation valuation equation de...
Approximate analytical solution (cont.)

y Outline
                                Fig. 2: Integration region for the vuln...
y Outline

Introduction

The model

Valuation equations

Valuation methods

Numerical examples
y Numerical

              ...
Numerical examples

y Outline
                                Table 1: A comparison of FDB vs VDB
                        ...
Numerical examples (cont.)

y Outline
                                Fig. 3: Vulnerable call values as a function of opti...
Numerical examples (cont.)

y Outline
                                Fig. 4: Vulnerable call values as a function of opti...
Numerical examples (cont.)

y Outline
                                Fig. 5: Vulnerable call values as a function of opti...
Numerical examples (cont.)

y Outline
                                Fig. 6: Vulnerable call values as a function of opti...
Numerical examples (cont.)

y Outline
                                Fig. 7: Vulnerable call values as a function of opti...
Numerical examples (cont.)

y Outline
                                Fig. 8: Vulnerable call values as a function of opti...
Numerical examples (cont.)

y Outline
                                Fig. 9: Vulnerable call values as a function of opti...
y Outline

Introduction

The model

Valuation equations

Valuation methods

Numerical examples

Conclusion
               ...
Conclusion

y Outline
                                This paper extends the vulnerable European option pricing
          ...
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Pricing Vulnerable European Options

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Pricing vulnerable European options when the option’s payoff can increase the risk of financial distress Peter Klein, Michael Inglis Journal of Banking & Finance

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Pricing Vulnerable European Options

  1. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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 Chaun-Ju Wang, November 1, 2007 10 / 35
  11. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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
  34. 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. 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

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