Optimal Early Exercise of Vulnerable American Options

March 15, 2008

This paper is preliminary and incomplete.

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Optimal Early Exercise of Vulnerable American Options

Abstract

We analyze the effect of credit risk on optimal early exercise policy for American options. We

find that the critical stock price at which early exercise of a vulnerable American put option is

optimal can be significantly higher than for an otherwise identical non-vulnerable American put.

In contrast, for American call options the critical stock price, if it exists, can be significantly

lower. We demonstrate that premature early exercise because of a pending credit event can

mitigate but not eliminate the effect of credit risk on the value of American options. Through

numerical examples we also demonstrate that premature early exercise can account for a

significant amount of the total reduction due to credit risk in the value of American options, even

when there is no write-down due to credit loss when premature early exercise occurs.

JEL classification: G13

Key words: Vulnerable American options; Credit risk; Derivative securities; Default barrier;

Default boundary; Pricing; Put options.

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1. Introduction

It is well-known that the value of an option can be affected by the credit risk of its writer. This

risk is negligible if the option is exchange traded but is an important consideration for options in

the over-the-counter market. In general, credit risk decreases the value of an option since the

payment that is actually received may be less than the payoff stipulated in the option contract.

There is a fairly extensive literature on the effect of credit risk on the pricing of vulnerable

European-style options (see, for example, Johnson and Stulz (1987), Hull and White (1995),

Jarrow and Turnbull (1995), Klein (1996) and Klein and Inglis (2001 and 1999)). This effect is

typically analyzed by reducing the payoff from an otherwise identical non-vulnerable option by

an amount which represents the expected credit loss. The percentage write-down depends on

the creditworthiness of the writer but can also vary with the magnitude of the payoff on the

option if the process driving financial distress is not independent of it.

The analysis for American-style options is more complicated because the presence of credit risk

may also affect the optimality of early exercise. For example, if the holder believes default is

pending, it may be optimal to exercise a vulnerable American-style option earlier, and at a

different price for the underlying asset, than would otherwise be the case. If such premature

early exercise is timed correctly it may allow the holder of the option to avoid the loss that would

typically arise when nominal claims are written down because of the occurrence of financial

distress. There is still an indirect effect on the value of the option, however, because the

timing of and payoff from premature early exercise are not the same as what would be optimal in

the absence of credit risk.

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The literature on the pricing of vulnerable American-style options has analyzed the effect of

credit risk on optimal early exercise policy only very generally and in combination with overall

pricing results (see Johnson and Stulz (1987), Hull and White (1995), Jarrow and Turnbull (1995)

and more recently Liu and Liu (2005) and Chang and Hung (2006)). These articles recognize

that credit risk may make early exercise more likely but they provide little concrete guidance on

how optimal early exercise policy differs from that for non-vulnerable options. The literature

also generally claims that the risk of credit loss is negligible for American-style options. This is

based on the assumption that the holder of the option is able to foresee accurately a pending

default and exercises accordingly. The loss in value due to premature early exercise itself,

however, has not been directly analyzed in the literature and has not yet been shown to be

insignificant.

These issues are important not only for the academic community but for industry practitioners as

well. Many options traders still set early exercise policy based on the assumption of no credit

risk, or under the popular assumption of independence as proposed by Hull and White (1995).

If credit risk induces significant differences in optimal early exercise policy the reasonableness

of these assumptions, as well as the exercise policy which they imply may be in question.

Properly recognizing the reduction in value due to premature early exercise as an indirect effect

of credit risk – even when full payout of the premature early exercise amount occurs – also has

important accounting and risk management implications.

The purpose of this paper is to analyze more closely the effect of credit risk on optimal early

exercise policy. It finds the underlying asset price at which early exercise is optimal is often

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very different for a vulnerable American option as compared to its non-vulnerable twin. Further,

the price of the underlying asset at which early exercise occurs is dependent on the degree of

credit risk, the correlation between the underlying asset and the assets of the option writer, and

the volatility of the assets of the option writer. This paper also finds the probability of early

exercise is higher for vulnerable American options and that the change in optimal early exercise

policy has a significant effect on value.

The outline of this paper is as follows. Section 2 reviews the existing literature on vulnerable

options. Section 3 presents results based on very general assumptions. Section 4 introduces a

structural model under more restrictive assumptions in order to allow the general results to be

illustrated in a series of numerical examples in Section 5. Section 6 provides a brief conclusion.

2. Literature review

While the impact of credit risk on the value of debt has been well studied since the early days of

modern option pricing theory (for example, see Black and Scholes (1973), Merton (1974), Black

and Cox (1976), Shimko et al. (1993) and Longstaff and Schwartz (1995)), it was not until much

later that the effect of credit risk on the value of options was first analyzed by Johnson and Stulz

(1987). They develop a model which values European options subject to default risk at the expiry

of the option. Under the assumption that the option is the only liability of the option writer and

the option holder receives all the assets of the option writer if default occurs, they demonstrate

that the effect of credit risk can be significant. They point out that, theoretically, a vulnerable

American call on a non-dividend-paying stock may be exercised early while a non-vulnerable

American call on non-dividend-paying stock will not. They also find that if prices are

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continuous the holder will always exercise before default and thus will never incur a loss. But

they do not provide numerical examples to demonstrate this point and they do not extend this

analysis to vulnerable American put options.

Compared with Johnson and Stulz (1987), Hull and White (1995) assume that in addition to the

option, other equal ranking claims on the assets of the option writer exist when the option writer

defaults. They examine the effect of credit risk on both American and European options.

Default can happen at any time before the option expires. The default boundary is fixed and

default is assumed to occur whenever the value of the assets of the option writer falls below that

fixed level. In the event of default, the option holder receives a proportion of the nominal

amount of the claim. Their numerical examples show that the impact of credit risk on

American options is much less than for European options. They give numerical examples of

vulnerable foreign currency options issued by a bank and find that as the correlation between the

foreign currency exchange rate and the assets of the bank decreases, the difference between the

percentage price reduction from default risk for American and European call options increases,

while the difference for vulnerable puts decreases. They also provide results in the special case

when the state variables determining the price of the option are assumed to be independent of the

state variables determining the occurrence of default and the payout rate. This assumption,

which is commonly made in industry, implies optimal early exercise policy is unaffected by the

credit risk of the option writer.

Jarrow and Turnbull (1995) consider both the effect on option prices of credit risk of the option

writer and also of the asset underlying the option. They apply a foreign currency analogy

approach to price options with credit risk. The payout ratio is not linked to the value of the assets

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of the counterparty, as in Hull and White (1995).

Klein (1996) extends the model of Johnson and Stulz (1987) by allowing the option writer to

have liabilities other than the option and by allowing the recovery rate to depend on the value of

the assets of the option writer. These assumptions are more appropriate when applying the

model to many business situations. He derives a closed-form solution for European options and

shows that the model is easy to calibrate.

Klein and Inglis (2001) further extend the literature by allowing both the claim of the option

holder and the other liabilities of the option writer to trigger default at the expiry of the option.

They also link the payoff received by the option holder in the event of default to the value of the

option writer’s assets.

[add brief review of Liu and Liu (2005) and Chang and Hung (2006) which are extensions of

Klein and Inglis (2001), and Klein and Yang (2007) which focuses primarily on pricing of

vulnerable American options and not on the effect of premature early exercise.]

[add a brief discussion of literature on early exercise policy for non-vulnerable American options;

this will give intuition to the proofs below. In general, this literature says to exercise when

stock price satisfies f = | S – X |.]

3. General results

We follow the approach of Johnson and Stulz (1987) and first analyze the effect of credit risk on

optimal early exercise policy under very general assumptions that do not depend on specific

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distributions or option pricing and credit risk models. We will, in a later section, provide

numerical examples based on a simple set of assumptions for the motion of the underlying asset

as well as the process driving financial distress.

3.1 Assumptions and Notation

Let f (t) denote the value at time t of a non-vulnerable American-style option written on

underlying asset S(t) with strike price X and time of expiry T. Define Sc as the critical price of

the underlying asset at which early exercise is optimal. It is well known that if f is an

American-style call option and S pays no dividends, early exercise is never optimal and thus this

critical value does not exist; otherwise this critical value depends on the various parameters

determining the value of the option such as term to expiry, volatility of the underlying asset,

strike price and the riskless interest rate.

Assume an otherwise identical option is written by a counterparty that is subject to credit risk.

We make two very general assumptions concerning the nature of this credit risk.

Assumption A1. The probability of default by the writer of the option is strictly greater than

zero.

Assumption A2. In the event of default, the option holder’s receipt is strictly less than it

would be if there was no default.

We note Assumption A1 is consistent with the definition of vulnerable options on page 279 in

Johnson and Stulz (1987), i.e., “options for which the probability of default is not zero”.

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Assumption A2 is needed to ensure the occurrence of default is indeed of some importance for

the option holder. We assume that all relevant variables are sufficiently well behaved that we

can use the risk neutrality approach developed by Cox and Ross (1976) and Harrison and Pliska

(1981).

In order to analyze the effect of credit risk on optimal early exercise policy we need to consider

the value of the vulnerable option in two different ways. The first, henceforth denoted as f *, is

the value of the vulnerable option when early exercise is determined by critical stock price Sc

i.e., when early exercise occurs at the same time, and only at the same time, as would be optimal

for the non-vulnerable option.

The second way in which the value of the vulnerable option needs to be considered, henceforth

denoted as f *^ , assumes the option holder may alter the timing of early exercise because of the

possibility of default. Define Sc^ as the new critical price of the underlying asset which can be

used to determine if early exercise of this vulnerable option is optimal. This critical value

depends on the various parameters determining the value of the option and also on the process

driving financial distress1. We seek to determine whether Sc^ = Sc and if f *^ < f .

For completeness, define f ^ as the value of the non-vulnerable derivative based on the early

exercise policy that is optimal for the vulnerable option f *^ , i.e., based on the critical asset value

Sc^ instead of Sc. Analyzing f ^ will be useful when conducting an attribution analysis of the

difference in values between f *^ and f .

1 These are the same as the theta and phi variables in Hull and White (1995)

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3.2 Distribution and model-free results

We are now able to derive some general results about the effect of credit risk on optimal early

exercise policy.

Proposition 1. Credit risk affects optimal early exercise policy; the critical stock price Sc^ at

which early exercise is optimal for a vulnerable American-style option is not equal to the critical

stock price Sc at which early exercise is optimal for its non-vulnerable twin.

Proof: Assumptions A1 and A2 imply f * < f . Assume Sc = Sc^ . This implies f * = f *^ .

By definition Sc satisfies S = | f – X | and Sc^ satisfies S = | f *^ - X | = | f * - X |. This implies

f * = f which contradicts f * < f . Thus Sc ≠ Sc^. □

This result means the independence assumption in Hull and White (1995) can never hold, i.e.,

that the phi variables and the theta variables can never be independent.

Proposition 2. Early exercise can mitigate but cannot entirely eliminate the effect of credit risk

on the value of vulnerable American-style options. Value is reduced either through the

probability of credit loss or through the reduced payoff from premature early exercise, or both.

Proof: Proposition 1 implies f > f ^ and f *^ > f * . Since f ^ ≥ f *^ we have f > f ^ ≥

f *^ > f * . □

At the level of generality of assumptions A1 and A2 we cannot prove f ^ strictly > f *^. This

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is because we have not yet imposed any assumptions with respect to the process driving the

payout rate when financial distress occurs. If it is driven by an asset price (presumably the

value of the assets of the option writer) and if this process (and hence payout rate) is continuous,

Johnson and Stulz show it is optimal to exercise at, or just before the default barrier is hit; in

such case there is no loss due to write-down but only due to the premature early exercise and f ^ =

f *^. In our model, if the reduction in the option holder’s receipt in Assumption A2 is arbitrarily

small then we should obtain the same result as in Johnson and Stulz. In practice, this

assumption is unreasonable, thus we should find a strict inequality. Also note Johnson and

Stulz did not identify the effect of the change in optimal early exercise policy on the value of the

vulnerable option.

Proposition 3a. The critical stock price at which early exercise is optimal for vulnerable

American puts is higher than it is for non-vulnerable American puts.

Proof: Consider an American put option and its vulnerable twin. Assume Sc > Sc^. This

implies there exists a value S between Sc and Sc^ for which X – S ≥ f and X – S < f *^,

which violates f > f *^. Since Sc ≠ Sc^ from Proposition 1 we have Sc < Sc

^. □

Proposition 3b. The critical stock price at which early exercise is optimal for vulnerable

American calls, if it exists, is lower than it is for non-vulnerable American calls.

Proof: [similar to that for 3a; to be added]

Note this result is consistent with the Johnson and Stulz result that for calls on non-dividend

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paying stock early exercise may occur, which implies that such a critical price may exist.

Proposition 4. A small reduction in value due to credit risk can have a very large effect on the

critical stock price at which early exercise is optimal for vulnerable American options. The

ratio between the reduction in value and the change in critical stock price is greater than 1/(1-|Δ|)

where Δ is defined in the standard way as ∂f /∂S and evaluated at Sc^ .

Proof: Sc satisfies S = | f – X | and Sc^ satisfies S = | f *^ - X |. Thus | Sc

^ - Sc | = | f (Sc )

- f *^ (Sc^ )| = | f (Sc ) – f (Sc

^ ) + α (Sc^ )| where α (Sc

^ ) ≡ f (Sc^ ) - f *^ (Sc

^ ) > 0 represents the

reduction in value due to credit risk evaluated at Sc^ . Since | Δ | is strictly decreasing in S, | Sc

^ -

Sc | > α (Sc^ ) / (1 - | Δ |) where Δ is evaluated at S = Sc

^ . □

Note since f is likely to be well in the money at Sc^ the absolute value of Δ is likely to be closer

to 1 than to 0. This implies the change in critical stock price is much greater than the reduction

in credit risk.

4. A structural model

We now introduce a simple structural model in order to demonstrate the significance of our

general results on optimal early exercise policy and valuation of vulnerable American put options.

The assumptions of our model are similar to those in Klein (1996) and Klein and Inglis (2001).

Assumption B1. Let V denote the market value of the assets of the option writer. The dynamics

of V are given by the diffusion process

VVV dZdtVdV σμ +=

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where Vμ is the instantaneous expected rate of return on the assets of the option writer, Vσ is

the instantaneous standard deviation of the return (assumed to be constant) on the assets of the

option writer and VZ is a standard Wiener process.

Assumption B2. Let S represent the market value of the asset underlying the option. The

dynamics of S are given by the diffusion process

SSS dZdtS

dS σμ +=

where Sμ is the instantaneous expected rate of return on the asset underlying the option, Sσ

is the instantaneous standard deviation of the return (assumed to be constant) on the asset

underlying the option and SZ is a standard Wiener process.

Assumption B3. The instantaneous correlation between d VZ and d SZ is ρ.

Assumption B4. The markets are perfect and frictionless, i.e., there are no taxes, transaction

costs or information asymmetries. Securities can be traded in continuous time.

Assumption B5. In an event of default, the nominal claim of the option holder is the value of the

otherwise identical non-vulnerable put option nP .

Assumption B6. At the expiry of the option, t=T, default occurs only if the value of the option

writer’s assets at time T, TV , is less than the threshold value *D + nTP , where *D is a fixed

amount which could correspond to the value of the other liabilities of the option writer.

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Assumption B7. Before the expiry of the option, t < T, default occurs only if the value of the

option writer’s assets at time t, tV , is less than the threshold value *D + ntP .

Assumption B8. At a given time of default, t, the option holder receives (1- tw ) times the

nominal claim, where w represents the percentage write-down of the nominal claim at time t.

Assumption B9. When default happens, the value of the option writer’s assets is subject to

deadweight bankruptcy/reorganization costs of α which is expressed as a percentage of the value

of the assets of the option writer.

Assumption B10. The percentage write-down on the nominal claim of the option holder upon

default is tw =1- (1-α) tV /( *D + ntP ) where the ratio tV /( *D + n

tP ) represents the value of the

option writer’s assets available to pay the claim expressed as a proportion of total claims at time

t.

Assumption B11. The asset underlying the put option pays dividends at continuous rate q. The

term structure is flat and the risk free rate is r.

Using the no-arbitrage approach, the price of a vulnerable put option P must satisfy the partial

differential equation given by Johnson and Stulz (1987):

tPrP

SVPVS

SPSqr

SPS

VPrV

VPV svsv ∂

∂=−

∂∂∂

+∂∂

−+∂∂

+∂∂

+∂∂ 2

2

222

2

222 )(

21

21 σρσσσ

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Note Johnson and Stulz assume q = 0 in the above equation. The boundary conditions implied

in the assumptions can be expressed as follows:

(1) At the expiry of the put option, t = T

*,| DVXSSX TTT >≤− + nTP

nTTT

nTn

T

T PDVXSPPD

V+≤≤

+− *

* ,|)1( α

0 otherwise

(2) Prior to the expiry of the put option, t < T

0),(lim =∞→

tSPS

If default does not happen prior to expiry

*,)(|)()),(( DVXtStSXttSP tfff >≤−= + ntP

1| )),(( −=∂∂

ttS fSP *,)(| DVXtS tf >≤ + n

tP

where )(tS f is the free boundary at t.

(3) If default occurs prior to expiry, t < T

nttt

ntn

t

t PDVXSPPD

V+≤≤

+− *

* ,|)1( α

The boundary condition set (1) characterizes the payoffs of the put option at expiry. The

boundary condition set (2) is similar to the boundary conditions of the non-vulnerable American

put option. The boundary condition set (3) expresses the amount which the put option holder will

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receive if the option writer’s assets hit the default boundary *D + ntP prior to expiry.

The above partial differential equation given the particular boundary conditions does not have an

analytic solution and must be solved numerically. We employ the three-dimensional binomial

tree approach to solve the partial differential equation as suggested by Hull and White (1990).

Note that in contrast with vulnerable European options, the payout rate will always equal 1 – α in

the limit as Δt → 0.

5. Numerical examples

5.1 American puts

In this section, we present some numerical examples to illustrate the factors which have impact

on the price of the underlying asset at which early exercise is optimal, the dependence of the

probability of early exercise on the degree of vulnerability, the attribution of the reduction in

value and various other properties of vulnerable American put options. In the base case2, the

parameters have been chosen to be similar to many business situations. The put option is at the

money. The option writer is, or has become with the passage of time, a highly leveraged firm

(90% debt-asset ratio). The correlation between the value of the option writer’s assets and the

value of the asset underlying the option is zero.

We first consider the effect of vulnerability on the critical stock price at which early exercise is

optimal. Our results are outlined in Tables 1, 2 and 3. As the degree of vulnerability increases,

2 The exact parameters are S=200, X=200, V=1000, Sσ =0.2, vσ =0.2, D*=900, T=2, r=0.05, ρ =0.0, α=0.25 and q=0.

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i.e., *D increases from 900 to 960, the stock price at which immediate exercise occurs

increases from 164 to 176, while the non-vulnerable put3 begins to be exercised immediately

when stock price is smaller than 158. This is because, as *D increases, the put option holder

becomes more concerned about the possibility of default in the future, and thus has a greater

tendency to exercise immediately. Similarly, as the volatility vσ increases or the correlation ρ

decreases, the stock price at which immediate exercise occurs also increases.

The dependence of the critical stock price on the degree of vulnerability implies that the

probability of early exercise at each non-terminal step in the tree must also depend on the degree

of vulnerability. Figures 1 and 2 show the probability of early exercise at each step for the

vulnerable and non-vulnerable put, respectively. For example, when S = 155, the probability of

early exercise is above .6 for a vulnerable option but is below .6 for a non-vulnerable option in

Figure 3. Note a probability equal to one at the first step means immediate exercise is optimal

at time zero.

As discussed above, premature early exercise can mitigate but not eliminate the effect of credit

risk, and the remaining reduction in value can be attributed to two sources: premature early

exercise with full payout; and write-downs of contractual amounts because of a credit event.

Table 4 illustrates these findings. In our example, the effect of premature early exercise, f - f ^ , is

always lower than the effect of write-down upon default, f ^ -f *^ , and f - f ^ can also be

significant. This implies the change in optimal early exercise policy can be an important

determinant of value.

3 The exact parameters are S=200, X=200, Sσ =0.2, T=2, r=0.05 and q=0.

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Figure 3 shows the relationship between the put option value and the stock price S for a

vulnerable American put option and a non-vulnerable American put option. The parameters are

the base case parameters except that α = 0.75 and the value of S is allowed to change. As the

stock price S decreases, the put option becomes more valuable. When the put option is deep

in-the-money (i.e. S < 100), the value of vulnerable put is considerably lower than that of

non-vulnerable put as expected; since S is sufficiently small, the option writer’s assets hit the

variable default boundary ntPD +* and default actually occurs at certain nodes in the tree (i.e.,

when ntt PDV +≤ * ). However, when the put option is not deep in-the-money (i.e. 110 < S <

150), default is less likely to occur and the effect of credit risk is less; in this case the value of the

vulnerable put is closer to that of the non-vulnerable put.

The relationship between the option value and the strike price X for vulnerable American puts

and non-vulnerable American puts is shown in Figure 4. The parameters are the base case

parameters except that α = 0.75 and the value of X is allowed to change. As the strike price X

increases, the put option will become more valuable. When the put is deep in-the-money (e.g.

X > 300), the value of the vulnerable put f *^ is considerably lower than that of non-vulnerable

put f, which means that the option writer’s assets hit the variable default boundary and default

actually occurs at certain nodes in the tree (i.e., when ntt PDV +≤ * ). In contrast, when the put

option is not deep in-the-money (e.g. 260 < X < 280), default will not happen ( ntt PDV +> * ) and

the value of vulnerable put is closer to that of non-vulnerable put.

[add discussion of additional tables and figures which will show effect of changes in other

parameter values on the size and attribution of the effect of credit risk.]

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Another interesting property is that the volatility of the stock price Sσ has a mixed effect on the

value of the vulnerable American put. Figure 8 shows the effect of Sσ on the value of the

vulnerable American put. The parameters are the base case parameters and the stock price S and

the volatility Sσ are allowed to change. The value of the vulnerable American put increases

with the value of Sσ if the stock price S is large. However, if the put option is in-the-money

(e.g. S = 100), the possibility of default increases, leading to the negative relationship between

the value of the vulnerable American put and Sσ . As expected, f * also has the same property

but f ^ does not.

The effects of changes in the other variables are more straightforward. Figure 5 depicts the

relationship between the put option value and the value of the option writer’s assets for

vulnerable American put options and non-vulnerable American put options. The parameters are

the base case parameters except that α = 0.75 and the value of V is allowed to change. The higher

the initial value of V, the less likely the variable default boundary will be hit in the future. In

the limit, if the value of V large enough (i.e., ntt PDV +> * is always true) the value of vulnerable

put equals that of non-vulnerable put.

Figure 6 shows the effect of the option writer’s other liabilities *D on the value of vulnerable

American put option. The parameters are the base case parameters except that α = 0.75 and the

price of stock S and option writer’s debt *D are allowed to change. As the value of *D

increases, the possibility of default increases and the value of V will be more likely to hit the

variable default boundary. Therefore, as *D increases, P decreases. At extremes, when *D is

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sufficiently small (e.g. *D < 700), ntt PDV +> * tends to be always true and the price of the

vulnerable American option is almost equal to that of the non-vulnerable American option. When

*D is large, the price of the vulnerable American option will approach zero.

Figure 7 shows the effect of deadweight costs α on the value of the vulnerable American put

option. The parameters are the base case parameters except that S and the deadweight costs α are

allowed to change. As the value of α increases, the payoff to the put option holder when the

option writer defaults decreases. Therefore as α increases P decreases.

Similarly, we can examine the effect of other four factors (volatility of the value of the assets of

the option writer Vσ , the correlation ρ, the time to expiry T, the risk-free interest rate r) on the

value of vulnerable American put. Although we do not present numerical examples, the effect

of Vσ is generally the same as the effect of Sσ on the value of vulnerable American put, the

value of the put is negatively related to ρ or r, and there is a positive relationship between T and

the value of the put.

5.2 American calls

So far our numerical examples have focused on vulnerable American puts although our model

can easily be extended to price American call options. Table 5 compares the percentage

reduction due to credit risk in our model for foreign currency American call values (i.e. q > 0)

with the results of Hull and White (1995). In general, the percentage reduction in our model is

somewhat higher than that for American options in their model. There are two reasons for this.

First, the payout ratio in our model is not exactly the same as p in the Hull and White model.

Even if the payout ratio is assumed to be zero in both models (i.e., α=1 and p=0), however, the

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percentage reduction is higher in our model. This is because in our model, the critical stock

price at which immediate or early exercise is optimal depends on the degree of vulnerability.

As shown in our Tables 1 and 2, for American put options, this effect is evident even when the

correlation between S and V is assumed to be zero. This implies that the independence

assumption that underlies the results in Tables 1 and 2 in Hull and White may not be appropriate

for American options.

6. Conclusion

We have analyzed the effect of credit risk on optimal early exercise policy for American options

and find that the critical stock price at which early exercise of a vulnerable American put option

is optimal can be significantly higher than for an otherwise identical non-vulnerable American

put. In contrast, for American call options the critical stock price, if it exists, can be

significantly lower. We demonstrate through numerical examples based on a specific structural

model that premature early exercise because of a pending credit event can mitigate but not

eliminate the effect of credit risk on the value of American options. We also demonstrate that

premature early exercise can account for a significant amount of the total reduction due to credit

risk in the value of American options, even when there is no write-down due to credit loss when

premature early exercise occurs.

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(1987), 267-280.

Klein, P. “Pricing Black-Scholes options with correlated credit risk.” Journal of Banking and

Finance, 50 (1996), 1211-1229.

Klein, P., and M. Inglis. “Pricing vulnerable European options when the options payoff can

increase the risk of financial distress.” Journal of Banking and Finance, 25 (2001), 993-1012.

Klein, P., and M. Inglis. “Valuation of European options subject to financial distress and interest

rate risk.” Journal of Derivatives, 6 (1999), 44-56.

Klein, P. and J. Yang, “Pricing vulnerable American options”, working paper, Simon Fraser

University, (2007).

Longstaff, F., and E. Schwartz. “Valuing risky debt: a new approach.” Journal of Finance (1995),

789-819.

Merton, R.C. “On the pricing of corporate debt: the risk structure of interest rates.” Journal of

Finance, 29 (1974), 449-470.

Shimko et. al. “The pricing of risky debt when interest rates are stochastic.” The Journal of Fixed

Income, 3 (1993), 58-65.

23

Table 1. The stock price at which exercise is immediate as a function of *D . Calculation of vulnerable put option prices are

based on the base case parameters. The numerical solution is based on a three-dimensional binomial tree using 100 steps. Calculations of non-vulnerable put option prices are also based on the base case parameters and the solution is obtained via a binomial tree using 100 steps. Prices in bold indicate immediate exercise is optimal.

S Vulnerable put option price Non-Vulnerable put

option price *D =900 *D =920 *D =940 *D =960

155 45 45 45 33.58 45

156 44 44 44 32.87 44

157 43 43 43 32.15 43

158 42 42 42 31.44 42.01

159 41 41 41 30.74 41.08

160 40 40 40 30.09 40.17

161 39 39 39 39 39.27

162 38 38 38 38 38.37

163 37 37 37 37 37.49

164 36 36 36 36 36.65

165 35.06 35.03 35 35 35.83

166 34.14 34.11 34 34 35.01

167 33.21 33.18 33 33 34.19

168 32.29 32.26 32 32 33.40

169 31.39 31.35 31 31 32.63

170 30.52 30.45 30 30 31.89

171 29.66 29.59 29 29 31.16

172 28.83 28.75 28 28 30.44

173 28.00 27.92 27 27 29.73

174 27.18 27.10 26 26 29.05

175 26.36 26.28 25 25 28.38

176 25.58 25.46 24.07 24 27.71

177 24.92 24.71 23.41 23.20 27.05

178 24.25 24.05 22.76 22.55 26.40

179 23.59 23.39 22.12 21.92 25.78

180 22.95 22.75 21.50 21.29 25.18

24

Table 2. The stock price at which exercise is immediate as a function of ρ. Calculation of vulnerable put option prices are based on the base case parameters. The numerical solution is based on a three-dimensional binomial tree using 100 steps. Calculations of non-vulnerable put option prices are also based on the base case parameters and the solution is obtained via a binomial tree using 100 steps. Prices in bold indicate immediate exercise is optimal.

S Vulnerable put option price Non-Vulnerable put

option price ρ=-0.8 ρ=-0.4 ρ=0 ρ=0.4 ρ=0.8

155 45 45 45 45 45 45

156 44 44 44 44 44 44

157 43 43 43 43 43 43

158 42 42 42 42 42 42.01

159 41 41 41 41 41.01 41.08

160 40 40 40 40 40.08 40.17

161 39 39 39 39 39.15 39.27

162 38 38 38 38.03 38.23 38.37

163 37 37 37 37.09 37.31 37.49

164 36 36 36 36.15 36.40 36.65

165 35 35 35.06 35.22 35.49 35.83

166 34 34.01 34.14 34.38 34.59 35.01

167 33 33.08 33.21 33.48 33.71 34.19

168 32 32.16 32.29 32.59 32.85 33.40

169 31.0722 31.28 31.39 31.72 32.00 32.63

170 30.2314 30.40 30.52 30.85 31.14 31.89

171 29.4048 29.53 29.66 29.99 30.29 31.16

172 28.5921 28.66 28.83 29.14 29.44 30.44

173 27.8139 27.81 28.00 28.29 28.59 29.73

174 27.0877 26.98 27.18 27.45 27.75 29.05

175 26.387 26.25 26.36 26.60 26.91 28.38

176 25.7009 25.57 25.58 25.77 26.07 27.71

177 25.0464 24.93 24.92 24.94 25.26 27.05

178 24.4321 24.50 24.25 24.19 24.93 26.40

179 23.8392 23.85 23.59 23.53 24.18 25.78

180 23.249 23.23 22.95 22.89 23.44 25.18

25

Table 3. The stock price at which exercise is immediate as a function of vσ . Calculation of vulnerable put option prices are based on the base case parameters. The numerical solution is based on a three-dimensional binomial tree using 100 steps. Calculations of non-vulnerable put option prices are also based on the base case parameters and the solution is obtained via a binomial tree using 100 steps. Prices in bold indicate immediate exercise is optimal.

S Vulnerable put option price Non-Vulnerable put

option price vσ =0.05 vσ =0.15 vσ =0.2 vσ =0.25 vσ =0.35

155 45 45 45 45 45 45

156 44 44 44 44 44 44

157 43 43 43 43 43 43

158 42 42 42 42 42 42.01

159 41.01 41 41 41 41 41.08

160 40.10 40 40 40 40 40.17

161 39.21 39 39 39 39 39.27

162 38.34 38 38 38 38 38.37

163 37.47 37.07 37 37 37 37.49

164 36.60 36.15 36 36 36 36.65

165 35.74 35.26 35.06 35.02 35 35.83

166 34.90 34.37 34.14 34.09 34 35.01

167 34.11 33.49 33.21 33.17 33 34.19

168 33.34 32.62 32.29 32.24 32 33.40

169 32.57 31.76 31.39 31.33 31 32.63

170 31.82 30.94 30.52 30.42 30.35 31.89

171 31.07 30.15 29.66 29.56 29.45 31.16

172 30.34 29.38 28.83 28.72 28.59 30.44

173 29.62 28.61 28.00 27.89 27.74 29.73

174 28.93 27.85 27.18 27.06 26.89 29.05

175 28.26 27.10 26.36 26.24 26.04 28.38

176 27.60 26.36 25.58 25.42 25.20 27.71

177 26.95 25.67 24.92 24.66 24.47 27.05

178 26.31 25.01 24.25 23.99 23.70 26.40

179 25.69 24.37 23.59 23.34 23.04 25.78

180 25.07 23.74 22.95 22.69 22.39 25.18

26

0

0.2

0.4

0.6

0.8

1

1.20 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96

step

Prob

abili

ty o

f ear

ly e

xerc

ise

S=130S=155S=180S=200

Figure 1 The probability of early exercise at each step as a function of steps in the three-dimensional binomial tree. Calculation of vulnerable put option prices are based on the base case parameters. The numerical solution is based on a three-dimensional binomial tree using 100 steps.

27

0

0.2

0.4

0.6

0.8

1

1.20 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96

step

Prob

abili

ty o

f ear

ly e

xerc

ise

S=130S=155S=180S=200

Figure 2 The probability of early exercise at each step as a function of steps in the binomial tree. Calculations of non-vulnerable put option prices are based on the base parameters. The solution is based on binomial tree using 100 steps.

28

Table 4 The reduction in put values due to credit risk. The exact parameters are S=200, X=200, V=1000, Sσ =0.2, vσ =0.2, T=2, r=0.05 and q=0. The non-vulnerable American put price is 15.4443. The numerical solution is based on a three-dimensional binomial tree using 100 steps. D* α =0.5 ρ 900 920 940 960 -0.8 f-f*^ 2.213 3.029 3.97 4.999 f^-f*^ 2.152 2.908 3.881 4.913 f*^-f* 0.06 0.071 0.088 0.097 -0.4 f-f*^ 2.728 3.69 4.066 5.464 f-f^ 0.406 0.205 0.377 0.148 f^-f*^ 2.321 3.485 3.69 5.316 f*^-f* 0.452 0.317 0.48 0.239 0 f-f*^ 3.092 4.114 4.901 5.768 f-f^ 1.104 0.393 0.4 0.296 f^-f*^ 1.988 3.722 4.501 5.472 f*^-f* 1.09 0.808 0.646 0.549 0.4 f-f*^ 4.121 4.495 5.621 6.143 f-f^ 1.22 1.25 0.398 1.153 f^-f*^ 2.9 3.245 5.224 4.99 f*^-f* 1.306 1.462 0.901 0.836 0.8 f-f*^ 4.751 5.813 6.377 7.29 f-f^ 1.432 0.925 1.596 0.138 f^-f*^ 3.319 4.888 4.782 7.152 f*^-f* 2.144 1.39 1.07 0.325 α =1 ρ 900 920 940 960

-0.8 f-f*^ 4.201 5.81 7.598 9.568

f-f^ 0.441 0.276 0.7 0.821 f^-f*^ 3.76 5.535 6.898 8.747 f*^-f* 0.275 0.317 0.432 0.499 -0.4 f-f*^ 4.725 6.666 7.193 10.07 f-f^ 1.14 1.172 1.222 1.025 f^-f*^ 3.584 5.495 5.971 9.043 f*^-f* 1.509 1.247 1.705 1.176 0 f-f*^ 4.966 7.137 7.479 10.51 f-f^ 1.193 1.221 3.724 1.066 f^-f*^ 3.774 5.917 3.755 9.444 f*^-f* 3.203 2.538 3.377 1.899 0.4 f-f*^ 6.568 7.358 10.01 10.92 f-f^ 5.308 3.249 1.339 4.013 f^-f*^ 1.26 4.109 8.67 6.91 f*^-f* 4.156 4.365 2.849 2.802 0.8 f-f*^ 6.498 8.157 10.2 11.66 f-f^ 5.129 3.768 7.052 4.548 f^-f*^ 1.369 4.389 3.145 7.114 f*^-f* 7.097 6.041 4.463 3.33

fffffffffffffff

f

fff

fff

fff

ffff

29

0

10

20

30

40

50

60

150 160 170 180 190 200 210 220 230 240 250

S

put v

alue

f*^ff*f^

0

20

40

60

80

100

120

140

160

50 60 70 80 90 100

110

120

130

140

150

160

170

180

190

200

210

220

230

240

250

S

put v

alue

f*^

f

f*

f^

Figure 3. Vulnerable American put values as a function of stock price S: comparison between vulnerable American put f*^, f*, f^ and non-vulnerable American put f. Calculations of vulnerable put option prices are based on the base case parameters except α = 0.75. The numerical solution is based on a three-dimensional binomial tree using 100 steps. Calculations of non-vulnerable put option prices are also based on the base case parameters and the solution is obtained via a binomial tree using 100 steps.

30

0

10

20

30

40

50

60

150 160 170 180 190 200 210 220 230 240 250

X

put v

alue

f*^ff*f^

0

20

40

60

80100

120

140

160

150

160

170

180

190

200

210

220

230

240

250

260

270

280

290

300

310

320

330

340

350

S

put v

alue

f*^

f

f*

f^

Figure 4. Vulnerable American put values as a function of strike price X: comparison between vulnerable American put f*^, f*, f^ and non-vulnerable American put f. Calculations of vulnerable put option prices are based on the base case parameters except α = 0.75. The numerical solution is based on a three-dimensional binomial tree using 100 steps. Calculations of non-vulnerable put option prices are also based on the base case parameters and the solution is obtained via a binomial tree using 100 steps.

31

0

2

4

6

8

10

12

14

16

18

500 600 700 800 900 1000 1100 1200 1300 1400 1500

V

put v

alue

f*^ff*f^

Figure 5. Vulnerable American put values as a function of the value of the option writer’s assets V: comparison between vulnerable American put and non-vulnerable American put. Calculations of vulnerable put option prices are based on the base case parameters except α = 0.75. The numerical solution is based on a three-dimensional binomial tree using 100 steps. Calculations of non-vulnerable put option prices are also based on the base case parameters and the solution is obtained via a binomial tree using 100 steps.

32

0

20

40

60

80

100

120

140

160

50 60 70 80 90 100

110

120

130

140

150

160

170

180

190

200

210

220

230

240

250

S

put v

alue D=0 f*^

D=900 f*^

D=9000 f*^

0

20

40

60

80

100

120

140

160

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250

S

put v

alue D=0 f*

D=900 f*

D=9000 f*

0

2040

60

80

100120

140

160

50 70 90 110

130

150

170

190

210

230

250

S

put v

alue D=0 f^

D=900 f^

D=9000 f^

Figure 6. The effect of the option writer’s debt D* on the value of vulnerable American put option. Calculations of vulnerable put option prices are based on the base case parameters except α = 0.75. The numerical solution is based on a three-dimensional binomial tree using 100 steps.

33

0

20

40

60

80

100

120

140

160

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250

S

put v

alue alpha=0 f*^

alpha=0.75 f*^

alpha=1 f*^

0

20

40

60

80

100

120

140

160

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250

S

put v

alue alpha=0 f*

alpha=0.75 f*

alpha=1 f*

0

20

40

60

80

100

120

140

160

50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250

S

put v

alue alpha=0 f^

alpha=0.75 f^

alpha=1 f^

Figure 7. The effect of deadweight costs α (i.e. alpha in the figure) on the value of vulnerable American put option. Calculations of vulnerable put option prices are based on the base case parameters. The numerical solution is based on a three-dimensional binomial tree using 100 steps.

34

0

20

40

60

80

100

120

140

20 60 100

140

180

220

260

300

340

380

S

put v

alue

sigmaS=0.01 f*^sigmaS=0.2 f*^sigmaS=0.4 f*^

0

20

40

60

80

100

120

140

20 60 100

140

180

220

260

300 340 380

S

put v

alue sigmaS=0.01 f*

sigmaS=0.2 f*

sigmaS=0.4 f*

0

50

100

150

200

20 60 100

140

180

220

260

300

340

380

S

put v

alue sigmaS=0.01 f^

sigmaS=0.2 f^

sigmaS=0.4 f^

Figure 8. The effect of the volatility Sσ (i.e. sigmaS in the figure) on the value of vulnerable American put option. Calculations of vulnerable put option prices are based on the base case parameters. The numerical solution is based on a three-dimensional binomial tree using 100 steps.

35

Table 5 Percentage reduction in call values due to credit risk, compared with Hull and White (1995, Tables 1 and 2). The exact parameters are S=1, X=1, V=100, Sσ =0.15, vσ =0.05, T=1, r=0.05, ρ=0.0, q=0.05. The numerical solution is based on a three-dimensional binomial tree using 100 steps.

ρ 90 92 94 96

-0.8 α =0.5 f*^ 0.17 0.18 0.40 3.42 α =0.5 f* 0.18 0.40 3.01 14.02 α =0.5 f^ 0.17 0.18 0.39 2.77 HW p=0.5 f*^ 0.00 0.00 0.10 1.73

-0.4 α =0.5 f*^ 0.21 0.33 1.04 4.16 α =0.5 f* 0.28 0.69 2.62 8.62 α =0.5 f^ 0.20 0.28 0.58 1.67 HW p=0.5 f*^ 0.01 0.09 0.59 3.03 0 α =0.5 f*^ 0.41 0.55 1.24 4.14 α =0.5 f* 0.44 0.72 1.88 5.98 α =0.5 f^ 0.38 0.43 0.64 1.27 HW p=0.5 f*^ 0.03 0.15 0.73 3.24

0.4 α =0.5 f*^ 0.24 0.30 0.66 2.43 α =0.5 f* 0.25 0.32 0.75 2.81 α =0.5 f^ 0.24 0.25 0.29 0.52 HW p=0.5 f*^ 0.01 0.06 0.40 2.07

0.8 α =0.5 f*^ 0.26 0.26 0.32 1.13 α =0.5 f* 0.26 0.26 0.32 1.14 α =0.5 f^ 0.26 0.26 0.26 0.26

HW p=0.5 f*^ 0.00 0.00 0.06 0.89

*D

36

Table 5 (Continued)

ρ 90 92 94 96 -0.8 α =1 f*^ 0.17 0.18 0.41 3.67

α =1 f* 0.18 0.63 5.84 27.87 α =1 f^ 0.17 0.18 0.40 3.53 HW p=0 f*^ 0.00 0.00 0.11 1.78

-0.4 α =1 f*^ 0.22 0.36 1.33 5.89 α =1 f* 0.37 1.20 5.04 17.04 α =1 f^ 0.21 0.31 1.07 3.37 HW p=0 f*^ 0.02 0.11 0.72 3.72 0 α =1 f*^ 0.42 0.65 1.73 6.38 α =1 f* 0.51 1.07 3.38 11.58 α =1 f^ 0.39 0.48 0.91 2.71 HW p=0 f*^ 0.04 0.22 1.10 4.93

0.4 α =1 f*^ 0.25 0.35 0.98 4.15 α =1 f* 0.26 0.40 1.27 5.38 α =1 f^ 0.24 0.26 0.39 1.16 HW p=0 f*^ 0.01 0.10 0.68 3.61

0.8 α =1 f*^ 0.26 0.27 0.38 2.00 α =1 f* 0.26 0.27 0.38 2.01 α =1 f^ 0.26 0.26 0.26 0.27 HW p=0 f*^ 0.00 0.01 0.12 1.78

*D