The of Subject to Default Risk

Shen-Yuan Chen

Department of , Ming Chuan University

No. 250, Chung-Shan N. Rd., Sec. 5, Taipei, Taiwan 110 Tel: 886-2-28824564 ext. 2390 e-mail: [email protected]

Abstract There is no sufficient settlement mechanism for unlisted options, thus the of the option writer must be considered when investors evaluate the price of unlisted options. This paper intends to extend option pricing model and derive a path-dependent valuation model for option subject to counterparty default risk. Numerical examples show that our model can effectively identify the impacts of counterparty credit risk on vulnerable option value. The most important result of sensitivity analysis is that the effect of the correlation between counterparty’s asset value and underlying stock price on vulnerable option value is obscured.

Keyword: option, option pricing, credit risk, default risk, path-dependent

1 I. Introduction

As the needs to hedge financial risks increase, a variety of non-exchange options or over-the-counter options are widely traded between financial institutions and their institutional costumers. In contrast to exchange listed options markets, there is no clearing organization, such as Options Clearing Corporation, in the over-the-counter markets requiring that the options writer pay an initial margin and be daily resettled. Thus it is unreasonable to assume that options investors are not exposed to possible default risk from their writer. Johnson & Stulz (1987) called this type of option a “vulnerable option.” The Black & Scholes (1973) option pricing model is not adequate to evaluate the options which are vulnerable to counterparty credit risk.

Several studies have focused on the default risk of vulnerable option, for example, Johnson & Stulz (1987), Hull & White (1995), Jarrow & Turnbull (1995), and Klein (1996). However, these papers consider the credit risk in their pricing models only on the option date. That is, at the expiration date, if the asset value of the option writer is greater than its debt, the option writer will make the necessary payment to the in-the-money option holders. On the contrary, if the option writer is insolvent, the in-the-money option holders can only claim a proportion of option payoff on the counterparty asset value.

Under this assumption, the default risk, whether the option writer is solvent or not, is only examined at the date of maturity and the path-dependent process of asset value is ignored. So if the asset value of the option writer is ever below its debt

(namely defaults) prior to expiration date but ends with asset value above debt, then the in-the-money option holders still can claim full payoff from the writer. In fact, this is seldom the real sutiation. Hence the value of vulnerable option is overpriced

2 by the Klein (1996) model.

The purpose of this paper is to extend the vulnerable option pricing model in

Klein (1996) and present a path-dependent valuation model for option subject to counterparty default risk. We also compare the pricing results of our model with

Black & Scholes (1973) default-free option model and Klein (1996) path- independent vulnerable option model.

The rest of this paper is organized as follows. The next section reviews the literature about the credit risk of options. Section III presents the methodology for pricing path-dependent vulnerable option. Section IV analyzes the difference of our model with those of Black & Scholes (1973) and Klein (1996). In section V, sensitivity analysis will be conducted to illustrate the impacts of key variables on vulnerable option values. The final section concludes the paper.

II. Literature Review

Default risk in securities has been discussed in previous research, for example, forward contracts in Kane (1980) and swaps in Cooper & Mello (1991).

However, Johnson & Stulz (1987) was the first paper covering default risk in options. They considered many options with writer’s default risk are written by private financial institutions that have limited assets instead of organized exchanges and there are no guarantee by a third party. Therefore, Johnson & Stulz (1987) assume the option is the sole debt claim of the writer and if the option writer is unable to make a promised payment, the vulnerable option holders receive all the assets of the option writer. They derive closed form solutions for the European options prices in some different cases by assuming stochastic processes both for the asset value of the option writer and the value of the underlying asset. Johnson &

Stulz (1987) also examine the impact of credit risk on option values and show that

3 the values of options subject to default risk are different from those of default-free options.

Following Johnson & Stulz (1987), Hull & White (1995) also present a model for valuing over-the-counter options when option holders are exposed to potential default risk. They extend the Johnson & Stulz (1987) model and allow the counterparty to have other equal ranking liabilities. When the counterparty defaults, option holders are assumed to obtain only a proportion of its no-default claims. Hull

& White (1995) not only propose a model for vulnerable option but also numerically simulate and compare the difference between European calls, American calls and ordinary call options under normal default risk.

Allowing both a for the default-free term structure and the term structure for risky debt, Jarrow & Turnbull (1995) provide a new technique for valuing and hedging derivative securities subject to credit risk. They derive a risk-neutral pricing procedure by applying arbitrage-free dynamics for these term structures. This methodology can also be applied to over-the-counter options, corporate debts and other securities.

For simplicity, both Hull & White (1995) and Jarrow & Turnbull (1995) are assuming independence between the asset underlying the option and the default risk of the counterparty. On the contrary, Klein (1996) indicates that this assumption is not realistic in many business situations. The default risk of vulnerable option occurs when the asset value of the counterparty declines as the value of the option itself increases as well. Thus Klein (1996) assumes the credit risk of the option writer and the asset underlying option to be correlated and derives a pricing formula which allows for the counterparty to have other debt claims and the existence of capital forbearance. Moreover, the proportion of nominal claims for option holders

4 in the case of the counterparty’s default is endogenous to the model.

The Klein (1996) model is more practical than those of Hull & White (1995) and Jarrow & Turnbull (1995); however, the author still ignores the possibility that the counterparty may default prior to the option expiration date. On the expiration date, once the counterparty’s asset value exceeds its liabilities, the in-the-money option holders can receive their option intrinsic value, no matter whether the asset value of the counterparty was ever lower than its debt claims during the option lifetime. Obviously, Klein (1996) assumes the vulnerable option price is path- independent, thus the default risk can be examined only on the expiration date.

In next section, this paper extends the Klein (1996) model and derives a more accurate path-dependent pricing model for vulnerable option subject to counterparty credit risk.

III. The Pricing Model for Vulnerable Option

Follow Klein (1996), it is assumed that both the stock price (S) underlying option and the asset value (V) of the option writer follow geometric Brownian motion. Using the Cox & Ross (1976) risk neutrality approach, the appropriate processes for S and V can be shown as:

dS = rdt + s dw (1) S S dV = rdt + s dz (2) V V where r denotes the riskfree rate and w and z follow standard Wiener processes.

sS and s V represent the instantaneous standard deviation of underlying stock and option writer’s asset respectively. Let S0 (V0) and ST (VT) be the value of the underlying stock (option writer’s asset) at initial time and at maturity date, respectively. These stochastic processes imply that lnST and lnVT are normally

5 2 2 distributed with mean of (r - 0.5s S )T and (r - 0.5s V )T , and standard deviation of

s S T and sVT , respectively. Moreover, the joint distribution for lnST and lnVT are bivariate normal as:

2 2 n2 (ln St + (r -0.5s S )T, lnVt + (r - 0.5s V )T, r) (3)

where n2 is the probability density function of standard bivariate normal distribution.

ρis the correlation coefficient between lnST and lnVT.

* If the option writer’s asset value Vt falls below some amount D anytime throughout the option lifetime,1 then we assume the option holders get nothing even

* though the option is in-the-money. If Vt never falls below D prior to maturity date

* and at the expiration date VT is greater than or equal to D , then the in-the-money option holders can fully claim their intrinsic value from the counterparty. However,

* if at the expiration date VT is smaller than D , then the option writer defaults and the in-the-money option holders can only get a proportion of their claims. Under these conditions, the value of a vulnerable option (C) can be stated as:

C = e -rT E{max( S - K, 0)[(1| "V ³ D* , 0 £ t £ T) T t (4) * * + ((1 -a)VT / D | "Vt ³ D , 0 £ t < T, and VT < D )]} or expressed as:

-rT * C = e E{max( ST - K, 0)Prob(VT ³ D ) * * - max( ST - K, 0)Prob(VT ³ D $Vt < D , 0 £ t < T) (5)

* + max(ST - K, 0)[(1-a)VT / D][Prob(the first time V < D occurs at maturity T] instead of

- rT * * C = e E[max( ST - K, 0)[(1| VT ³ D ) + ((1-a)VT / D | VT < D )] (6) in Klein (1996), where E denotes risk neutral expectation over ST and VT, K is the

1 As in dealing with problem financial institutions, we allow D* to be smaller than the option writer’s outstanding liabilities D. In other words, there is capital forbearance for the counterparty.

6 of the option; and α is bankruptcy cost expressed as a percentage of the asset value of the option writer. Equation (4) means that option holders can’t get full claims from the counterparty at maturity day if default occurs. This is an important difference between a vulnerable option and a non-vulnerable option.

Besides, Equation (4) differs from equation (6) in modeling the possibility of default event. Equation (6) examines the default condition only on the option expiration date and doesn’t consider the default possibility prior to maturity. On the other hand, the second term of equation (5) is ignored by equation (4) and the third term of equation (5) is also different from the second of equation (4). In fact, equation (6) is the value of a path-independent vulnerable option and equation (4) or (5) is the value of a path-dependent vulnerable option.

Following Rubinstein and Reiner (1991a, 1991b) and Briys, Bellalah, Mai, and Varenne (1998), equation (5) can be solved as follows:

-rT C = S0 N 2 (a1 ,a2 , r)- Ke N 2 (b1,b2 , r)

2mv 2mv 2 2 D sV -rT D sV - S 0 ( ) N 2 (x1 , x2 , r) + Ke ( ) N 2 (y1, y2, r) (7) V0 V0

V (1- a) (r+ rs s )T V (1-a) + 0 S e s v N (c ,c ,-r )h(T ) - 0 KN (d ,d ,-r)h(T) D 0 2 1 2 D 2 1 2

1 S 2 1 V 2 where a = éln 0 + çær + 1 s ÷öTù, a = éln 0 + çær - 1s + rs s ÷öTù 1 ëê K è 2 s ø ûú 2 ê D* è 2 v v s ø ú s s T s v T ë û

S 0 2 V0 1 2 ln + çær - 1s ÷öT ln + çær - s ÷öT K è 2 s ø D * è 2 v ø b1 = , b2 = s s T s v T

* 1 S0 1 2 1 D 1 2 x1 = [ ln + (r + s s )T ], x2 = [ ln + (r - s v + rs ss v )T ] s s T K 2 sv T V0 2

* 1 S0 1 2 1 D 1 2 y1 = [ln + (r - s s )T ], y2 = [ln + (r - s v )T ] s s T K 2 s v T V0 2

7 S 0 2 V0 1 2 ln + çær + 1 s + rs s ÷öT ln + çær + s + rs s ÷öT K è 2 s s v ø D * è 2 v v s ø c1 = , c2 = s s T s v T

1 é S ù 1 é V ù d = ln 0 + çær - 1s 2 + rs s ÷öT , d = ln 0 + çær + 1s 2 ÷öT 1 ê K è 2 s s v ø ú 2 ê * è 2 v ø ú s s T ë û s v T ë D û * - ln D V * h(T ) = 0 exp{-[ln( D ) - (r - 1 s 2 )T ]2 /(2s 2T )} V 2 v v s vT 2pT 0

In equation (7), N2 is bivariate standard normal cumulative distribution function and h(T) is the density function for the first time T when the option writer’s asset value falls below D* .2 Note that the value of vulnerable option C is not only influenced by the price of underlying security (S), the of security return

(s s ), the strike price (K), the lifetime of the option (T), and riskfree rate (r), but also

the asset value (V), the asset volatility (s v ), and the liability (D) of the option writer, and the correlation coefficient (ρ) between the underlying security and the option writer’s asset.

Comparing equation (7) with the Klein (1996) model, we can find that the first line of equation (7) is the same as the latter but the last two lines of the former are different from the latter. The second line of equation (7) and h(T) of the third line in equation (7) are ignored in the latter, thus resulting in the Klein (1996) model overestimating the value of the option with counterparty credit risk. In the next section, we will compare the path-dependent vulnerable option pricing model (7) with Black & Scholes (1973) model and Klein (1996) path-independent model.

IV. Models Comparison

In order to compare our model with those of Black & Scholes (1973) and

Klein (1996), we take same numerical examples as in Klein (1996). The calculation

8 results are shown in Table 1. All calculations of options values in Table 1 are based

on following parameter values: s s =0.3, s v = 0.3, ρ= 0.5, r = 0.04833, T =

* 0.3333, K = 40, S 0 = 40, V0 = 5, D = 5, D = 5, unless otherwise noted. Black &

Scholes (1973) values and values calculated by Klein (1996) are also reported in the last three columns for comparison.

Table 1 Values of vulnerable options and non-vulnerable options

Case α= 0 α= 0.5 B & S Klein (α= 0) Klein(α=0.5) Base case 0 0 3.0697 3.0049 2.6654

s s = 0.2 0 0 2.1641 2.1160 1.8674

s s = 0.4 0 0 3.9760 3.8954 3.4686

s v = 0.2 0 0 3.0697 3.0312 2.7167

s v = 0.4 0 0 3.0697 2.9772 2.6240

ρ= -0.5 0 0 3.0697 2.7068 1.7320 ρ= 0 0 0 3.0697 2.8801 2.2114 T = 0.0833 0 0 1.4603 1.4439 1.2723 T = 0.5833 0 0 4.1806 4.0681 3.6235 K = 30 0 0 10.5744 10.1628 8.4089 K = 50 0 0.4341 0.4305 0.4069

S 0 = 30 0 0 0.1487 0.1478 0.1418

S 0 = 50 0 0 10.9319 10.5427 8.8250

V0 = 3 -3.2619 -3.2140 3.0697 2.0873 1.0546

V0 = 10 3.0691 3.0691 3.0697 3.0697 3.0697 r = 0.02833 # 0 0 2.9394 2.8742 2.5371 r = 0.06833 0 0 3.2031 3.1387 2.7975

#: The option values in the last two rows are not reported in Table 1 of Klein (1996).

In order to emphasize his model can identify the effect of default risk on option value, Klein (1996) specially let the base value of counterparty’s asset be

2 See Briys et al. (1998).

9 equal to its debt value, V0 = D = 5; namely, the counterparty is at the bankruptcy margin when selling the vulnerable option. Hence the probability of counterparty default is extremely high and Klein (1996) can differentiate his model from Black &

Scholes (1973) model. For example, in the base case, the values of Black & Scholes and Klein are 3.0697 and 3.0049 respectively. (See the forth and the fifth columns in

Table 1) When considering a more realistic case, for example, let the base value of counterparty’s asset be 10 instead of 5, we can see that both prices of Klein model and Black & Scholes model are equal. In other word, when the counterparty is solvent enough to protect option holders while writing the over-the-counter option, then Klein (1996) model is unable to effectively differentiate credit risk from Black

& Scholes (1973) model due to the low possibility of defaulting to make promised payment by the counterparty.

On the other hand, in base case, after deducting the value of default possibility prior to expiration date, which is ignored by Klein (1996), our model shows the value of path-dependent vulnerable option with credit risk to be 0, which is substantially lower than 3.0049 for Klein (1996) model. As Table 1 shows, except

for V0 = 3 and 10 cases, our model consistently indicates that, once V0 = D = 5, the vulnerable option values are zero while Klein (1996) model still reports the options are worthy, no matter the changes in each variables. In fact, under the base case, the default possibility of counterparty before option maturity date is very high, the vulnerable option should be worthless. Thus Klein (1996) path-independent pricing model will overestimate the vulnerable option value.

Similarly, in V0 = 10 case, our model indicates the vulnerable option value is

3.0691 instead of 3.0697 both in Black & Scholes and Klein. This means, when the probability of counterparty default is relatively low, the model presented in this

10 paper still can point out the credit risk effect in vulnerable option while Klein (1996) model doesn’t work.

Take another example, V0 = 3 case in Table 1, we can find the vulnerable option value should be –3.2619 instead of 2.0873 reported in Klein (1996). This special case represents the counterparty to be insolvent in the beginning, then the reasonable value of vulnerable option should be worthless even negative, not positive as in Klein. Just because we want to consider the credit risk embedded in vulnerable options, no rational investors will buy insolvent option and the option value would be minus. This case shows that the reasonable pricing model for options subject to credit risk must consider the default probability of the counterparty within the option lifetime.

For illustrating the difference between our model and Klein (1996), we further calculate some numerical examples reported in Table 2. From Table 2, we can easily find that both prices in the last two columns are always equal in each case.

The result indicates if the counterparty’s capital is adequate (V0 = 10 and D = 5 ), there will be no difference between Klein (1996) path-independent vulnerable option pricing model and Black & Scholes (1973) non-vulnerable option pricing model. This is because Klein (1996) ignores the default risk occurring prior to option maturity and hence overprices the vulnerable option. In contrast, our model shows the values of path-dependent vulnerable option are lower than those of Klein

(1996) and Black & Scholes (1973), except for the case of s v = 0.2, which means the default possibility of counterparty is extremely low.

The above results can be reconfirmed in Table 3, which we let the counterparty’s asset value be 8 instead of 10 and other variables be the same as in

Table 2. Again, the price differences between Klein (1996) and Black & Scholes

11 (1973) are either zero or very small. On the contrary, our model still can effectively differentiate the credit risk from the non-vulnerable option pricing model and Klein model. In next section, we will perform sensitivity analysis for the path-dependent vulnerable option pricing model (7).

Table 2 Values of vulnerable options and non-vulnerable options (V0 =10)

Case α= 0 B & S Klein Base case# 3.0691 3.0697 3.0697

s s = 0.2 2.1638 2.1641 2.1641

s s = 0.4 3.9752 3.9760 3.9760

s v = 0.2 3.0697 3.0697 3.0697

s v = 0.4 3.0485 3.0697 3.0697

ρ= -0.5 3.0691 3.0697 3.0697 ρ= 0 3.0695 3.0697 3.0697 T = 0.0833 1.4603 1.4603 1.4603 T = 0.5833 4.1572 4.1806 4.1806 K = 30 10.5735 10.5744 10.5744 K = 50 0.4338 0.4341 0.4341

S 0 = 30 0.1485 0.1487 0.1487

S 0 = 50 10.9309 10.9319 10.9319 r = 0.02833 2.9388 2.9394 2.9394 r = 0.06833 3.2026 3.2031 3.2031

#: All calculations of options values based on same parameter values as in Table 1 except for V0 = 10.

Comparing with Klein (1996) model, Klein ignores the possibility that the counterparty may default within the option lifetime and hence overprices the vulnerable option with credit risk.

12 Table 3 Values of vulnerable options and non-vulnerable options (V0 =8)

Case α= 0 B & S Klein Base case# 3.0285 3.0697 3.0697

s s = 0.2 2.1369 2.1641 2.1641

s s = 0.4 3.9196 3.9760 3.9759

s v = 0.2 3.0694 3.0697 3.0697

s v = 0.4 2.8373 3.0697 3.0693

ρ= -0.5 3.0316 3.0697 3.0675 ρ= 0 3.0504 3.0697 3.0692 T = 0.0833 1.4603 1.4603 1.4603 T = 0.5833 3.9012 4.1806 4.1802 K = 30 10.4942 10.5744 10.5741 K = 50 0.4194 0.4341 0.4341

S 0 = 30 0.1418 0.1487 0.1487

S 0 = 50 10.8430 10.9319 10.9317 r = 0.02833 2.8946 2.9394 2.9394 r = 0.06833 3.1653 3.2031 3.2031

#: All calculations of options values based on same parameter values as in Table 1 except for V0 = 10.

V. Sensitivity Analysis

In this section, we analyze the key variables in model (7) how to affect the path-dependent vulnerable option prices. The results of sensitivity analysis are presented in Figures 1 to 7, which all options values are calculated with the same

parameter values in Table 1, except for V0 is 8 instead of 5 and α is 0.

In Figure 1, when initial asset value (V0 ) of option writer increases, leads both the counterparty’s debt ratio and default risk to decrease, therefore the path-dependent vulnerable option value increases. At the same time, the price differences between our model and those of Black & Scholes (1973) and Klein

(1996) are getting smaller due to the lower default probability.

13 We can find that, in Figure 2, as the volatility of counterparty’s asset value increases, the value of path-dependent vulnerable option decreases. This is due to the fact that larger asset volatility implies higher default possibility of counterparty, which will reduce the vulnerable option value.

Figure 1 The impact of counterparty asset value on vulnerable option value

Figure 2 The impact of standard deviation of counterparty asset value on vulnerable option value

14 The analysis results of ρis presented in Figure 3. Our model indicates that the effect of the correlation between counterparty’s asset value and underlying stock price on vulnerable option value is indeterminate. In the case of negative correlation between counterparty’s asset value and underlying stock price, the increase in ρ will cause an increase in vulnerable option value. On the other hand, in the case of positive correlation, as ρ is increasing, the vulnerable option value decreases.

This finding is substantially different from the result of Klein (1996) model which reports that the higher the ρis, the higher the vulnerable option value is. To confirm our finding, we also calculate alternative numerical examples and get the robust result. 3

Figure 3 The impact of coefficient of correlation between S and V on vulnerable option value

As in the properties of non-vulnerable , Figures 4 to 7 show the

3 We have changed the parameter values of S, s s , K, T, r, V, and s v , respectively, and consistently obtain the indeterminate effect of ρon vulnerable option value, that is a parabolic

15 vulnerable option value to be positively correlated with following variables: the price of underlying asset, the volatility of underlying stock, the lifetime, and the riskless rate. In other word, the higher the price of underlying asset, the greater the volatility of underlying stock, the higher the riskless rate, and the longer the lifetime will result in the higher the vulnerable option value. All these results are consistent with Black & Scholes (1973) option pricing model.

Figure 4 The impact of underlying stock price on vulnerable option value

From Figures 1 to 7, we can also clearly find the fact that the price gaps between Klein (1996) model and Black & Scholes (1973) model are always very small even equal to zero, no matter how the values of key variables change. These results show that, once under the assumption of well capital adequacy for the option writer, Klein (1996) model can’t truly reflect the effect of counterparty’s credit risk on vulnerable option. In contrast to Klein (1996) model, the model in this paper can

relation between vulnerable option value and ρas shown in Figure 3.

16 effectively differentiate the default risk from non-vulnerable option in all numerical examples except for the case of change in underlying stock price.

Figure 5 The impact of standard deviation of underlying asset value on vulnerable option value

Figure 6 The impact of riskless rate on vulnerable option value

17

Figure 7 The impact of lifetime on vulnerable option value

VI. Conclusions

Considering the credit risk of the option writer, this paper derives a path

-dependent vulnerable option valuation model which can effectively improve Klein

(1996) pricing formula quality and can truly reflect the impacts of counterparty credit risk on vulnerable option value. After putting the possibility of default risk prior to maturity into pricing model, we can find that Klein (1996) model overestimates the vulnerable option value. The theoretical value for a path

-dependent vulnerable option is lower than those of Black & Scholes (1973) model and Klein (1996) model.

From sensitivity analysis, we find that as counterparty’s asset value rises, the vulnerable option value also rises. However, the increase in the volatility of the counterparty’s asset value will lead to a decrease in vulnerable option value. On the other side, the most important result is that the effect of the correlation between

18 counterparty’s asset value and underlying stock price on vulnerable option value is uncertainty, which is different from the result of the positive relation in Klein (1996) model. In according with the properties of non-vulnerable call option, we also see that the price of underlying asset, the volatility of underlying stock, the riskless rate, and the option lifetime have positive effects on the vulnerable option value.

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