The Valuation of Option Subject to Default Risk Shen-Yuan Chen Department of Finance, 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 margin settlement mechanism for unlisted options, thus the credit risk 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 expiration 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 derivative 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 stochastic process 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: -rT * C = e E{max( ST - K, 0)[(1| "Vt ³ D , 0 £ 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.