| This paper is an overview. Vulnerable options, simply speaking, means to contain the credit risk of the options, options is one of the financial derivatives, credit derivatives are important tools to transfer credit risk, and pricing theory of credit risk is theoretical basis of credit derivatives transaction. Market can be divided into complete market and incomplete market, regardless of the East and the West, market is inherently incomplete. In this paper, we overview a class of vulnerable European call option model in complete market and incomplete market respectively, and review their respective advantages and disadvantages.Vulnerable options was defined by Johnson and Stulz in 1987. However, the general view, Merton(1974) pricing structure model of credit risk was the originator, this model applied Black-Scholes option pricing model directly. First of all, this paper introduced the Merton(1974) structure model in complete market. However, because its assumptions were too harsh, so a number of conditions were not equate with the fact.secondly, this paper introduced two theoretical approaches to model this effect when pricing bonds subject to credit risk. And we list some paper. The first defined an event of default as occurring when the value of the assets of the firm is less than some boundary, for example the value of outstanding debt, at the maturity of the bond under consideration. The second approach to modelling the effect of credit risk on bond prices allows greater flexibility in the timing of an event of default. Then we mainly introduce some papers stand out in the area.In 1987, Johnson and Stulz followed the first approach when deriving pricing formulas for vulnerable European options. Their research was in fact a promotion of Merton model, it was a pity that Johnson and Rene Stulz did not derive analytical expression. But they pointed out the substantial characteristics of vulnerable options.In 1995, Hull and White(1995) took the second approach to modelling the effect of credit risk on American and European options. The model assumed competing claims can exist which rank equally with the option in default. Default could occur at any time prior to maturity of the option if the value of the assets of the option writer failed below a specified default boundary. They assumed that in default only a proportion of the nominal amount of a claim is paid out. They allowed this proportion to be a general function of a number of variables, but did not relate it directly to the value of the assets of the counterparty. At last, they derived analytical expression of pricing vulnerable options.In 1995, Jarrow and Turnbull(1995) also took the second approach when considering the effect of credit risk on fixed income and other options. They modeled not only the effect of credit risk of the option writer, but made the further contribution of modelling the effect on option prices of credit risk in the asset underlying the option. They also assumed for simplicity that the payout ratio is exogenous. The exogenous payout ratio assumed in these models implied that the value of the assets of the financially distressed firm and the nominal amount of the claims on that firm were fixed at the time of default. This implication is inappropriate, because the imposition of an exogenous payout rate may overstate the effect of credit risk on option values.In 1996, Peter Klein(1996) followed the first approach to modelling credit risk. This approach allowed for the presence of other liabilities and a proportional recovery of nominal claims in default, but had the advantage of explicitly relating the payout ratio to the value of the assets of the counterparty. It also allowed correlation between the assets of the counterparty and the asset underlying the option. And they showed credit spread through the price of tradable bonds for the first time:where r-r* represented the difference in yield between risky zero coupon bonds of the counterparty and a similar term riskless zero coupon bond, N1 was standard normal cumulative distribution function,And they finally obtained the analytical solution of option prices. But the assumption that the border were not dependent on the value of the option itself was clearly unreasonable. In 2001, Peter Kleinå’ŒMichael Inglis extended the model of Peter Klein(1995),they allowed the existence of other debt in the capital structure of options seller, and referred that the growth of the value of the option itself will lead to the occurrence of an event of default. However, because the assumption is too strong, they finally obtained a approximate solution.Finally, this paper also introduces the vulnerable options in incomplete market. Since in the incomplete market, the existence of non-tradable assets, the hedge does not apply to the meyhod of option pricing. In 2000, Cochranc and Sa(?)-Requejo proposed the" discount factor" asset-pricing method to deal with the evaluation of uncertain payoffs. In 2005, Mao-Wei Hungå’ŒYu-Hong Liu obtained a similar vulnerability option pricing formula in the incomplete markets by the method. But they also made the same unreasonable assumption that the border were not dependent on the value of the option itself. |
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