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We derive analytical formulas for European call and put options on underlying assets that are exposed to double defaults risks which include exogenous counterparty default risk and endogenous default risk. The endogenous default risk leads the asset price to drop to zero and the exogenous counterparty default risk induces a drop in the asset price, but the asset can still be traded after this default time. A novel technique is developed to evaluate the European call and put options by first conditioning on the predefault and the postdefault time and then obtaining the unconditional analytic formulas for their price. We also compare the pricing results of our model with default-free option model and counterparty default risk option model.

Over the past few decades, academic researchers and market practitioners have developed and adopted different models and techniques for pricing European option. The path-breaking work on option valuation was done by Black and Scholes [

In the financial market, a counterparty default usually has important influences in various contexts. In terms of credit spreads, one observes in general a positive jump of the default intensity which is called the contagious jump (see Jarrow and Yu [

The explicit valuation of European options with assets exposed to exogenous counterparty default risk was partly given by Ma et al. [

The rest of this paper is organized as follows. In the next section, we introduce the financial model and change from the actual probability measure

In this section, we consider a financial market model with a risk asset (stock) subject to double default risks. We denote the stock by

Assume

According to Bielecki and Rutkowski [

Assume that

Let us define the

Consider a European option, expiring at time

If the dynamic of stock price process follows model (

Let

Combining the distribution function of the stock price at expire time

The risk-neutral price of the European call option at time 0 under model (

According to the distribution function of

(1) If

(2) If

Call option prices versus stock prices for double defaults and counterparty default models with

Call option prices versus stock prices for double defaults and Black-Scholes models with

The risk-neutral price of the European put option at time 0 under model (

According to the following identity,

In this paper, we have derived explicit analytical formulas for the price of European call and put options when the underlying asset is subject to double defaults risks. The external counterparty default risk induces a drop in the price of the stock, and the stock price drops to zero when the stock itself defaults. Double defaults risks cause difficulty in deriving the distribution function of the stock price at expire time

No data were used to support this study.

The author declares that there are no conflicts of interest regarding the publication of this paper.

The work was supported by the NSFC (Grant no. 11301257).