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This paper gives analytical formulas for lookback and barrier options on underlying assets that are exposed to a counterparty risk. The counterparty 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 valuate the lookback and barrier options by first conditioning on the predefault and the postdefault time and then obtain the unconditional analytic formulas for their prices.

Lookback and barrier options are among the most popular path-dependent options traded in exchanges and over-the-counter markets. A standard floating lookback call (put) option gives the holder the right to buy (sell) an asset (e.g., stock, index, exchange rate, and interest rate) at its lowest (highest) price during the life of the contract. In other words, the payoffs of a floating lookback call/put are, respectively,

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, e.g., Jarrow and Yu [

The explicit valuation of vanilla European options with this counterparty default risk was partly given by Ma et al. [

The outline of the paper is organized as follows. In Section

In this section, we consider a financial market model with a risky asset subject to counterparty risk: the dynamics of the risky asset is affected by the possibility of the counterparty defaulting. However, this stock still exists and can be traded after the default.

Let

The stock price process is governed by the following dynamic:

Assume that

In this section, we derive an analytical formula for pricing a floating strike lookback option, whose payoff is the difference between the maximum asset price over the life time and the asset price at expiration.

Using Ito’s lemma, the solution of the SDEs (

The risk-neutral price of the lookback option at time 0 under models (

The risk-neutral price of the lookback option (

We shall use the independence lemma (see [

In this section we derive an analytic formula for pricing barrier options under the counterparty risk models (

Consider an up-and-out barrier call option with expiry date

The risk-neutral price of an up-and-out barrier call option at time 0 under models (

We rewrite (

Then we compute the second term of (

Substituting (

This paper derives analytic formulas for lookback and barrier options when the underlying asset is subject to counterparty risk. The counterparty default risk induces a drop in the price of the risky asset (stock), and the stock can still be traded after this default time. This kind of counterparty risk causes difficulty in deriving the density of the first passage time for the maximum asset price. The conditional density approach, which is developed by Jiao and Pham [

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

The work was supported by Science and Technology Research Projects of Chongqing Education Committee (Grant no. KJ15012004). The author thanks Mr. Xiaosi Shen for making partial contributions to the paper.