The Pricing Of Permanent American Options And Game Options In The Presence Of Event Risk

The pricing of options is the central problem for research in mathematical finance. For the result of pricing, the closer to the fact the better. In a financial market, credit risk events often occur. Consequently, it is more close to the fact that we consider a financial market model in the presence of credit risk. Suppose the risky event occurs at random time τ, usually τ is unobservable. That's to say, τ is not a stopping time with respect to the asset filtration F = {F_t}_t∈R⁺. In order to obtain a specified pricing formula, usually we need to make some reasonable hypothesis in financial market.In this paper, as in Kusuoka(1999), the hypothesis(H) is assumed to hold: Any F-square integrable martingale is a G-square integrable martingale. This hypothesis is very nature. Where G is the minimal enlargement of F such that τ is a stopping time with respect to the enlarged filtration G. In the sense of this hypothesis, credit risk is called event risk. For European contingent claims in a event risk setting,pricing and hedging of European contingent claims has studied and solved by Lando(I998), Elliott et al.(2000), Blanchet ,Jeanlanc(2004), Collin Dufresne,Hugonnier(2000). Alex Szimayer(2005) has studied the valuation of American options in the presence of event risk and solved its optimal stopping problem. On the basis of their work, this paper introduces and extends the mathematical tools which we need in study of this model. Besides, we introduce a new derivative security — Game option and its pricing. Game options are contracts which enable both their buyer and seller to stop them at any time and then the buyer can exercise the right to buy (call option)or sell(put option) a specified securityfor certain agree price. If the contract is terminated by the seller he must pay certain penalty to the buyer.In this paper, we mainly use the properties of (H)hypothesis and snell envelope to study the valuation of permanent American options and Game options in the presence of non-hedgeable event risk ; for a given equivalent martingale measure, the optimal stopping problem of the permanent American option is solved. As a main result, no-arbitrage bounds for permanent American option values in presence of event risk are derived, as well as hedging strategies corresponding to the no-arbitrage bounds.