Pricing On Seller-Defaultable Perpetual American Option

In this paper we are going to discuss a special kind of options, which embed a seller-default clause in plain vanilla options(it allows the option sellers freedom to walk away bypaying a certain amount of penalty). We hereinafter call them/Seller-Defaultable Option0.Based on the types of corresponding options, the question comes into4part: Seller-Defaultable European Call Option, Seller-Defaultable European Put Option, Seller-DefaultableAmerican Call Option and Seller-Defaultable American Put Option. The first3type can beeasily solved by breaking each into two plain vanilla options. As for the Seller-DefaultableAmerican Put Option, the question gets a little complicated. So we are going to work it out.These are our basic assumptions:1. The price of underlying asset obeys:dS (t)=rS (t)dt+σS (t)dW(t).where r andσdenote the interest rates and volatility rates of underlying asset. W(t) repre-sents a Brownian motion under risk neutral measure P.2. Base option is Perpetual American Call option, with a strike price K.3. The option seller has the right to pay the penalty and walk away at anytime, includingthe moment buyer exercising the option. P is a positive constant and it denotes the amountof penalty.4. Efcient market, no dividend.And specially,x denotes S (0).Diferent from plain vanilla options, in this discuss we can’t study option buyer or sellerseparately. Because that although the put option owned by buyer and the default optionowned by seller are both literally Perpetual, the excising of either option will immediatelyterminate the opponent’s period of validity.We will start from studying the option seller’s strategy.Theorem1As the seller of a seller-defaultable perpetual American option, the opti-mal strategy is passively waiting for the excising of buyer, instead of following an actively default strategy.So the excising time of both option will be decided by the buyer’s strategy. Furthermore, the premium of seller-defaultable perpetual American option v*can be defined as: where S denotes the set of all stopping times.Different from the case of plain vanilla American put option, we have got an additional min[K一S(T),P]here.Our goal is to work out the optimal strategy of the option buyer. Beforehand, we have to work out the expected return of a random strategy.Theorem2If the option buyer’s strategy is:excising the put option once the price of underlying asset gets L. Then for a random L, the strategy’s expected return is:Then we will begin to discuss the optimal strategy of the option buyer, the existence of default option will cause some problems in the process. But thanks to the monotony of vL(x) in [K-P,K), there still exits a valid result.Theorem3The optimal target price of the option buyer is: it is in [K-P,K). Therefore,Substituting the L*in (1) with (2),then we will get the final answer:Theorem4The value of seller-defaultable perpetual American option is:In the end, we caculate the Delta and Gamma of seller-defaultable perpetual American option. It can be useful when people want to manage the overall risk via Delta and Gamma hedging.Theorem5...