Pricing Of Power Options And Exchange Options With Default Risk

An option gives the buyer the right but not the obligation to buy(or sell)a certain amount of underlying assets for a predetermined price in a period of time(American options)or at a certain date(European options)in the future.Buyers can get this right after they pay the option premium to the seller.Credit risk,or default risk,means that the option holder may suffer an economic loss for the counterparty not obey the option contract.With the rapid development of the world economy,the phenomenon of default in financial transactions has become more and more serious because of the financial crisis and other uncertain factors.In recent years,the over-the-counter market has great credit risk with it's rapid development,since there is no guarantee from the third party.So the study of options pricing with credit risk has become one of the most popular research topics.This paper focuses on the pricing of the Power option and the Exchange option with credit risk.The credit risk model used in this paper is a structural model.Let constant L be the default level.If the counterparty company's asset value was less than L at some time prior to the expiration date,the counterparty could default at exercise date.At this case,the income of the option holder is the product of the value recovery rate and the non-default income.When the underlying asset value and the counterparty company's asset value follow special differential equations,we can derive the price formulas of the Power option and the Exchange option with credit risk.Firstly,assuming that both the underlying asset value and the counterparty company's asset value follow the Geometric Brownian processes,we can get the prices of the European call and put power option with credit risk at time 0 by using the change of measure.Secondly,assuming that the underlying asset value follows the jump-diffusion process and the asset value of the counterparty company follows the Geometric Brownian process,we can get the prices of the European call and put power option with credit risk at time0 by using the change of measure.Lastly,assuming that the underlying asset value and the counterparty company's asset value follow the Geometric Brownian processes,we can get the price of the European exchange option with credit risk at time 0 by using the change of measure.