European Vulnerable Option Pricing In Vasicek Environment With Double Fractional Jump Diffusion

Many domestic and foreign researches on option pricing are based on geometric Brownian motion or fractional Brownian motion.Double fractional Brownian motion,which not only has adaptability and long-term memory,but also has incremental non-stationary,is more in line with the actual situation of the stock.When an option is traded on the market,since strict laws and regulations of relevant departments as well as exchange,the two parties almost never default.However,there is a big difference in over-the-counter trading,because there is no institution like the clearing house to supervise the seller's due performance obligations of the option,and the buyer of the option has to bear the risks of market risk and credit risk.This paper investigates the pricing of vulnerable options with both risks.This paper assumes that the stock price is a stochastic process,which driven by the double-fraction Brownian motion and the jump process.The short-term market interest rate and the default intensity obey the more general double-fraction Brownian motion under the Vasicek model.The quasimartingale method combined with the multi-dimensional Girsanov's theorem is used to measure the transformation.European vulnerable options have been studied for pricing.Firstly,the problem of European fragile option pricing under the default strength model is studied.Because the traditional process of transform can not be applied to the jump process,the jump process is subjected to measure transformation and its expression after the measure transformation is calculated.Thus it can be used under the new measure.Using the Girsanov theorem to measure the transformation,the pricing formula of the European vulnerable put option is obtained.Then,combining the default strength model with the structural model,we assume that the company's asset price also obeys the double-fraction Brownian motion.Therefore,using the measure transformation and the pseudo-rhythm method to calculate,we obtain the hybrid model.The pricing formula for European vulnerable put options is desired.