Abstract
A new framework for pricing European vulnerable options is developed in the case where the underlying stock price and firm value follow the mixed fractional Brownian motion with jumps, respectively. This research uses the actuarial approach to study the pricing problem of European vulnerable options. An analytic closed-form pricing formula for vulnerable options with jumps is obtained. For the purpose of understanding the pricing model, some properties of this pricing model are discussed in the paper. Finally, we compare and analyze the pricing results of different pricing models and discuss the influences of basic parameters on the pricing results of our proposed model by using numerical simulations, and the corresponding economic analyses about these influences are given.
Highlights
Options are popular financial derivatives that play essential roles in financial markets
In 1973, a famous option pricing model was introduced by Black and Scholes [1], which paved the way for the wide application of options in financial markets
Assume that the stock price pays a continuous dividend yield q; the pricing formulae of vulnerable (41) options can be given when S0 is replaced with S0e− qT in (17) and (18), and r is replaced with r − q in the first set of parameters to the bivariate normal distribution function
Summary
Options are popular financial derivatives that play essential roles in financial markets. Wang [12] carried out research into the pricing problem of vulnerable options by assuming that the underlying asset price and the value of counterparty asset both followed jump-diffusion processes and the default barrier was stochastic. Considering the long-range dependence of the underlying asset returns, Li and Wang [43] studied the valuation of the bid and ask prices of European options under the MFBM environment and obtained the explicit formulae for the bid and ask prices by using WANG-transform as a distortion function. Considering the long-range dependence and abnormal fluctuations of financial assets, a mixed jump fractional Brownian motion pricing model for vulnerable options combining the MFBM and jump process is constructed. Ji(t) is jump size percent at time t which is a sequence of lognormal, independent, and identically distributed variables with mean ln(1 + θi) − (σ2Ji/2) and variance σ
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