Abstract
This paper considers the pricing of the CatEPut option (catastrophe equity put option) in a mixed fractional model in which the stock price is governed by a mixed fractional Brownian motion (mfBM model), which manifests long-range correlation and fluctuations from the financial market. Using the conditional expectation and the change of measure technique, we obtain an analytical pricing formula for the CatEPut option when the short interest rate is a deterministic and time-dependent function. Furthermore, we also derive analytical pricing formulas for the catastrophe put option and the influence of the Hurst index when the short interest rate follows an extended Vasicek model governed by another mixed fractional Brownian motion so that the environment captures the long-range dependence of the short interest rate. Based on the numerical experiments, we analyze quantitatively the impacts of different parameters from the mfBM model on the option price and hedging parameters. Numerical results show that the mfBM model is more close to the realistic market environment, and the CatEPut option price is evaluated accurately.
Highlights
As European put options, catastrophe equity put option provides the insurance company the right to sell a specified amount of its stock to investors at a predetermined price at the expiration date of the option in case the total accumulated losses of the insured surpass a given trigger level of losses during the life time of the option. e CatEPut option is a catastrophe risk management tool and is firstly issued by RLI Corporation in 1996 to cover the potential losses caused by catastrophes such as typhoons, hurricanes, storms, waves, and earthquakes
A CatEPut option differs from a European put option in two ways
Using the fractional Girsanov transform approach and the fractional Ito formula, we proposed the pricing models for the CatEPut options with constant or stochastic interest rates driven by the mixed fractional Vasicek model
Summary
As European put options, catastrophe equity put option (hereafter referred to as the CatEPut option) provides the insurance company the right to sell a specified amount of its stock to investors at a predetermined price at the expiration date of the option in case the total accumulated losses of the insured surpass a given trigger level of losses during the life time of the option. e CatEPut option is a catastrophe risk management tool and is firstly issued by RLI Corporation in 1996 to cover the potential losses caused by catastrophes such as typhoons, hurricanes, storms, waves, and earthquakes. Cox et al [1] first applied arbitrage-free approach to obtain a closed-form solution for the price of this option when the underlying asset price process satisfies the famous Black–Scholes model with the aggregate catastrophe losses of an insurance company following a Poisson process. Jaimungal and Wang [2] extended the work of Cox et al [1] to analyze the pricing and hedging of the CatEPut option in a stochastic interest rate framework suggested by Vasicek [3] and assumed that the aggregate catastrophe losses followed a compound Poisson process and the underlying asset price process depended on the total loss level rather than on only the total number of losses.
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