The Actuarial Pricing Method Of Two Types Options

Option is one of financial derivatives. The option pricing is one of the core problem of financial mathematics. Traditional option pricing is based on the financial market there is no arbitrage, equilibrium and complete. This can not explain the real financial market so its application is subject to certain restrictions. In 1998 Bladt and Rydberg first put forward the actuarial option pricing approach, which does not involve in any economic assumption and can be used in the arbitrage, incomplete and non-equilibrium market. Bladt and Rydberg’s basic idea is that risk-free assets is discounted by the risk-free interest rate and risk assets is discounted by expected return. They think the actuarial price of European call option is CBR(S,t)= E[(eu(T-t)ST-e-r(T-t)K)1{e u(T-t)ST>e-r(T-t)K}]. where μdenote the rate of return of ST, γ denote the risk-free interest rate. In 2008, Zheng Hong and Guo Ya-jun give another actuarial price of European call option CZG(S,t)= E[(e-μ(T-t)ST-e r(T-t)1 {ST>K}}. According to these two definitions some scholar discussed the price of some options. This paper will study the following content:Firstly, this paper analyses the two actuarial definitions by pricing European call option. From obtained results we think the definition of Bladt and Ryberg is more reasonable.Secondly, under the assumption that risk-free interest rate follows the Vasicek model, we derive the pricing formula of European call and put options.Lastly, By the help of change of measure and martingale method, we derive the pricing formulas on Barrier options with exponential barrier.