Actuarial Pricing Of Option Under Uncertainty Theory

With the rapid development of financial derivatives market,people pay more attention to price the option.Traditional pricing methods are mostly based on probability theory,while the samples are needed to estimate the probability distribution.But when no samples are available,we have to invite some domain experts to evaluate the belief degree of the event occurred.In order to rationally deal with belief degrees,Professor Liu Baoding establishs uncertainty theory in 2007.Uncertainty theory is a mathematical branch based on normality,duality,subadditivity and product axioms,which opens up new ideas for option pricing.The classic option pricing model assumes that the market is a complete,arbitragefree and equilibrium,but in the real world it does not exist.In 1998,Mogens Bladt and Tina Hviid Rydberg put forward the actuarial approach to price option.They turn the option pricing into fair premiums.The basic idea is that risk-free asset is discounted by the risk-free rate,and the risk asset is discounted by the rate of expected return.This method does not make any assumption on financial markets and is suitable for a wide range of real financial markets.In this paper we study the pricing of standard European options and lookback options based on the uncertainty theory by the actuarial approach.The main contents are as follows:Firstly,we give the definition of actuarial prices for standard European call and put option and show that put-call parity holds between call and put option.Secondly,assuming that the underlying asset price follows the geometric Liu process with time-varying parameters,we derive the analytical pricing formulas of the standard European call and put option by using the properties of the uncertain Liu process.The result extends the previous one.Thirdly,assuming that the underlying asset price follows the geometric process with jumps,we obtain the analytical pricing formulas of the standard European call and put option by using the properties of the uncertain Liu process and the renewal process.Lastly,we give the definition of actuarial prices for partial lookback options and fraction lookback options.We assume that underlying asset prices follow the geometric Liu process with constant parameters,and give the price of these options by using the properties of the uncertain Liu process.