Researches On Pricing And Variance-Optimal Hedging Of Path-Dependent Derivatives

The path-dependent derivative is one of the most important research objects in financial mathematics,and its unique design can help people manage risk or obtain extra revenues.In this paper,we consider the pricing problem of path-dependent derivatives with discrete dividend payments.Taking up-and-out call barrier option as an example and fixing the amount of the discrete dividend,an affine function of logarithmic variables is obtained through Taylor series expansion.We derive a new approximation pricing formula of barrier option with single fixed-amount dividend and with multiple fixed-amount dividends.Moreover,this formula only involves one-dimensional integrals and can improve computing speed,which is practical for the time saving and cost saving during trading.In addition,this method can be applied to pricing other derivatives,such as look-back options and so on,which has instructional significance for trading options in real markets.When it comes to the variance optimal hedging of path-dependent derivatives,we first list three typical path-dependent derivatives,which are Asian options,volatility swaps and target volatility options.Then we assume that the price of the underlying assent follows the discrete time process with independent increments,a F?llmer-Schweizer decomposition of the price of the underlying assent is presented.In this way,we derive variance optimal hedging strategies of three derivatives,and the corresponding variance optimal hedging error respectively.Furthermore,this method can be used in the two factors model and other non-stationary independent increments processes.It will be provided for investors and financial institutes with more efficient measurement models,and enhance people’s understanding on risk management.