The Optimal Hedging Strategy Of Contingent Claim In The Incomplete Market

There is no arbitrage opportunity in the classic no-arbitrage theory, so the contingentclaim can be perfectly hedged, but in the incomplete market which has many opportunitiesthe contingent claim can’t be perfectly hedged. A given contingent claim can have differentmethods to achieve a sense of the optimal hedge, this paper let the strategy which requirethe minimal initial capital and achieve a given terminal payoff be the optimal hedgingstrategy, and study the optimal hedging strategy under the condition that the asset pricemodel defined by Bessel process is continuous time markov process.Firstly, this paper defines the asset price model and the trading strategies，and get“market price of risk”(MPR) which generalizes the concept of Sharpe ratio to severaldimensions in the market model，then the “stochastic discount factor”(SDF) which isrepresented by “market price of risk”(MPR) will be given. By Markovian MPR weconstruct the hedging price function. Secondly, we prove that the hedging price function canbe represented by an ito process and simplify it to be a wealth process, the trading strategyof the wealth process is the optimal hedging strategy, furthermore, this paper rely onnecessary assumptions and corresponding theories to derive the sufficient conditions for thedifferentiability of the hedging price function. finally, we construct a new probabilitymeasure to simplify the dynamics of the stock price process and the reciprocal of the SDF，then the computations of the optimal hedging price function and the optimal hedgingstrategy is also simplified.At the end of this paper it will provide a asset price model which contains the auxiliaryn-dimensional Bessel process with drift and not to get the specific optimal hedging strategy.according to the results which have be proved, we construct the hedging price function byMarkovian SDF and simplify it to the form of wealth process to get the optimal strategy,by comparing with existing examples, this paper confirms that the strategy is the optimalhedging strategy.