Valuation Of Game Options And Some Applications

Option Pricing has been a research hot-spot and also been a cutting-edge issue ofmathematical finance region so that the research on it has a profound theoretical andpractical significance. Besides, with the development and needs of the market, more andmore novel options are presented. In this dissertation, we mainly study the problems ofpricing and hedging Game options. And for some special type of Game options, expressionsof the price and optimal portfolios of the holder and writer are obtained under di?erentconditions. In addition, we also consider some applications of the Game option and themain work and results include:1. The problem of pricing game option in complete market is considered. We intro-duce barrier in the game option for the first time, and when the interest rate is constant,the closed-form expressions for the prices of the barrier option of game type and someother game options are obtained for the cases that the volatility are constant and noncon-stant respectively. Especially for the research of Russian game option, which improve theexisting results. while for the case of random interest, we extend the results of Ekstr¨omand obtain the expressions for the price of game option and conditions under which theoptimal strategy for the holder exists. Finally we give two examples to illustrate thevalidity and practicability of the result.2. The problem of pricing game option in complete market is considered. Herethe so-called incompletion of the market means that the risky value process has jumps.By introducing the restriction to the filtration of Radon-Nikodym derivative of everyequivalent martingale measure with respect to primary measure, we obtain the analyticexpressions of some game options discussed above. Furthermore, we obtain the price ofsome convertible bonds by applying this result(method) to it.3. The problem of hedging game contingent claims is considered. First we systemat-ically give the definitions of upper-hedging price, lower-hedging price and arbitrage. Thenwe extend the result about American option of Karatzas to game option and obtain theexpressions of the upper-hedging price and lower-hedging price. In addition, an arbitrage-free interval is also obtained. Similarly we give the upper-hedging price and lower-hedgingprice of game contingent claims with constrained portfolios under proportional transac-tion costs. While for the problem of hedging game contingent claims in general incompletemarket, we also obtain the expressions of the upper-hedging price and lower-hedging price by using optional decomposition of super-martingales of Kramkov.