Several Researchs On Pricing Formulates And Hedging Stratagems Of Contingent Claims

Block-scholes pulished a originality paper about option pricing in1973, since that time, numerous researchers have made important expansion andget extensive of application, such as marton,cox and rebinstein, option pricingtheories and application always be developing constantly and fertilizing enduringly,have alreadly extended as the more abstract theories for contingent claim and themore extensive sense for assete pricing theories, the paper followed evolution anddevelopment of contingent claim's pricing formulate and hedging stratagems over 30years as clue, by using the methods of measure theories and stochastic analysis, dis-cussed and analyzed several questions of pricing formulate and hedging stratagemsfor contingent claims, as well optimal stopping of American option and perpetualAmerican option.The paper's main results and innovation:In the first place, bsed on the assumption that there were time-dependent in-terest rateγ(t),expeced rate of retureμ(t),volatilityσ(t),divident yieldρ(t)existed in the market, Pricing formulas and stratagems of hedging and preservingvalue for European contingent claims are discussed with no risk neutral valuation.By using the distribution of the process of option price and the equivalent martingalemeasure, pricing formulas for generalized European option is given in the two cases:having dividend and having no dividend, and a put-call party of European optionsis also obtained, using Ito formula, some stratagems of hedging and preserving valuefor European buy and sale options are posed, This results of this paper generalizethose with risk neutral valuation.Secondly, Bsed on the assumption that underlying asset's lognormal distribu-tion and multiple risky aseets tread with constant transaction costs, the definitionof the preferred hedging is at first introduced; by using the methods of auxiliarymartingales, we obtained pricing interval[H₂^*, H₁^*] and hedging stratagems for pre-ferred hedging Aermican contingent claims under transaction costs, the least priceh_low the seller would accept and the greatest price h_up the buyer can afferd are alsoobtained.Thirdly, convert the problems of how to price and hedge contingent claims intothe problems of how to project a vector of Hilbert space into its closde sup-space, byusing the Galtchouk-Kunita-Watanabe theory to decompose contingent claims un- der gived martingale measure, applied the theory of orthogonal projection to solvethe optimal hedging strategems of derivative asset under the incomplete market,with the assumption that the price process of the underlying asset is martingale.Then, by the way of extending orthogonal projection, the paper explored the prob-lems of how to price and hedge the derivative asset, based on the knowledge ofmarket price information of underlying asset and other assets, the optimal hedgingmixed portfolio strategems and its corresponding approximate price of the assetsare obtained too. finally, by using the variance approximation theory to find theconcrete portfolio strategemsβof contingent claims with discrete state under theincomplete market.Finally, consider return utility function under specific conditions U(x)=(X_t-K)⁺, by uing methods of optimal stopping theories obtain optimal stopping for-mulas of American option bsed on the assumption that the process of underly-ing asset's pricing followed jump, make sure that American option's best treadtime is Expiration Date T, as the same time American options became Euro-pean options, and options'initial value is C₀^*=E^* X_T-K/e^-γT; Next by using the meth-ods of martingales, discussed optimal stopping formulas of perpetual Americancontingent claims h(X_T)=(K-(П_j=1^N_t(1+U_j))^eX_t)⁺bsed on the assumptionthat the process of underlying asset's pricing followed jump, obtain its best treadtime isγ^*=inf(t≥0:σW_t=x^*-(r-1/2σ²-λE(U₁))t}, its initial value isC^*=e^-γ2x*(K-(П_j=1^N_t(1+U_j))^ex*)⁺. followed the two questions'discussing, inorder to find the equivalent martingale measure of the process of underlying asset'spricing, it shoud have sufficient conditionμ=γ-λE(U₁).