Option pricing and hedging has been a focus of finance since the publication of the BlackScholes formula in 1973.Option pricing theory as developed by Black and Scholes rests on an arbitrage argument: by continuously adjusting a portfolio consisting of a stock and a riskfree bond, an investor can exactly replicate the returns to any option on that stock. The value of the option must, therefore, equal the value of the replicating portfolio. Option must, therefore, equal the value of the replicating portfolio. In the presence of transactions costs, the arbitrage argument used by BlackScholes to price option no longer can be used: because continuous trading would be ruinously expensive, no matter how small transactions costs might be as a percentage of turnover. Leland (1985) considered Option hedging strategies in the presence of transactions costs and developed a modified option replicating strategy which depends of the size of transactions costs and frequency of revision. Leland considered hedging strategies which involve adjusting a replicating portfolio at fixed equal time intervals. We generalize the Leland hedging strategy to include the case of Varying time intervals between rebalancing.Such strategies can be parameterized with a smooth, positive, strictly increasing function f(t) viaDifferent function f(t) yield different distributions of the rehedge time t_i,Note that taking f(t) = t would correspond to the constant interval caseconsidered in Leland (1985).We assume the stock value follows a stochastic differential equation:Where Bt is a standard Brownian motion , y. is the drift in the stock price ,and a is the volatility.We define the value of the stock to be held at the rehedge times ti to beWhere the function c(t,x) prescribes the composition of the portfolioHedging according to c(t,x) yields a random payoff u{XT) at expiry T ofThe first term on the right is the initial wealth , the second term is the number of shares of stock held between ti and ti+1 multiplied by the stock price change ,the third term is the expenditures in transaction costs duce to rebalacing, denote by Tc.We obtain the c(t,x) which yields perfect replication in a limit in which both the rehedge intervals ti+1 r, and p tend to zero at prescribed rates . They are theorem 1 and theorem 2 .Theorem 1 Assume that the payoff function u(x) has strictly positive secondderivative and that f(t) a 0 is strictly increasing with continuous secondderivative on [0,T]. Define v(t) s 7/'(0 > and let C(t,x) be the solution toWith c(T,X) = u(x) .Take the limit, 0,p 0 so that p = 2us(xT)*u(xT)andBoth limits are in L2Theorem 2 Under the conditions of theorem 1,e(v) = E{(us(XT)u(XT))2}Following these ,this paper talk about the optimization criteria , our goal is toprescribe an initial portfolio value c(0,x0) and to then find the hedging strategywhich minimizes the replication error. We defineThe replication error can how be writtenThe goal is to find extremals of e(z) subject to the boundary conditions z(0) = 20 and z(T) = 0 .The associated EulerLagrange equation is...
