Narrow Results

We present the no-arbitrage price and the hedging strategy of an European contingent claim through a representation formula which is an extension of the Clark-Ocone formula. Our formula can be interpreted as a second order Taylor formula of the no arbitrage price of a contingent claim. The zero order term is given by the mean of the contingent claim payoff, the first order term by the stochastic integral of the mean of its Malliavin derivative and the second order term by the stochastic integral of the conditional expectation of the second Malliavin derivative. A Taylor series expansion is also provided together with a bound to the approximation error obtained by neglecting the second order term in the Taylor formula.

We assume that the call option's value is correctly priced by Black and Scholes' option pricing model in this paper. This paper derives an exact closed-form solution for implied standard deviation under the condition that the underlying asset price equals the present value of the exercise price. The exact closed-form solution provides the true implied standard deviation and has no estimate error. This paper also develops three alternative formulas to estimate the implied standard deviation if this condition is violated. Application of the Taylor expansion on a single call option value derives the first formula. The accuracy of this formula depends on the deviation between the underlying asset price and the present value of the exercise price. Use of the Taylor formula on two call option prices with different exercise prices is used to develop the second formula, which can be used even though the underlying asset price deviates significantly from the present value of the exercise price. Extension of the second formula's approach to third options value derives the third formula. A merit of the third formula is to circumvent a required parameter used in the second formula. Simulations demonstrate that the implied standard deviations calculated by the second and third formulas provide accurate estimates of the true implied standard deviations.