Convergence with respect to the parameter of a series and the differentiability of barrier option prices with respect to the barrier
Анотація
We obtain the weak convergence of measures generated by the price process and prove the continuity of the price of a barrier call option with respect to the parameter of a series for the BlackâScholes model of a complete market. The explicit form of the price of the barrier option is not required. The result obtained allows one to prove the continuity of a solution of the corresponding boundary-value problem for the parabolic partial differential equation with respect to the parameter of a series. Applying the Malliavin calculus, we establish the existence of a bounded continuous density of the distribution of a Wiener integral with shift restricted to an arbitrary âpositiveâ ray and prove the differentiability of the fair price with respect to the barrier (the differentiability with respect to other parameters is a classical result).
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