PDE, Linear algebra, Probability.

PDE, Linear algebra, Probability. Partial di?erential equations 1. Consider the Black-Scholes problem @V @t + 12 "2 S2 @2V @S2 + r S @V @S - r V = 0, S>0, t<T, V (S, T) = f(S), (1) where " > 0, r 2 R and T > 0 are constants and f : R+ ! R is a given function. (i) Given that V (S, t), the solution of (1), is infinitely di?erentiable in S > 0 and t < T, show that V1(S, t) = S @V @S (S, t) also satisfies the partial di?erential equation in (1). (ii) Assume that V (S, t) and all its t and S-partial derivatives are di?erentiable with respect to the parameter r, for all S > 0 and t < T, and any value of r. Deduce that ?(S, t) = @V @r (S, t) satisfies the problem @? @t + 12 "2 S2 @2? @S2 + r S @? @S - r ? = V - S @V @S , S>0, t<T, ?(S, T) = 0. (2) (iii) Given that (2) uniquely determines ?(s, t) and assuming that both V and S @V/@S are order o ! 1/(T - t) " as t ! T-, show that for S > 0 and t ? T ?(S, t) = (T - t) ? S @V @S (S, t) - V (S, t) ? . (iv) You may assume that if K >0 is a constant and d = log(S/K) + (r - 12 "2)(T - t) p "2(T - t) , "(x) = 1 p2? Z x -1 e-p2/2 dp, then V (S, t) = e-r(T-t)"(d) is a solution of the Black-Scholes equation in (1), for S > 0, t < T, " > 0. For this particular V , and assuming S > 0 and t < T, find f(S) = lim t!T PLACE THIS ORDER OR A SIMILAR ORDER WITH US TODAY AND GET AN AMAZING DISCOUNT :)