The Black-Scholes pricing problem for a European binary put optionis given by the partial differential equation and payoff condition

2. The Black-Scholes pricing problem for a European binary put option is given by the partial differential equation and payoff condition av ) = { 1 ,0 trs at S1 SE V (S,t) is the option value, S is the underlying asset price, t is time, T is expiry, r is the constant risk-free interest rate, o is the constant volatility and E is the strike price. a. By considering the following change of variables C log 2 E t' (T-1), w(x, t') w (x, t') V(S,t), e-at'u (x, t') where a = 2r/o2 and finally z = x + (a 1)'T=t', show that the resulting diffusion equation and final condition are given by au #z2 U (3,0) = { 1 2 0 b. The solution of the above diffusion equation is given in terms of the fundamental solution by u(2,7) 21a Sf(u)e+(3-57%/+dy, u(3,0) = f(2) = { 1 2 0 Use this to calculate the option value and show that it is given by V(S,t) = e ="(T-1) N(d2). 2. The Black-Scholes pricing problem for a European binary put option is given by the partial differential equation and payoff condition av ) = { 1 ,0 trs at S1 SE V (S,t) is the option value, S is the underlying asset price, t is time, T is expiry, r is the constant risk-free interest rate, o is the constant volatility and E is the strike price. a. By considering the following change of variables C log 2 E t' (T-1), w(x, t') w (x, t') V(S,t), e-at'u (x, t') where a = 2r/o2 and finally z = x + (a 1)'T=t', show that the resulting diffusion equation and final condition are given by au #z2 U (3,0) = { 1 2 0 b. The solution of the above diffusion equation is given in terms of the fundamental solution by u(2,7) 21a Sf(u)e+(3-57%/+dy, u(3,0) = f(2) = { 1 2 0 Use this to calculate the option value and show that it is given by V(S,t) = e ="(T-1) N(d2)