1. The Black-Scholes pricing problem for a European binary put option is given by the PDE and final condition av 1 + 02 82 Ət 2 მS2 +rS -rV=0, V (S,T) = 1 S<E av as 0 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, σ is the constant volatility and E is the strike price. a. By introducing the following change of variables x = log E t' = ½ o² (T− t), w(x,t) = V(S,t), show that the problem can be reduced to θω J²w θω + (a 1) მე2 - αω, ω дх ; (x, 0) = { 1 x <0 0x0 It' = where a = 2r/02. b. Show that the term -aw can be eliminated by a further transfor- mation of variable -at' น w(x, t') = e¯atu (x, t') c. Finally show that the drift term will disappear by putting z=x+(a1)t', T=t', resulting in the diffusion equation and final condition ди J²u Эт Oz2 и (2,0) 1 <0 = 0 z>0 d. The solution of the above diffusion equation is given in terms of the fundamental solution by 1 u(z,T) = = f(y)e-(-2)²³/4 dy, u(2, 0) = ƒ (²) = 1 2<0 2√7 пт 0 20 Use this to calculate the option value and show that it is given by V(S,t) er(T-)N(-d₂). =