3. This question is on option pricing using Black-Scholes The Black-Scholes pricing problem for a European binary call op- tlon is given by the partial differential equation and nal condition 6V 1 282V 6V 1 S > E 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, a is the constant volatility and E is the strike price. a. [25 Marks] By introducing the following change of variables 8' :1; =108 (E) , t' = $02(T t), w(a:,t') = V(S,t), ShOW that the problem can be reduced to 6w 6%) 8w 1 :1: > O 6t16?+(a1)%'-awa \"(3320):{0 $30 Where a = 2r/02. [20 Marks] b. [10 Marks] Show that the term -aw can be eliminated by a further transformation of variable 4 w (as, t') = (\"u (2:, t') c. [15 Marks] Finally show that the drift term will disappear by putting z=x+(a-1)t', T=t', resulting in the diffusion equation and nal condition 6a 821 1 z > 0 5029 \"(MD{0 230 d. [30 Marks] The solution of the above diffusion equation is given in terms of the fundamental solution by lz>0 1 2 = ____ (yz) /41- = _ Use this to calculate the option value and show that it is given by m: t) = ere-0W2)