2 (c) The Black-Scholes equation for the value of an option V(S, t), where S is the value of the underlying asset and t is time, can be transformed into the diffusion equation for a new variable u(x, 7) using S = Ee", t=T 27/02, k = 2r/o2 and - 2 V = Eu (, T) exp . x - exp(- * = ;*_=790 (k 1) x 2 (k + 1)^T 4 where r is the risk-free interest rate, o is the volatility, E is the strike price, and T is the expiry date. The diffusion equation has the solution 1 u(x, 7) = T 2VTA = 2. o(s)exp (13-17) din L Up () 1 (x s)? ) ds. 4T Determine, showing all steps, the function uo(s) for a European asset- or-nothing call with the pay-off A(S) = SH(S E). = 2 (c) The Black-Scholes equation for the value of an option V(S, t), where S is the value of the underlying asset and t is time, can be transformed into the diffusion equation for a new variable u(x, 7) using S = Ee", t=T 27/02, k = 2r/o2 and - 2 V = Eu (, T) exp . x - exp(- * = ;*_=790 (k 1) x 2 (k + 1)^T 4 where r is the risk-free interest rate, o is the volatility, E is the strike price, and T is the expiry date. The diffusion equation has the solution 1 u(x, 7) = T 2VTA = 2. o(s)exp (13-17) din L Up () 1 (x s)? ) ds. 4T Determine, showing all steps, the function uo(s) for a European asset- or-nothing call with the pay-off A(S) = SH(S E). =