(i) Verify that f(S, t) = e (2r+3σ 2 )(T −t)S 3 is a solution of the Black-Scholes partial differential equation ft + rS

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(i) Verify that
f(S, t) = e
(2r+3σ
2
)(T −t)S
3
is a solution of the Black-Scholes partial differential equation
ft + rSfS +
1
2
σ
2S
2
fSS = rf
1
2
with f(0, t) = 0 for 0 ≤ t ≤ T and f(ST , T) = S
3
T
(ii) Consider a derivative with underlying asset whose price S follows the Ito process
dS = µSdt + σSdB and which provides a single payoff at time T > 0 in the
amount of S
3
T
, where ST is the underlying asset price at time T. What is the
price of this derivative at time 0 ≤ t < T? (Hint: The answer is not far away.)
(i) Verify that f(S. t) = e(21+302)(n-1) is a solution of the Black-Scholes partial differential equation Sa+rSfs +25° fss = rf 1 2 with f(0,0) = 0 for 0 St ST and S(Sr. T') = SF (ii) Consider a derivative with underlying asset whose price S follows the Ito process dS = Sdt + SdB and which provides a single payoff at time T > 0 in the amount of S, where Sr is the underlying asset price at time T. What is the price of this derivative at time 0 St<T? (Hint: The answer is not far away.)