Assume we have a non-dividend-paying stock governed by the stochasticdifferential equation dSt = µSt dt + σSt dzt and a risk-free asset carrying a fixed interestrate r The value B of the risk-free ass

To find A(T) and α, we need to use the fact that the price of the European contract satisfies the Black-Scholes equation:

∂P/∂t + rS∂P/∂S + (1/2)σ^2S^2∂^2P/∂S^2 - rP = 0

where r is the risk-free interest rate and σ is the volatility of the stock.

Since the European contract pays the kth power of the stock at time T, we have P(T,S) = S(k) if S > 0 and P(T,S) = 0 if S ≤ 0.

Using the terminal condition P(T,S) = S(k), we can find A(T) as follows:

A(T) = e^(αT)P(T,S) / S(k)

Substituting this into the Black-Scholes equation, we get:

∂P/∂t + rS∂P/∂S + (1/2)σ^2S^2∂^2P/∂S^2 - rP = 0

∂P/∂t + rS∂P/∂S + (1/2)σ^2S^2∂^2P/∂S^2 - r*e^(αT)P(T,S)/S(k) = 0

Let's make the change of variables u = ln(S) and v(t,u) = e^(-rt)P(t,S) to simplify the equation. Using the chain rule, we get:

∂v/∂t = -r*e^(-rt)P(t,S) + e^(-rt)∂P/∂t

∂v/∂u = e^(-rt)∂P/∂S * 1/S

∂^2v/∂u^2 = e^(-rt)∂^2P/∂S^2 * 1/S^2 - e^(-rt)∂P/∂S * 1/S^2

Substituting these into the Black-Scholes equation and simplifying, we get:

∂v/∂t + (1/2)σ^2∂^2v/∂u^2 - α∂v/∂u - r*v = 0

where α = r - (k-1)σ^2/2.

This is a heat equation with a source term. Using the method of separation of variables, we can assume that v(t,u) = T(t)U(u) and substitute into the equation to get:

T'(t)/T(t) + (1/2)σ^2U''(u)/U(u) - αU'(u)/U(u) - r = 0

Since the left-hand side depends only on t and the right-hand side depends only on u, they must be equal to a constant λ. Therefore, we have two ordinary differential equations:

T'(t)/T(t) - λ = -r

(1/2)σ^2U''(u)/U(u) - αU'(u)/U(u) - λ = 0

The solution for T(t) is:

T(t) = e^(λ-r)t

The solution for U(u) satisfies the differential equation:

U''(u)/U(u) - 2αU'(u)/(k-1)σ^2U(u) - 2λ/((k-1)σ^2) = 0

This is a Cauchy-Euler equation, which has the general solution:

U(u) = c1u^m1 + c2u^m2

where m1 and m2 are the roots of the characteristic equation:

m^2 - (2α/(k-1)σ^2)*m - 2λ/((k-1)σ^2) = 0

Solving for the roots, we get:

m1 = (1/2)*[(2α/(k-1)σ^2) + sqrt((2α/(k-1)σ^2)^2 + 8λ/((k-1)σ^2))]

m2 = (1/2)*[(2α/(k-1)σ^2) - sqrt((2α/(k-1)σ^2)^2 + 8λ/((k-1)σ^2))]

Therefore, the general solution for U(u) is:

U(u) = c1u^m1 + c2u^m2

Using the terminal condition P(T,S) = S(k), we have:

v(T,u) = e^(-rT)P(T,S) = e^(-rT)S(k)

Substituting v(t,u) = T(t)U(u) and solving for the constants c1 and c2 using the boundary condition v(T,u) = e^(-rT)S(k), we get:

c1 = (e^(rT)/2)*[S(k)/U(T,ln(S(k))) - U(T,-∞)]

c2 = (e^(rT)/2)*[S(k)/U(T,ln(S(k))) + U(T,-∞)]

Therefore, the solution for P(t,S) is:

P(t,S) = e^(r-t)S(k) * [f(u,t;α,λ,k) * S^(m1) + g(u,t;α,λ,k) * S^(m2)]

where u = ln(S), m1 and m2 are the roots of the characteristic equation, and f(u,t;α,λ,k) and g(u,t;α,λ,k) are given by:

f(u,t;α,λ,k) = (e^(α+λ)(e^(2(k-1)σ^2(t-T)) - 1) + 2(k-1)σ^2(e^α - 1))/(2(k-1)σ^2(e^α - 1))

g(u,t;α,λ,k) = (e^(α+λ)(1 - e^(2(k-1)σ^2(t-T))) - 2(k-1)σ^2(e^α - 1))/(2(k-1)σ^2(e^α - 1))

Finally, we can find A(T) and α using the fact that P(T,S) = S(k):

P(T,S) = S(k) = A(T)*S(k)^k * e^(-α(T-t))

Therefore, we have:

A(T) = 1/e^(α(T-t))

α = r - (k-1)σ^2/2

Note that α is the risk-neutral drift rate, which is different from the actual drift rate µ in the original stochastic differential equation. The risk-neutral drift rate is used to price the European contract under the assumption of no arbitrage