Question

Let aſt),o(t), and r(t) be non-random adapted processes. For a stock process that follows: dS(t) = a(t) S(t)dt + o(t)S(t)dW(t) and a discount factor given by: D(t) = e-Soudu derive the formula at any time 0 <t< T for a forward agreement to purchase the stock at time T for price K.

Answer 1

s(t): represent the sigma(t)

dS(t) = a(t)S(t)dt + s(t)S(t)dW(t)

dS(t)/S(t) = a(t)dt + s(t)dW(t)

Integrating on both sides we get

$\int_{0}^{T}dS(t)/S(t) = \int_{0}^{T}(a(t)dt + s(t)dW(t))dt$

$ln(S(T)/S(0)) = \int_{0}^{T}(a(t)dt + s(t)dW(t))dt$

S(T) = S(0) *e * elit (aſt)dt + s(t)dw (t))dt)

Forward price: F(T)

F(T) = S(T)/D(T)

$F(T) = S(0)*e^(\int_{0}^{T}(a(t)+s(t)dW(t))dt)/e^(-\int_{0}^{T}r(u)*du)$

$F(T) = S(0)*e^(\int_{0}^{T}(a(t) + s(t)dW(t))dt+\int_{0}^{T}r(u)*du)$

$F(T) = S(0)*e^(\int_{0}^{T}(a(t) + s(t)dW(t)+r(t))dt)$