Calculating futures price

Question

Consider a world as follows:

$$\frac{dB}{B} = r_tdt$$ $$\frac{dS}{S} = r_tdt - 0.05dW_1 + 0.5dW_2$$ $$dr_t = 0.2 dW_1$$

where $r_0=0$. The Wiener processes $W_1$ and $W_2$ are independent. The price of any asset in this world is $$P_0 = E_0\left[\exp\left(-\int_0^T r_t dt\right)P_T\right ] $$

Calculate the futures price of a two-year futures contract on $S$.

My questions:

The futures price is just given by: $E_0\left[S_T\right ]$

But I am having trouble computing the above expression for the futures price.

Answer 1

So the first thing is to note that using Fubini (see here) $$ \int_0^T r(t) dt = \int_0^T \int_0^t dr(u) dt = \int_0^T \int_u^T dt dr(u) = 0.2 \int_0^T (T-u) dW_1(u) $$ such that $$ \int_0^T r(t) dt \sim \mathcal{N}\left( 0, 0.2^2 \, \int_0^T (T-u)^2 du = 0.2^2 \frac{T^3}{3} \right) $$ From that observation, in the expression $$ S_T = S_0\exp\left(- (0.05^2+0.5^2)\frac{T}{2}\right) \exp\left( \int_0^T r_t dt - 0.05W_{1}(T) + 0.5W_{2}(T) \right) $$ The last term on the RHS is a lognormal with mean $$ \mu = E_0\left[ \int_0^T r_t dt - 0.05W_1(T) + 0.5W_2(T) \right] = 0 $$ and variance (Itô isommetry + independence of $W_1$ and $W_2$) \begin{align} \sigma^2 &= \Bbb{V}_0 \left[ \int_0^T (0.2(T-u)-0.05) dW_1(u) + 0.5W_2(T) \right] \\ &= \int_0^T (0.2(T-u)-0.05)^2 du + 0.5^2 T \\ &= 0.2^2 \frac{T^3}{3} + 0.01 T + 0.05^2 T + 0.5^2 T \end{align} Now using the fact that the expectation of a lognormal with parameters $(\mu,\sigma^2)$ is $\exp(\mu+\sigma^2/2)$ you get \begin{align} F(0,T) &= \Bbb{E}_0[S_T] \\ &= S_0\exp\left(- (0.05^2+0.5^2)\frac{T}{2}\right) \exp\left( 0.2^2 \frac{T^3}{6} + (0.01 + 0.05^2 + 0.5^2)\frac{T}{2} \right) \\ &= S_0\exp\left(0.005 T + 0.04 \frac{T^3}{6}\right) \end{align}