Deriving a Futures Price with a Stochastic Interest Rate and Correlated Risk
Summary
The document derives the futures price for an asset whose price process and short rate are driven partly by the same Brownian motion. Since the futures price is the expected future spot price, the solution first expresses the integrated short rate as a stochastic integral using a change in integration order. This yields a normally distributed term. The log of the terminal asset price can then be written as a sum of stochastic integrals, with the shared Brownian driver creating covariance between the integrated rate and the asset’s return.
Using Itô isometry and independence of the second Brownian motion, the derivation computes the variance of that combined normal term. The expected value of the resulting lognormal variable gives the futures price, with the final expression depending on the initial spot level, time, and the model’s volatility and rate parameters. The example demonstrates why simply treating the short rate as constant misses effects from stochastic rates and covariance. It assumes the stated diffusion model, zero initial short rate, and independent Brownian drivers; the result is specific to those assumptions rather than a general pricing formula.
Key ideas
- A futures price in the stated setup is the expected terminal spot price.
- Rewriting the integrated short rate as a stochastic integral makes its distribution tractable.
- The shared Brownian driver creates covariance between interest-rate accumulation and the asset return.
- Itô isometry gives the variance needed to evaluate the expectation of the terminal lognormal price.
- The resulting pricing expression depends on the specific diffusion assumptions in the example.
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Full text
# Calculating futures price
# Calculating futures price
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 by Quantuple (score 2, accepted)
https://quant.stackexchange.com/a/63813
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}Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.