# Dynamics of an option on a future

I have trouble understanding why $$V_s=exp(\int_s^t r_u du) \int_s^t exp(−\int_t^v r_u du)\theta_v dW_v$$ is the solution to the following SDE

$dV_s=\theta_s dW_s+r_s V_s ds$.

I tried of course with Ito's Lemma but I do not really know what function f I should use.

Thank you for your help!

• Why don't you start by explaining your two formulae in words? – will Jul 2 '18 at 8:13
• ok. So r describes the interest rate, which is in this context also a stochastic process. Than we have of course our brownian motion W and our strategy, which should be admissible so we don't get any problems with arbitrage opportunities. – Tipumba Jul 2 '18 at 8:49
• Seems to be an Ornstein-Uhlenbeck equation, with variable interest rate. But the solution looks off, is there a "+" between the factors? – userid is i Jul 2 '18 at 14:07
• In my lecture notes there is no plus, but it could be possible that there is some mistake. – Tipumba Jul 2 '18 at 14:33
• I don't think that $(r_t)_{t \geq 0}$ is a stochastic process rather a deterministic term structure. Otherwise could you provide the SDE solved by $(r_t)_{t \geq 0}$. The solution is indeed off, to solve the original SDE (assuming $r(t)$ is indeed a deterministic term structure), just let $Y_t = \exp(-\int_0^t r_u du) V_t$ and apply Itô's lemma. – Quantuple Jul 3 '18 at 10:01