# Application of Itô's lemma - Forward process

How would be applied the itô's lemma in the following equation:

And we know that:

• What is r_c? Why does it depend on T in your dS process? – ilovevolatility Mar 20 at 18:27
• r_c is the discrete spot interest rate compounded continuously, on date t, with expiration in T. c denotes that is continuously – JB1 Mar 20 at 18:34
• Discrete compounded continuously? I am a bit confused, but anyway lets take r to be the short rate, T is irrelevant in the dS. Since r is stochastic, that's how you specified it, then you cannot write P that way. – ilovevolatility Mar 20 at 18:38
• @Sanjay yees, because i need a formula of the variance of the forward process – JB1 Mar 20 at 19:29
• yes, I added the correlation in the question section – JB1 Mar 20 at 19:38

Start by defining the function $$f(t,s,r):=se^{r(T-t)}$$ where $$T$$ is just a parameter here. The derivatives of $$f$$ is: $$f_t(t,s,r)=-se^{r(T-t)} \text{ , } f_s(t,s,r)=e^{r(T-t)} \text{ , } f_r(t,s,r)=se^{r(T-t)}(T-t)\\ f_{s,s}(t,s,r)= 0 \text{ , } f_{r,r}(t,s,r)=se^{r(T-t)}(T-t)^2 \text{ , } f_{s,r}(t,s,r) = e^{r(T-t)}(T-t)$$ Just to avoid conflict with mathematical formality redefine$$F$$ to be function of $$t,S,r$$ and $$r(t)=r_c(t,T)$$: $$dF(t,S(t),r(t))= \\ f_t(t,S(t),r(t)) dt + f_s(t,S(t),r(t)) dS(t)+ f_r(t,S(t),r(t)) dr(t)+\frac{1}{2}f_{r,r}(t,S(t),r(t)) (dr(t))^2+f_{r,s}(t,S(t),r(t)) dr(t)dS(t)$$ Let's use the shorthand notation:

$$dF= f_t dt + f_s dS+ f_r dr+\frac{1}{2}f_{r,r} (dr)^2+f_{r,s} drdS (1)$$ Note $$drdS=\sigma_sS\sigma_rr\rho dt$$ $$(dr)^2=\sigma_r^2r^2dt$$

Now you have the relevant information to simplify equation 1 and after (a lot of) symbol-jiggling you should reach a decent expression for $$dF$$.

Please let me know if my (some how nonchalant) notation is confusing or not understable.

• Then, the variance of the forward process is: (sigma_s)^2+(T-t)[(T-t)*((sigma_r)^2)+2*ssigma_srsigma_rrho] It is correct? I'm not sure – JB1 Mar 21 at 15:10
• @Sanjay if $r$ is stochastic, how can you write $P = e^{-r(T-t)}$ as you do in your derivation? – ilovevolatility Mar 22 at 8:01
• So guys,what would be the variance of the forward process? Because I not sure in what manners changes in comparison to be under normal assumption in terms of interest rate – JB1 Mar 23 at 11:29
• I gave you the answer below. All you have to do is calculate $d(S/P)$ as explained in my answer. Which part is not clear? – ilovevolatility Mar 23 at 14:20

If $$r$$ is the stochastic short rate then first of all you cannot write $$P = e^{-r(t,T)(T-t)}$$. The zero-coupon bond price will be $$P(t,T) = E_t \left[ e^{- \int_t^T r_u du} \right]$$ Now finding the dynamics of $$P(t,T)$$ given the dynamics of $$r_t$$ is, as far as I know, relatively easy only in so-called affine term structure models (ATS models). For your particular problem I think it's fine to start with supposing the dynamics of $$P$$ is given by $$dP = rP dt + \sigma_P P dW_r$$ Note that the zero coupon $$P$$ dynamics is driven by the same $$dW_r$$ as the one that drives $$r$$ since $$P$$ depends on $$r$$. The risk-neutral drift of $$P$$ is $$r$$ as $$P$$ is a tradable asset. The $$\sigma_P$$ we will leave unspecified.

We can apply Ito now: $$d(S/P) = (1/P)dS + S d(1/P) + dS d(1/P)$$ with $$d(1/P) = (-1/P^2) dP + (1/P^3) (dP)^2$$ Now if you work this out you'll see what the drift is and the volatility is of $$S/P$$ under the risk-neutral measure $$\mathcal{Q}$$. To make the forward price driftless you do a measure change to work under the forward measure $$\mathcal{Q}^T$$. If $$\sigma_P$$ is zero due to constant or deterministic short rate then $$(dP)^2 = 0$$ and $$dS d(1/P) = 0$$, and the forward price is already driftless under $$\mathcal{Q}$$.