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

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And we know that:

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  • $\begingroup$ What is r_c? Why does it depend on T in your dS process? $\endgroup$ – ilovevolatility Mar 20 at 18:27
  • $\begingroup$ r_c is the discrete spot interest rate compounded continuously, on date t, with expiration in T. c denotes that is continuously $\endgroup$ – JB1 Mar 20 at 18:34
  • $\begingroup$ 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. $\endgroup$ – ilovevolatility Mar 20 at 18:38
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    $\begingroup$ @Sanjay yees, because i need a formula of the variance of the forward process $\endgroup$ – JB1 Mar 20 at 19:29
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    $\begingroup$ yes, I added the correlation in the question section $\endgroup$ – JB1 Mar 20 at 19:38
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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.

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  • $\begingroup$ 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 $\endgroup$ – JB1 Mar 21 at 15:10
  • $\begingroup$ @Sanjay if $r$ is stochastic, how can you write $P = e^{-r(T-t)}$ as you do in your derivation? $\endgroup$ – ilovevolatility Mar 22 at 8:01
  • $\begingroup$ 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 $\endgroup$ – JB1 Mar 23 at 11:29
  • $\begingroup$ 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? $\endgroup$ – ilovevolatility Mar 23 at 14:20
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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 $\mathcal{Q}$.

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