Vasicek interest rate model: $$dr_t = α(θ−r_t)dt + σdW_t$$

Price at time t of a 0-coupon bond maturing at T is given by: $$dp(t,T) = α_{t,T} . p(t,T)d_t + β_{t,T} . p(t,T)dW_t$$ $$βt,T = −σB(t,T).$$

T-forward price

$$ F_t = \frac{P_{t,S}}{P_{t,T}} $$

Show that the dynamics of the T-forward $F_t$ price with respect to the T-forward measure in the Vasicek model is given by

$$dF_t = σ (B(t,T)−B(t,S))F_tdW_t$$

Solution provided: $$ d(\frac{P_{t,S}}{P_{t,T}}) = \frac{dP_{t,S}}{P_{t,T}} + P_{t,S} d(\frac{1}{P_{t,T}}) + (dP_{t,S}) d( \frac{1}{P_{t,T}} ) $$ $$= \frac{dP_{t,S}}{P_{t,T}} −P_{t,S}\frac{dP_{t,T}}{P^2_{t,T}} + (···)dt$$ $$= (β_{t,S} −β_{t,T}) \frac{P_{t,S}}{P_t}dW_t + (···)d_t. $$

Can anyone explain how to obtain the first line of the solution please. Initially I thought you could just use the product rule here i.e $u (dv/dx) + v (dv/dy)$ but if I use that I don't get the first term $\frac{dP_{t,S}}{P_{t,T}}$


For the first line of the solution it's just Ito's lemma. \begin{align} d f(X_t,Y_t,t)&= \frac{\partial }{\partial t} f(X_t,Y_t,t)dt+\frac{\partial }{\partial X_t} f(X_t,Y_t,t) dX_t +\frac{\partial }{\partial Y_t} f(X_t,Y_t,t) dY_t \\ &+\frac12 \frac{\partial^2 }{\partial X_t^2} f(X_t,Y_t,t) d\langle X_t,X_t\rangle+\frac12 \frac{\partial^2 }{\partial Y_t^2} f(X_t,Y_t,t) d\langle Y_t,Y_t\rangle\\ &+ \frac{\partial^2 }{\partial X_t\partial Y_t} f(X_t,Y_t,t) d\langle X_t,Y_t\rangle \end{align} In your case you have $X_t=P_{t,S}$ and $Y_t=\frac{1}{P_{t,T}}$ so that the first derivative of $f(X_t,Y_t,t)=X_tY_t$ in $t$ and second derivatives in $X_t$ or $Y_t$ are all 0. So you are left with the first line of the solution. \begin{align}dX_tY_t&=Y_t dX_t + X_t dY_t +d\langle X_t,Y_t \rangle\\ &= \frac{1}{P_{t,T}}dP_{t,S}+P_{t,S}d\left(\frac{1}{P_{t,T}}\right)+d\langle P_{t,S},\frac{1}{P_{t,T}}\rangle \end{align} By the way I suggest you go back to your stochastic calculus class notes because Ito's lemma is a must in quantitative finance.

  • $\begingroup$ Very useful, thanks a lot, managed to prove the final answer using Ito lemma. $\endgroup$ – Calypso Apr 1 '17 at 11:24

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