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I have a perpetual bond price dynamics under a martingale measure $Q$: $$ dp(t,T)=p(t,T)r(t)dt+p(t,T)v(t,T)dW(t) $$ where $W$ is a vector-valued $Q$-Wiener process. I also have a bond pricing formula: $$ C(t) = \int_t^\infty p(t,s)ds $$ How can I prove that the dynamics of $C(t)$ is: $$ dC(t)=(C(t)r(t)-1)dt+\sigma_C(t)dW(t) $$ where $\sigma_C(t)=\int_t^\infty p(t,s)v(t,s)ds$?

I think we might use the idea of the derivation of HJM drift condition, but I don't have any further clues.

Cyan
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1 Answers1

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Your first equation is not a perpetual bond price dynamics. It is the well-known HJM SDE for the zero coupon bond that matures at $T\,.$ Your $C(t)$ is in fact a perpetual bond that pays a continuous coupon. Writing the HJM SDE for $p(t,T)$ in in integral form and plugging it into the definition of $C(t)$ we have \begin{align} C(t)&=\int_t^\infty\left\{ p(0,s)+\int_0^tp(u,s)\,r(u)\,du+\int_0^t p(u,s)\,v(u,s)\,dW(u)\right\}\,ds\,. \end{align} Applying Fubini and stochastic Fubini we get \begin{align} C(t)&=\int_t^\infty p(0,s)\,ds+\int_0^t\int_t^\infty p(u,s)\,r(u)\,ds\,du+\int_0^t\int_t^\infty p(u,s)\,v(u,s)\,ds\,dW(u)\,. \end{align} This can be written as \begin{align} C(t) &=\int_t^\infty p(0,s)\,ds\\ &\quad+\int_0^t\int_u^\infty p(u,s)\,r(u)\,ds\,du+\int_0^t\int_u^\infty p(u,s)\,v(u,s)\,ds\,dW(u)\\ \tag{1} &\quad-\int_0^t\int_u^t p(u,s)\,r(u)\,ds\,du-\int_0^t\int_u^t p(u,s)\,v(u,s)\,ds\,dW(u)\,. \end{align} The first two double integrals are obviously \begin{align}\tag{2} \int_0^tC(u)\,r(u)\,du+\int_0^t\sigma_C(u)\,dW(u)\,. \end{align} Using again (stochastic) Fubini the last two double integrals can be written as \begin{align} -\int_0^t\int_0^s p(u,s)\,r(u)\,du\,ds-\int_0^t\int_0^s p(u,s)\,v(u,s)\,dW(u)\,ds\,. \end{align} Using the HJM SDE for $p(u,s)$ this is \begin{align} -\int_0^t\underbrace{p(s,s)}_{1}-p(0,s)\,ds=-t+\int_0^tp(0,s)\,ds\,. \end{align} Therefore, (1) becomes \begin{align} C(t)=\underbrace{\int_0^\infty p(0,s)\,ds}_{C(0)}+\int_0^tC(u)\,r(u)\,du+\int_0^t\sigma_C(u)\,dW(u)-t\,. \end{align} In differential form this equation is $$ \boxed{ dC(t)=C(t)\,r(t)\,dt+\sigma_C(t)\,dW(t)-dt\,.} $$

Kurt G.
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