= Forward-rate equation for a Markov short-rate diffusion
{c}
{title2=$f_t(T)=F(T-t,r_t)$}
Suppose $dr_t=a(r_t)dt+b(r_t)dW_t$ under a risk-neutral measure and $f_t(T)=F(T-t,r_t)$. By <Itô formula>, the forward-rate volatility is $\sigma_t(T)=b(r_t)F_r(T-t,r_t)$. The <Heath-Jarrow-Morton model> drift restriction is equivalent to
$$
F_\theta=aF_r+\frac12b^2F_{rr}-b^2F_r\int_0^\theta F_r(s,r)ds,\qquad F(0,r)=r.
$$
To verify it directly for bonds, put $G(\theta,r)=\int_0^\theta F(s,r)ds$. Integration gives $F-r=aG_r+\tfrac12b^2G_{rr}-\tfrac12b^2G_r^2$. The <zero-coupon bond> price $P(t,T)=e^{-G(T-t,r_t)}$ therefore satisfies
$$
\frac{dP(t,T)}{P(t,T)}=r_tdt-b(r_t)G_r(T-t,r_t)dW_t.
$$
Its bank-account-discounted price is a positive <local martingale>. A common equivalent <local martingale> measure excludes admissible <arbitrage> for any finite collection of these traded maturities.
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