= Solution
For the <quadratic Ornstein-Uhlenbeck state-price density>, apply the <Itô's formula> under the reference probability measure. Since $f'(x)=x$ and $f''(x)=1$,
$$
d\zeta_t=e^{-\alpha t}\sigma X_t\,dB_t-e^{-\alpha t}\left[\alpha a-\tfrac12\sigma^2+(\lambda+\tfrac12\alpha)X_t^2\right]dt.
$$
The <stochastic integral> is a true <martingale> on finite horizons by the finite moments of the <Ornstein-Uhlenbeck process>. \b[A uniform sufficient condition is $\boxed{\alpha\geq\sigma^2/(2a)}$], making the finite-variation drift nonpositive. This is also the condition for nonpositive drift at every state.
Use the positive <state-price density> $\zeta$ to price a terminal payoff $H$ by $\Pi_t(H)=\mathbb E[\zeta_TH\mid\mathcal F_t]/\zeta_t$. The riskless account must cancel the drift of $\zeta$, giving the <short rate>
$$
\boxed{r_t=\frac{\alpha a-\sigma^2/2+(\lambda+\alpha/2)X_t^2}{a+X_t^2/2}=\alpha+\frac{\lambda X_t^2-\sigma^2/2}{a+X_t^2/2}.}
$$
This pricing system is consistent with a money-market <risk-neutral measure>: with $M_t=e^{\int_0^tr_udu}$, the process $\zeta_tM_t/\zeta_0$ is a <stochastic exponential> whose diffusion coefficient $\sigma X_t/f(X_t)$ is bounded. The <Novikov condition> makes it a true density <martingale>. The original <Brownian motion> is for the reference measure, not automatically for this new pricing measure.
For $X_0=x_0$, the OU second moment is $\mathbb E X_T^2=x_0^2e^{-2\lambda T}+\sigma^2(1-e^{-2\lambda T})/(2\lambda)$. Thus \b[the <zero-coupon bond> price is]
$$
\boxed{P(0,T)=e^{-\alpha T}\frac{a+\tfrac12x_0^2e^{-2\lambda T}+\frac{\sigma^2}{4\lambda}(1-e^{-2\lambda T})}{a+\tfrac12x_0^2}.}
$$
Conditionally at time $t$, replace $T$ by $T-t$ and $x_0$ by $X_t$.
The <short rate> is nonnegative under the condition above, increases with $X_t^2$, and lies between $\alpha-\sigma^2/(2a)$ and $\alpha+2\lambda$. The model is tractable but imposes fixed rate bounds, identical rates for opposite values of $X$, and a restricted one-factor term structure. Its long-maturity yield is $\alpha$. These structural restrictions, including exclusion of negative rates under the supermartingale condition, may be unsuitable for a general interest-rate fit.
Back to article page
