# Where is the Quadratic Variation Coming from in this One-Factor Cheyette Model?

I am having difficulty switching from a general interest rate model (the quasi-gaussian or cheyette model) and a specific version of this model. In particular, I assume the following instantaneous forward vol separation:

$$\sigma_f(t, T, \omega) = g(t,\omega)h(T)$$

and define $$h(t) = \exp\left(\int_0^t \kappa(u)\,du\right) \hspace{1cm} G(t,T) = \int_t^Te^{\int_t^s \kappa(u)\,du} \,ds \hspace{1cm} \sigma_r(t,\omega) = g(t,\omega) h(t)$$

The model is driven by stochastic processes $$x$$ and $$y$$ with dynamics given by

$$\,dx_t = (y_t - \kappa(t)x_t) \,dt + \sigma_r(t, \omega) \,dW_t$$ $$\,dy_t = (\sigma_r(t,\omega)^2 - 2\kappa(t)y_t)\,dt$$ and $$x(0) = y(0) = 0$$. From here I know that we can recover discounted ZCB prices as $$P_1(t,T, x_t, y_t) = \frac{P(0,T)}{P(0,t)} \exp \left(-G(t,T)x_t - \frac{1}{2}G(t,T)^2y_t\right)$$

This is all well and good, my problem is that I have a specification of the model with $$\,dx(t) = \sigma_r(t,\omega) \,dW_t$$ (no problem) but with discounted bond prices as $$P_2(t,T, x_t, \langle x \rangle_t) = \frac{P(0,T)}{P(0,t)} \exp \left( -(H_T - H_t) x_t -\frac{1}{2} (H_T^2 - H_t^2)\langle x \rangle_t\right)$$ where $$H_t = \int_0^t e^{\int_0^s \kappa(u)\,du} \,ds$$. My question is, how are we recovering the dynamics of $$y_t$$ when $$\,d\langle x \rangle_t = \sigma_r^2\,dt$$ and the factor of $$2\kappa(t)y(t)$$ is removed? ie. how do we go from $$P_2$$ to $$P_1$$?

We have that $$G(t,T) = (H_T - H_t)e^{\int_0^t \kappa(s)\,ds}$$. At this point I have spent enough time on this problem that I feel silly and hope I am just missing something obvious. Help is very much appreciated.

• 1) For the $P_1$-ZCB formula to hold you must ensure that $(x_t,y_t)$ is a Markov process. In the one-factor (i.e. no stochastic vol) Cheyette literature this is achieved by choosing $\sigma_r(t,\omega)=\eta(t,x(t,\omega),y(t,\omega))\,.$ 2) How is $\sigma_r(t,\omega)$ specified in your $dx(t)=\sigma_r(t,\omega)\,dW_t$ model? Is it a function of $x(t)$ only or of $y(t)$ and $x(t)\,$? Nov 15, 2021 at 12:22
• @KurtG. 1) Yes, I'm assuming Markovian state variables. 2) $\sigma$ is dependent only on $x(t)$, ie. $dx(t) = \sigma(t, x(t)) \,dW_t$. Nov 16, 2021 at 10:06
• Unlike in the Cheyette model $y(t)$ is not part of $dx(t)$ and $x(t)$ is a one dimensional Markov process. Without further details I don't know why $\langle x\rangle_t$ occurs in $P_2\,.$ I would expect only $x_t$ as in every other Markovian short rate model (Vasicek, CIR, Black Karasinski). What paper is your model from? Nov 16, 2021 at 14:24