# Use Discrete ARMA(1,q) Process to Model Short Rate for Term Structure Fitting

I'm new to this field but I'm reading related literature lately and quite obsessed with the topic. I come to know that people like to model short rate under risk-neutral measure $$Q$$, because under $$Q$$ the price of a zero-coupon bond can be expressed as $$P(t,T)=\mathbb{E}_Q\left[e^{-\int_t^Tr_sds}|\mathscr{F}_t\right],$$ while under the real-world measure $$Q^0$$ the formula is complicated. And if one does not care about how to travel between $$Q^0$$ and $$Q$$ or the form of $$\lambda$$, one can directly model the short rate dynamics under $$Q$$ where $$\lambda$$ becomes implicit.
So here's my idea. By the fundamental theorem of asset pricing, no arbitrage is equivalent to existence of a risk-neutral measure $$Q$$. Assume no arbitrage. Then under this particular $$Q$$, I suppose the short rate to follow a discrete ARMA(1,1) process $$r_{t+1}=\gamma_1r_t+\gamma_0+\epsilon_{t+1}+\epsilon_t.$$ Given the model is discrete, I therefore use a Riemann approximation to represent $$\int_t^Trsds$$, $$\hat{r}=\frac{1}{2}r_th+\sum_{j=1}^{m-1}r_{j+t}h+\frac{1}{2}r_{m+t}h,$$ with $$T-t=mh$$. In fact, if the rates are observed on a daily basis, take $$h=\frac{1}{365}$$. Then, by proving $$\hat{r}|\mathscr{F}_t$$ follows a normal distribution, use the moment generating function to derive $$\hat{P}(t,T)=\mathbb{E}_Q\left[e^{-\hat{r}}|\mathscr{F}_t\right]=e^{\mathbb{E}[-\hat{r}|\mathscr{F}_t]+\frac{1}{2}Var[-\hat{r}|\mathscr{F}_t]},$$ which is now an approximation of $$P(t,T)$$.
I understand the above is definitely not elegant, because we all like continuous differential models. But is the above at least theoretically OK? And if there are flaws, can it be tackled with to save the idea?

Edit: My major concern is about $$Q$$. I see lines about Girsanov change of measure and Radon-Nykodym derivatives when people derive $$Q$$ from $$Q^0$$, and I'm not really confident about simply assuming $$Q$$ exists by no arbitrage and placing my model under it.

You are describing something called Geometric Brownian Motion, and in the realm of short rates, you are describing the discretization short rates. For the Vasicek model, $$R_t = aR_{t-1} + b +\epsilon$$ where $$a=e^{-\lambda*dt}$$ and $$b=\mu(1-e^{-\lambda*dt})$$. You can use OLS to calibrate your short rate process using this identity.
• Thanks for your comment! I understand that if we apply Euler discretization to the O-U process in Vasicek model we can get AR(1), which can be estimated through OLS. But as for ARMA(1,q) where $q\ge1$, isn't that OLS cannot work and we usually use conditional or unconditional maximum likelihood? Moreover, I'm still not so sure about whether my idea is theoretically coherent as I'm not so confident about the choice of $Q$, because I see so much about Girsanov change of measure and radon-nykodym derivatives when people derive $Q$ for Vasicek-like models, while I simply assume such a $Q$ exists? Feb 19, 2021 at 15:00