How to incorporate momentum in Ornstein Uhlenbeck to capture overshooting in financial markets?

Question

In modelling asset prices, it is a good idea to model it using a fair value or target price concept. Recently Carr & Prado explored this idea to find optimal stop loss/take profit levels when the asset price is pulled towards a target price or fair value (see Determining Optimal Trading Rules Without Backtesting). In this model, the price is modelled as an Ornstein Uhlenbeck (OU) process and when the price is below or above the target price, the drift term of OU process pulls the price towards target price. The model is given as follows where $P_t$ is the asset price and $E[P_T]$ is the target price:

$$𝑃_{𝑡} = (1 − \theta) 𝐸[𝑃_{𝑇}] + \theta 𝑃_{𝑡−1} + \sigma \varepsilon_{t}$$

We also know that the price sometimes overshoots the target price when there is a momentum effect. Assume a case where the current price is below the target price. Then there will be a positive drift that pulls the price upwards towards the target price. This typically leads to momentum effect which attracts some other investors and due to herd behavior the price may overshoot and exceed the target price first and then reverts back the target price.

In this setting there are two forces. First one is the pullback towards target price. And the second one such that if the recent momentum is strong there is a tendency of continuing this momentum.

The first pullback effect can be easily captured by an OU process. My question is about how to incorporate the second momentum effect into this OU process. Which type of stochastic differential equation can capture such two effects?

In discrete time these two effects can be captured using the following regression, where $R$ is the asset return:

$$R_t= \alpha + \beta_1 R_{t-1} + \beta_2 (E[P_T]-P_{t-1}) +\varepsilon_t $$

with $\beta_1 >0$ and $\beta_2>0$.

What would be the correct form of SDE that captures the same effect in continous time?

Edit: Actually this is the difference between a mean reverting process vs an oscilliation process. The mean reverting process is always expected to go towards the mean. But an oscilliating process may overshoot in the short term but go towards mean in the long term

Edit 2: As a related phenomena in asset prices, past momentum typically continues for the next period, but if the momentum becomes very strecthed it reverts back. Here is a very important paper on this. I am trying to find an SDE that captures both momentum and reversal (or regression towards target price) effects

Answer 1

If the current price $P_{t-1}$ is below the target price $E[P_T]$ is the drift not $\theta(P_{t-1}-E[P_T])$ (negative) ? Seems to me the authors have a sign error. Regarding the possiblity to overshoot. In a OU process this is usually achieved by the stochastic term which is $\epsilon$ in your case. In short: the expected process is pulled towards the long term mean, the indivdual paths are not.

Answer 2

Consider the following Ornstein-Uhlenbeck dynamics in continuous time with $P\equiv E(P_T)$ and $\sigma>0$: $$\text{d}P_t=\theta(P-P_t)\text{d}t+\sigma\text{d}W_t$$ where $W$ is a Brownian Motion. If $\theta>0$ then whenever $P_t$ gets away from its long-term mean, the price is pulled-back towards it (mean-reversion). If on the other hand $\theta<0$, then price trends reinforce themselves, which can be viewed as momentum.

Now instead of considering a fixed value $\theta\in\mathbb{R}^*$, introduce a process $(\theta_t)_t$ with $\theta_0=0$ and following Ornstein-Uhlenbeck dynamics with zero drift; use this process as the mean-reversion for the price series $(P_t)_t$: \begin{align} \text{d}P_t&=\theta_t(P-P_t)\text{d}t+\sigma\text{d}W_t \\[3pt] \text{d}\theta_t&=-\gamma\theta_t\text{d}t+\eta\text{d}B_t \\[3pt] \text{d}\langle W,B\rangle_t&=\rho\text{d}t \end{align} with $B$ a Brownian Motion, $\gamma,\eta>0$ and $-1\leq\rho\leq1$. The stochastic mean-reversion $(\theta_t)_t$ is pulled back towards zero, alternating periods of mean-reversion ie. $\theta_t>0$ with moments of momentum ie. $\theta_t<0$.