Threshold and Cooldown Rules with Mean-Reverting Processes

# Mathematician needs help from the Quants: Is there an interpretation of this stochastic model to stock market trading?


# Mathematician needs help from the Quants: Is there an interpretation of this stochastic model to stock market trading?












I'm building a stochastic model for a noisy physical system. I suspect this model might have an interpretation in the context of stock market trading and this would be very useful in communicating the model to a more general crowd. Let me first explain a simplified version of the model.

Model:

Consider a noisy system where the noise is modeled by the Ornstein-Uhlenbeck process $$ \mathrm dX_t=\omega(\mu-X_t)\,\mathrm dt+\omega\sigma\,\mathrm dW_t, $$ with $X_0=x\in\Bbb R$. In my application $X_t$ is a noisy voltage. At time $t=0$ the system has a stored reference voltage $r\in\Bbb R$. The system also has two adjustable parameters $\alpha,\beta>0$ that are set at time $t=0$. Combining the reference voltage and the specified parameters defines an interval: $I_r=(r-\alpha,r+\beta)$.

At some future time $t=T$ the process $X_t$ exits $I_r$. It's possible for $T$ to be zero if $X_0\notin I_r$. At time $T$ two things happen:

- The reference voltage is updated to $X_T$ which defines a new interval $I_{X_T}=(X_T-\alpha,X_T+\beta)$.

- The system enters a dead state of $\rho$ seconds where it is unable to do any updates to the reference voltage.

At the end of the dead state (at time $t=T+\rho$), the process repeats using the updated interval $I_{X_T}$ and initial voltage $X_{T+\rho}$.

Observations:

After simulating this process it seems like there may be a nice analogy to trading strategies in the stock market. For example, $X_t$ might represent the price of a mean reverting security and the parameters $\alpha,\beta$ along with the rule for updating the reference voltage may define a strategy for buying or selling the security. The parameter $\rho$ might define a period of time following a trade where no other trades can be made. The trading strategy is set and forget. The rules for making a trade are determined at time $t=0$ and not altered in the future as new information becomes available.

Question:

Is there any connection of this model to trading strategies? Does such a trading strategy have a name? Are there any financial interpretations of the parameters $\omega,\sigma,\alpha,\beta,\rho$?

Edit:

Thank you everyone for your feedback on this question. I thought it was worthwhile to include a picture/schematic depicting the model and in particular the interval updating. Some of the notation is slightly different from the post ($r$ is $Z_0$ and the $\tilde\theta$'s are $\alpha$ and $\beta$), but I thought this was useful for anyone reading this post.

## Answer by Jan Stuller (score 5, accepted)

https://quant.stackexchange.com/a/82084

The Vasicek model is basically the Ornstein-Uhlenbeck process.

It is used to model interest rates. Using your notation and writing in long-hand:

$$X_t=X_0+\int_{h=0}^{h=t}\omega(\mu-X_h)\,\mathrm dh+\int_{h=0}^{h=t}\sigma\,\mathrm dW_h,$$

The interpretation is as follows:

- $X_0$ is the current level of interest rates

- $\mu$ is the long-run average towards which (by assumption) the interest rates converge

- $\omega$ is the speed at which the rate process $X_t$ converges to the long run average

- $\sigma$ is the volatility of the interest rates (can be estimated by the standard deviation of historical returns, where returns are defined as $\frac{X_{i}-X_{i-1}}{X_{i-1}}$ over some historical period $i\in\{0, 1, ..., n\}$, $n<t_0$)

Think of it like this: the central banks increase rates when inflation is high or lower rates when they want to stimulate the economy (these are the "shocks" coming from the $dW_h$ term), but in the long run, the steady-state of interest rates is (or should be) around 2% to 3% (this is the $\mu$ parameter).

As you undoubtedly know, the solution to the model is:

$$X_t=X_0e^{-\omega t}+\mu(1-e^{-\omega t})+\sigma e^{-\omega t}\int_{h=0}^{h=t}e^{\omega h}dW_h$$

So the solution is normally distributed, centered on $X_0e^{-\omega t}+\mu(1-e^{-\omega t})$.

Obviously the model has many deficiencies, for example it's continuous in time, but rates change at discrete points in time, etc.

The difference to your model is that the volatility term is not multiplied by $\omega$.

Trading strategy: if the the model would be used to simulate a security price, the strategy that would make sense to me would be to buy the security when it's lower than the band $I_r=(r-\alpha,r+\beta)$ and sell when it's higher. The reasoning is that if we believe in the model (i.e. we believe that the security is mean-reverting), then in the long term, this should make money.

Obviously, people have thought of strategies that are much more complex. This would be a super basic model. Additionally, most securities are not mean reverting in practice.

## Answer by Andrea (score 6)

https://quant.stackexchange.com/a/82081

I have a couple of (possible) interpretations of the 2nd part of the model (the reference range) in finance:

- Delta hedging ($X_t$ = delta): one only wants to re-hedge when the option delta has moved away from the last hedge level by $(\alpha, \beta)$ and never hedge more frequently than $\rho$ (1 minute?)

- Market making ($X_t$ = quoted price): only update the price to the client if it has moved (maybe asymmetrically) and never more than 1 every second

Does it make sense at all?