# Trading over a Ornstein/AR process

For a OU/AR(1) process is there anyway to analytically calculated most probable period of time the process is likely to diverge from the average, before turning to converge.

Basically I am looking to derive a proof that under an OU/auto-regressive process a profitable moving average crossover strategy exists.

Many thanks

Sam

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Consider an (arithmetic) Ornstein-Uhlenbeck process as a model of the asset price $X_t$:

$$dX_t = \kappa(\mu-S_t)dt + \sigma dW_t$$

where $\mu$ is the mean-reversion level, $\sigma$ is a volatility parameter, $W_t$ is Brownian motion, and $\kappa$ is the reversion speed. An Ornstein-Uhlenbeck process will revert to the mean infinitely often if $\kappa > 0$ and the characteristic time scale for reversion is $\kappa^{-1}$.

The exact solution over an interval $[0,t]$ is

$$X_t = X_0 e^{-\kappa t} + \mu (1 - e^{-\kappa t}) + \sigma\int_{0}^{t} e^{\kappa(s-t)} dW_s,$$

with expected value

$$E(X_t) = X_o e^{-\kappa t} + \mu (1 - e^{-\kappa t}).$$

To address your question, there are a number of metrics that characterize the time frame of reversion -- the most obvious being the expected first passage time. For a mean reverting process the expected time to revert from level $A> \mu$ to level $B<A$ will, of course, be smaller than the expected time to migrate from level $B$ to $A$.

The formulas are complicated, and in my view of limited practical value (as I discuss below).

Here is a reference:

L. M. Ricciardi and S. Sato, First-passage-time density and moments of the Ornstein-Uhlenbeck process, J. Appl. Probab. 25 (1988) 43—57; MR0929503 (89b:60189).

In theory, a profitable trading strategy certainly exists for an asset that exhibits a stable mean-reverting behavior. There are many variations such as putting on a trade when the price deviates from the expected reversion level by a fixed amount in a binary fashion, or entering and taking profit in a continuous fashion.

However, in real markets, assets seldom exhibit stationary mean-reversion with predictable reversion levels. Mean reversion may persist for a period of time and then suddenly break down. The assumption of a stable reversion level is, generally, the weak link in the chain -- leading eventually to a large draw-down.

In practice, a more robust trading strategy allows the expected reversion level to adjust gradually as a moving-average. Position size may also be adjusted based on historical volatility. The graph below shows trading profits (without transaction costs) for a simple mean-reversion strategy applied to the EURSEK currency pair. In this case, position size is proportional to the deviation of the exchange rate from a 1-month or 3-month moving average (using daily NY market closing levels). The position size (long or short) is not allowed to go above a cap based on the percentage deviation form the moving average.

The 3-month moving average adjusts more slowly than a 1-month average -- so the position sizes tend to be bigger -- yielding a bigger mean return but with much bigger drawdown when the mean reversion breaks down (late 2008, for example).

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Hi RRL, thanks for the detailed response it is a great help! I guess the strategies using the deviation from the mean are similar (maybe?) to pairs trading, which have some analytical solutions. For the purpose of my work, which is purely theoretical, I need a process for which there exists a profitable strategy, which as you have explained OU should be good! –  Sam Palmer Aug 12 '14 at 18:45
For OU is there anyway of estimating the expected time a deviation from the mean will occur for (with out knowing the level A), i'm guessing probably not? I was thinking perhaps of finding the expected frequency of oscillations, so an FFT on the expected path from monte carlo samples? –  Sam Palmer Aug 12 '14 at 18:47
Basically the aim of this work is to try and find an analytical optimal trading strategy and compare this to numerical optimisation results (for an OU process). –  Sam Palmer Aug 12 '14 at 18:48
Jurek, Jakub W. and Yang, Halla, Dynamic Portfolio Selection in Arbitrage (April 2007). EFA 2006 Meetings Paper. Available at SSRN: ssrn.com/abstract=882536 or dx.doi.org/10.2139/ssrn.882536 -- they look at optimal strategies for the OU process –  RRL Aug 12 '14 at 18:52
The expected passage time is always conditioned on something even if it is just the starting level. So maybe you want the expected time to return to the mean or the expected number of crossings, etc. –  RRL Aug 12 '14 at 19:01