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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
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).
