# Threshold calculation for buying a mean-reverting asset

I am trying to figure-out an optimal policy for buying a unit when its price follows a mean-reverting price process (Ornstein–Uhlenbeck), when I have a finite time deadline for buying the unit.

I tried to search the literature for it, but couldn't find anything. I would very much appreciate any help.

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An interesting paper could be: Yingdong Lv & Bernhard K. Meister: Application of the Kelly-Criterion to Ornstein-Uhlenbeck Processes. But they don't have the a finite time deadline. However, you could use their results to estimate the optimal trading strategy and do some Monte Carlo to incorporate the finite time horizon. However, I have to say that I didnt read the paper in detail. –  philippe Feb 27 '13 at 11:07

## 4 Answers

you find theoretical results for the Ornstein-Uhlenbeck process if you search for "pairs trading". In pairs trading it is assumed that the ratio of the pair is mean reverting. Then one often models this ratio as Ornstein–Uhlenbeck process.

You find something on page 11 here

Further theoretical results that might be of interest can be found here.

All these results are theoretical and you can play with them. I don't know how much they help you in practice.

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mean-reversion may be the underlying assumption of those that employ "pairs-trading" approaches, however, other than that there is no relation at all to the topic under discussion. A Pairs trading strategy exhibits completely different dynamics than a mean-reverting process, even though each individual asset may follow a mean-reverting process. Its like someone asks about Black Scholes and I answer with using a completely unrelated stochastic model just because both are driven by Brownian motions. –  Matt Wolf Jan 28 '13 at 0:20
@Freddy I disagree, Alon looks for rules for the Ornstein-Uhlenbeck process and this is what I deliver. The headline "pairs trading" does not change this. –  Richard Jan 28 '13 at 8:05
I think it does matter, the dynamics of the OU-process driving a single asset are entirely different than the dynamics that describe a combined short and long position of a pairs trading approach even when each underlying asset is driven by an OU-process. I am sure your referenced papers describe the properties of OU-processes, however that is something the wiki page describes as well. You yourself pointed out OP is asking a specific question in regards to an optimal trading strategy, and I agree with you. But it does not concern pairs trading and goes beyond the OU-process itself. –  Matt Wolf Jan 28 '13 at 8:45
The OU-process has $3$ parameters - what can be different if I speak of theoretical results concerning stopping times and theresuch? OU process is OU process no matter the context - at least if I speak of stopping times - of course the parameters will be different and all "caveats" too. –  Richard Jan 28 '13 at 9:53
But most importantly - let us here what @Alon needs. Are you with us, Alon? –  Richard Jan 28 '13 at 9:54

I think a good way to think about your problem is the example of finding an optimal VWAP trading strategy. You basically have a finite point in time by which you must have performed your transaction and you trade a similar asset than the one you are considering, one with the same underlying assumptions of mean-reversion (I make such assumption in the same way than you make the assumption of mean-reversion).

With this assumption in mind and given you must at some point in time transact you are now faced with the following optimization problem: By how much does the asset have to traverse away from whatever you define as mean point in order to induce you to transact and in what size?

Also, contrary to a pairs trading strategy you do not want to transact at the point where an asset moved away from its mean but in the same direction as your order direction. You believe in mean-reversion and assume you can transact the asset more optimally at a later point.

I cannot provide an optimization function (because its very closely related to something I have been working on in the past and do not want to make it public) but here couple points I would consider:

• Does the asset really mean-revert more often than it trends in order to overcome each and every cost of execution, including commission, market impact,...? Does it pay to consider implementing a hybrid strategy in which you measure regime changes and only employ a mean-reversion approach when price dynamics favor such approach?
• Get a firm grasp at how volatile the asset is. By how many standard deviations does the asset trade away from its mean?
• Are you willing to take on more proprietary risk in that you are willing to potentially transact the full size of your order at once given the asset diverted sufficiently much away from your defined mean? Or do you want to split the order into many child orders and trade smaller sizes at smaller diversions?

I would first try to answer and consider those points before proceeding. Please note I am sharing my own experience here and do not present an academic approach. I implemented a VWAP strategy with systematic proprietary overlay that performed at a close to zero tracking error over a longer period of time in several Asian equity markets, including names that were generally not considered to be executed through standard DMA engines either for lack of liquidity, or other anomalies.

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I think the question is worth a better answer. Your answer is at most a comment. He asks for rules for the Ornstein-Uhlenbeck process. This is a clear question. –  Richard Jan 27 '13 at 20:24
ok, let me elaborate then, fair point made, I tried to improve on the answer. –  Matt Wolf Jan 28 '13 at 0:22
thanks for improving the answer. –  Richard Jan 28 '13 at 8:18

Following references from the answer provided by @Richard, we see that the optimality condition for a continuous process in general (and therefore an OU process in particular) is covered in Section 2 concluding on page 6 of Thompson 2002, where he also represents the solution in terms of the Hamilton-Jacobi-Bellman equations.

If you change the limits of the integral on the top of that page (and its antecedents) to $\min( T,H_S \wedge H_B )$ and then solve (which I don't think is necessarily possible in closed form) then you will have your optimum for the finite time horizon $T$.

If you actually try to trade this, pay close attention to the practical issues raised by @Freddy.

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The link to Thompson 2002 is corrupted. It would be appreciated if you cold update it. –  Hans Jul 18 '14 at 3:30

I'm seriously trying to figure out the exact same thing for my dissertation. I can easily solve for reservation (threshold) prices when offered prices are independent, but I haven't yet solved for the case of mean reversion.

There's an example in Bertsekas (1987) page 83 with an autocorrelated asset sale model, but it's too brief for me to follow all the way.

Here are my first steps. The asset must be sold before period $T$. We know the final reservation price is zero: $RP_{T} = 0$. In the next to last period, the agent compares the payoff with selling in period $T-1$ or waiting until period $T$. The value function is

$J(T-1) = \max[P_{T-1},\beta E[P_T|P_{T-1}]$,

where $\beta$ is a discount factor. The threshold price at time $T-1$ is the value that makes the asset holder indifferent to selling in either of the two periods. Substituting the expected value of the OU process,

$P_{T-1}= \beta(\mu+e^{-\eta}\left(P_{T-1}-\mu\right))$,

Where $\eta$ is the level of mean reversion. Solving for $P_{T-1}$ yields the reservation price:

$RP_{T-1}=\frac{\beta \mu (1-e^{-\eta})}{1-\beta e^{-\eta}}$.

(check the algebra, but I think it's correct). Then, I derived the remainder of the reservation prices using the equation

$J(t) = \max[P_t,\beta E[J(t+1)|P_t]]$

where

$E[J(t+1)|P_t] = \mbox{Pr}\left(P_{t+1}\geq RP_{t+1}\right)\times\left(E\left[P_{t+1}|P_{t+1}\geq RP_{t+1}\right]\right) + \mbox{Pr}\left(P_{t+1}<RP_{t+1}\right)\times\left(RP_{t+1}\right)$.

For the OU process,

$P_{t+s}|P_{t}\sim N\left(\mu+e^{-\eta s}\left(P_{t}-\mu\right),\frac{\sigma^{2}}{2\eta}\left(1-\exp\left(-2\eta s\right)\right)\right).$

I used R's etruncnorm function to calculate the probabilities in the value equation.

I have more details in my dissertation, pages 35-41: http://people.clemson.edu/~campbwa/dissertation/WAC_dissertation_3-15-2013.pdf

I have derived a full set of reservation prices, but they're too high. If I shift them down in the simulation model, profits increase!

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