Mean Reversion in the Markets and Trading Strategy
Team Latte July 04, 2012
Recently we came across an excellent blog post by one Ernie Chan who runs a blog Quantitative Trading. Here the author discusses the role of a "stop loss" strategy as employed by many traders and he links it to the notion of "mean reversion" in the market. By all accounts the author's arguments – and findings – are very interesting and original.
Traders frequently use a stop loss strategy to avoid incurring big losses. If you buy an asset at 100 with the objective of profiting from its upside then you may want to put a stop loss at 96, in case, you are proven wrong and after a certain period of time the market reverses direction. The choice of this stop loss and how it will impact a trader’s overall position will depend on whether a trader follows a trending strategy or a mean reverting strategy.
Broadly speaking, over a certain period of time, financial markets either trend in one direction or revert to some long term mean. The million dollar question, of course, is over what “period of time”. This is where most financial theorists – and surprisingly, many practitioners – are either silent or vague. The "time" element is purposefully kept nebulous. This is the only reason why many traders and market practitioners whole heartedly endorse the mean reversion school of thought. Mean reversion over what period? That’s always the standard refrain of traders.
Another very relevant point is the regime in which a mean reversion is being modeled. A regime could be an inflationary or a deflationary economy or some other scenario that is governed and influenced by macroeconomic variables. Mean reverting – or, a trending – market has relevance within a particular regime. If we are using a mean reversion model then with a regime switch this model will break down. Or at least, the parameters of the modeled will be greatly altered.
This is where the author seems to come up with an answer, albeit in the context of using stop loss strategy for traders. It is possible, the author argues, to estimate whether a particular mean reversion model is correct by focusing on the dimension of "time". Here is an excerpt from the said blog:
On the other hand, if you employ a meanreverting strategy, and instead of reverting, the market sticks to its original direction and causes you to lose money, does it mean you are wrong? Not necessarily: you could simply be too early. Indeed, many traders in this case will double up their position, since the latest entry signal in this case is in the same direction as the original one. This raises a question though: if incurring a big loss is not a good enough reason to surrender to the market, how would you ever decide if your meanreverting model is wrong? Here I propose a stop loss criterion that looks at another dimension: time.
The notion of "mean reversion" originated within the context of interest rates in mid 1970s. From mid 1990s onwards the idea has gone on to dominate other fields within quantitative finance such as volatility models, equity trading and spread models.
In 1977 Vasicek introduced a continuous time mean reverting random walk, which was arithmetic in nature, to model the interest rate. In this model he proposed modellingthe the term structure of rates through the changes in the spot rate. Vasicek's process is an extremely important stochastic process because it is one of the first models to have laid the foundations for modeling interest rates as a mean reverting process. The idea of mean reversion – the notion that the underlying variable, the rate, reverts to a long term average value – finds strong acceptance amongst many economists and central bankers who take the Vasicek model as the classical model for short rates.
The stochastic differential equation for a Vasicek process is given by:
The above is an example of an Ornstein Uhlenbeck (OU) process. In the above stochastic process, the short rate, is being modeled as a mean reverting arithmetic Brownian motion (ABM) where, is the long term average value of the short rate and is the speed of mean reversion. Since the process follows an arithmetic Brownian motion, the short rate can become negative.
The mean reverting parameter, , which is also known as the speed of mean reversion, can be estimated from a linear regression of the daily change of the variable, versus the variable, itself.
However, the mean reverting parameter, , is not an intuitive concept at all when we are trying to figure out as to how long it takes an factor – or, an interest rate or any other variable that is being modeled – to revert to its long term goal. A far more intuitive concept is that of "halflife", something that is borrowed from nuclear physics. Halflife is defined as the time it takes for the variable, say, an interest rate or a spread, to move half the distance towards its objective, i.e. the long term average given by in the above equation. Mathematically, halflife is given by:
Where, the unit of is in years. For example, if we say that the mean reversion speed, , is 0.5 and the long term average of 6 month USD LIBOR, , is 5% then the statement may not mean much to us except saying that some long term average value of USD LIBOR is 3%. But if we calculate the halflife of this LIBOR rate then we get,
Thus, in the above example, the halflife of 6m USD LIBOR is 1.39 years. This means if the current value of the 6m USD LIBOR is then it will take this rate 1.39 years to move half the distance to 2.5%.
Thus, the halflife of the rate, or any variable that is being modeled using the Vasicek / OU model, gives us the "time" context within which we can meaningfully talk about interest rates.
One of the financial instruments that is frequently modeled using a Vasicek / OU mean reverting model is the spread between two interest rates or bond yields. In the above mentioned blog the author gives examples of commodity ETF spreads which are modeled as mean reverting OU process. Say, in a particular mean reversion model of spreads (between any two assets) the long term average is zero. This means that the assets should converge. What the author argues is that if we enter into a meanreverting position to trade this spread and say, after 5 or 6 halflife’s later the spread has not reverted to zero then it would be a strong signal that that perhaps the regime has changed, and the meanreverting model is invalid.Or, it could also be the case that the the long term average value of spread (which was assumed to be zero) and given by in the above equationhas changed
Therefore, to use a viable stop loss while engaging in a mean reverting trading strategy it is essential to have a well calibrated meanrevering OU model and also the halflife of the asset that is being traded.
Reference: Chan, Ernie, What is Your Stop Loss Strategy? Quantitative Trading, Blog, January 15, 2007
http://epchan.blogspot.hk/2007/01/whatisyourstoplossstrategy.html
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