{"id":229215,"date":"2025-08-20T11:29:20","date_gmt":"2025-08-20T15:29:20","guid":{"rendered":"https:\/\/ibkrcampus.com\/campus\/?p=229215"},"modified":"2025-08-20T13:56:59","modified_gmt":"2025-08-20T17:56:59","slug":"trading-mid-frequency-and-rv-like-a-pro-part-2-how-to-scale-trades","status":"publish","type":"post","link":"https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/trading-mid-frequency-and-rv-like-a-pro-part-2-how-to-scale-trades\/","title":{"rendered":"Trading Mid-Frequency and RV Like a Pro (Part 2): How to Scale Trades"},"content":{"rendered":"\n

Read Part I of this series to discover common use cases for mean reversion<\/em>: Trading Mid-Frequency and RV like a Pro (Part 1): Identifying Mean-Reversion with Statistical Tests<\/a>.<\/em><\/em><\/p>\n\n\n\n

Introduction<\/h2>\n\n\n\n

Mean reversion trading can be as simple as forecasting the level (or spread) we want to trade, and allocating relative to the forecast, i.e., in the simple case of an AR(1) model (or Ornstein-Uhlenbeck process), we have a model such as<\/p>\n\n\n\n

<\/path><\/path><\/path><\/path><\/path><\/path><\/path><\/path><\/path><\/defs><\/use><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/g><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/g><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/g><\/g><\/g><\/svg><\/p>\n\n\n\n


where <\/path><\/path><\/defs><\/use><\/g><\/use><\/g><\/g><\/g><\/g><\/svg> is the price level (or spread) at time <\/path><\/defs><\/use><\/g><\/g><\/g><\/svg> and <\/path><\/path><\/path><\/path><\/path><\/path><\/path><\/defs><\/use><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/g><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/g><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/g><\/g><\/g><\/g><\/svg>, is the backward differences, <\/path><\/defs><\/use><\/g><\/g><\/g><\/svg> is the long-term mean,  <\/path><\/defs><\/use><\/g><\/g><\/g><\/svg> is the autoregressive coefficient, and <\/path><\/path><\/defs><\/use><\/g><\/use><\/g><\/g><\/g><\/g><\/svg> is a white noise error term (although we may have other terms in the model, such as a trend, or a seasonal, or some AR terms to capture the short-run dynamics).<\/p>\n\n\n\n

In such a model, if we recast it as<\/p>\n\n\n\n

<\/path><\/path><\/path><\/path><\/path><\/path><\/path><\/path><\/path><\/path><\/path><\/path><\/path><\/defs><\/use><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/g><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/g><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/g><\/use><\/g><\/g><\/g><\/svg><\/p>\n\n\n\n

<\/p>\n\n\n\n

where <\/path><\/path><\/path><\/path><\/path><\/defs><\/use><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/rect><\/g><\/g><\/g><\/svg>, we can see that the model is mean-reverting if <\/path><\/path><\/path><\/path><\/path><\/defs><\/use><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/g><\/g><\/svg>, and it mean reverts to the long-run level <\/path><\/defs><\/use><\/g><\/g><\/g><\/svg>, with the speed of mean reversion<\/path><\/defs><\/use><\/g><\/g><\/g><\/svg>.<\/p>\n\n\n\n

With this model we can forecast the level as:<\/p>\n\n\n\n

<\/path><\/path><\/path><\/path><\/path><\/path><\/path><\/path><\/path><\/path><\/path><\/defs><\/use><\/g><\/use><\/g><\/g><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/g><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/g><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/g><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/g><\/g><\/use><\/g><\/g><\/g><\/svg><\/p>\n\n\n\n

where <\/path><\/path><\/defs><\/use><\/g><\/use><\/g><\/g><\/g><\/g><\/g><\/svg> and  <\/path><\/path><\/defs><\/use><\/g><\/use><\/g><\/g><\/g><\/g><\/g><\/svg> are the estimated coefficients from the model, (i.e., they are just found via the OLS regression of <\/path><\/path><\/path><\/path><\/path><\/defs><\/use><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/g><\/g><\/g><\/g><\/svg> on <\/path><\/path><\/defs><\/use><\/g><\/use><\/g><\/g><\/g><\/g><\/g><\/svg>). As we mentioned, we could have also considered adding more AR terms, or a trend or a seasonal component.<\/p>\n\n\n\n

No matter how complex the model, it is possible to forecast one (as we have done above) or more periods ahead, by recursively applying the model or \u201cbootstrapping\u201d the forecast. We note however that forecast quality often deteriorates as we forecast further ahead.<\/p>\n\n\n\n

Optimal Trading Strategies<\/h2>\n\n\n\n

In all of our discussions, we will assume that returns <\/path><\/path><\/path><\/path><\/path><\/path><\/path><\/defs><\/use><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/g><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/g><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/g><\/g><\/g><\/g><\/svg> are just the PnL of the underlying asset. We could have assumed that they were log-returns, returns, etc. It matters most in how we describe or scale our holdings since in the end whatever strategy we come up with should have PnL denominated in USD.<\/p>\n\n\n\n

 We define a strategy as a set of rules that determines how we allocate at time t<\/em> given our forecast of the level (or spread) for time t + 1<\/em>. A (self-financing)strategy will have a PnL at time T<\/em> of:<\/p>\n\n\n\n

<\/path><\/path><\/path><\/path><\/path><\/path><\/path><\/path><\/path><\/path><\/path><\/path><\/path><\/path><\/path><\/path><\/path><\/path><\/defs><\/use><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/g><\/g><\/g><\/use><\/g><\/g><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/g><\/g><\/use><\/g><\/use><\/g><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/use><\/g><\/g><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/g><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/g><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/use><\/g><\/g><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/g><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/g><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><\/p>\n\n\n\n

where, the strategy returns at time t <\/em>are given by the holdings at time t <\/em>multiplied by the return at time t + 1<\/em>, i.e.,<\/p>\n\n\n\n

<\/path><\/path><\/path><\/path><\/path><\/path><\/path><\/path><\/path><\/path><\/path><\/path><\/defs><\/use><\/g><\/use><\/g><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/g><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/g><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/g><\/use><\/g><\/g><\/g><\/svg><\/p>\n\n\n\n

<\/p>\n\n\n\n

The holdings at time t<\/em> should depend on our expectations at time t<\/em> of expected returns over the following period(s). (In more exact terminology, we would say they are adapted to the filtration<\/em> at time t<\/em>, which is the information available at time t<\/em>). Given that we let our returns be just PnL, our holdings are denominated in USD notional, and if we scale them by the point-value of the given asset, we can think of them as the number of units of the asset we hold at time t<\/em>.<\/p>\n\n\n\n

Optimal Mean Reversion Trading with no costs<\/h2>\n\n\n\n

One standard (and economically justifiable) method for forming a strategy, given the forecasts, is known as mean-variance optimization (MVO). In the simplest case, we assume that we want to maximize the utility of our holdings, which is a function of the expected return and the risk (variance) of the returns.:<\/p>\n\n\n\n

<\/path><\/path><\/path><\/path><\/path><\/path><\/path><\/path><\/path><\/path><\/path><\/path><\/path><\/path><\/path><\/path><\/path><\/path><\/path><\/defs><\/use><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/g><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/rect><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/g><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/use><\/g><\/g><\/g><\/g><\/svg><\/p>\n\n\n\n

<\/p>\n\n\n\n

Where:<\/p>\n\n\n\n