The occurrence of jumps in asset prices is a well-known phenomenon that has many implications in financial models. In his 1976 paper, Robert Merton extended the Black-Scholes-Merton model for option pricing, which assumes continuous Geometric Brownian Motion (GBM), to account for jumps in the underlying asset price. Interest rate models for the short rate often include the possibility of jumps. Jumps add another dimension of complexity to hedging derivatives. Jump models also have applications in predicting credit default risk.

In each of these examples, a model is *developed *based on some underlying process for the evolution of prices, rates, probabilities of default, etc., and then the model is *used *by inputting parameters either estimated from market data or calibrated to match observable quantities in the market. Decisions are then based on predictions from the model.

This can be a classic case of “garbage in, garbage out” if we are not careful. The parameters used in the models are generally estimated using historical data, which means that jump frequencies and sizes must be determined from what we have observed in the market. There are many research studies that focus on detecting jumps and estimating these parameters, with mixed results (a good survey of some methods can be found here).

Many of these applications seem disconnected from the realm of tactical asset management, which is more concerned with reacting to prices and market conditions to optimize, either implicitly or explicitly, a risk-return profile for investors. Certain methods may forecast future volatility and correlations (somewhat manageable) or returns (much more difficult), but I have not seen any that attempt to *predict* a jump in a stock price. If your tactical model relies on forecasting a jump as a basis for a positive or negative return, I certainly will not invest with you. So how do jumps come in to play?

Let’s look at some data. The chart below shows the daily closing price of Coventry Health Care (CVH) in 2012, along with the realized volatility calculated using an exponentially weighted moving average (EWMA) over the previous 63 trading days (a typical predictor of current volatility).

Notice the jump in price and spike in volatility in August? On August 20, 2012, Aetna entered into a deal to purchase CVH. But take a look at what happens to the realized volatility graph over the subsequent time, once the EWMA takes weight off of the larger return from the day of the news announcement. It decays back to its previous value about 30 days after August 20^{th}. Although the volatility was elevated for some time, the exponential weighting eventually filtered out the effect of the price change on volatility.

Now let’s look at Citi (C) during the financial crisis.

We should not take for granted the fact that specific, public information drove these changes and that we can venture guesses at the time as to whether volatility will in fact increase or will revert to the previous level with the stock price merely shifted to a new level. However, we must be careful to avoid hindsight bias when looking at these results.

The graph below shows two hypothetical stock price paths generated using GBM with the same volatility up to day 200. On day 200, the price of both stocks jumps up by 15% and the volatility of Stock Y doubles thereafter. From that point on, we observe two different behaviors in the realized volatilities:

- Stock X exhibits what we saw in Coventry Health Care with the realized volatility spiking for the 63 day window and the decaying to the previous level.
- Stock Y is a bit like Citi, with the realized volatility increasing at the jump and remaining elevated even after the jump is not used heavily in the realized volatility calculation.

It takes 25 days after the jump for the realized volatilities to differ by 5 percentage points, and since the volatility of a more realistic stock would likely not have such a stable volatility, the decay in realized volatility of Stock X would not be so smooth. Thus, it would take some time after day 200 to detect that the volatility of Stock Y did actually increase.

As I stated before, specific news about companies, sectors, or markets as a whole can cause the price jumps, and in theory I could look on Yahoo Finance to see if larger jumps could be attributable to what I find. But this kind of oversight compromises the quantitative integrity around which we center our models.

Also, imagine how hard this would be when backtesting strategies (refer to our white paper on backtesting for other issues that could arise) or trying to explain apparent jumps in ETFs, mutual funds, or other combinations of assets.

At Newfound Research, our dynamic models are able to adapt to changes in the speed of information flow in the market. Since the models utilize volatility adjusted momentum, accurately estimating volatility from this information is essential, which necessitates identifying and handling jumps in an appropriate manner that maintains quantitative integrity. How to handle the jumps once they are detected is a topic that will be further discussed in a subsequent blog post.

Source: All stock data was obtained from Yahoo Finance.

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