Jumps Identification

Before a price jump can be accounted for in an estimation stage, it first has to be identified. Surprisingly, but the literature up to now does not offer a consensus on how to identify price jumps properly. Jumps are identified with various techniques that yield different results.

Generally, a price jump is commonly understood as an abrupt price movement that is much larger when compared to the current market situation. But this definition is too general and hard to define and test. The best way to treat this definition is to define the indicators for price jumps that fit the intuitive definition.

The most frequent approach in the literature is based on the assumption that the price of asset St follows stochastic differential equation, where the two components contribute to volatility:

dS[ — fxt dt C It dWt C Yt dJt

Table 1 A comparison of two modelling approaches

Jump-diffusion models

Infinite activity models

Must contain Brownian component

Do not necessarily contain Brownian component

Jumps are rare events

The process moves essentially by jumps

Distribution of jump sizes is known

“Distribution of jump sizes is known” do not exist: jumps arrive infinitely often

Perform well for implied volatility smile interpolation

Give a realistic description of the historical price process

Easy to simulate

In some cases can be represented via Brownian subordination, which gives additional tractability

where fit is a deterministic trend, St is time-dependent volatility, dWt is standard Brownian motion and YtdJt corresponds to the Poisson-like jump process (Merton 1976). The term StdWt corresponds to the regular noise component, while YtdJt corresponds to price jumps, both terms together form the volatility of the market. Based on this assumption for the underlying process, one can construct price jump indicators and theoretically assess their efficiency. Their efficiency, however, deeply depends on the assumption that the underlying model holds. Any deviation of the true underlying model from the assumed model can have serious consequences on the efficiency of the indicators.

Another approach to describe price formation mechanism is to use models with infinite number of jumps in every interval, which are known as infinite activity Levy models. Cont and Tankov (2004) provide detailed description of ways to define a parametric Levy process. Table 1 compares the advantages and drawbacks of these two approaches.

Since the price process is observed on a discrete grid, it is difficult if not impossible to see empirically to which category the price process belongs. The choice is more a question of modelling convenience than an empirical one (Cont and Tankov 2004).

The key role price jumps play in financial engineering triggered interest in the financial econometrics literature, especially how to identify price jumps. Numerous statistical methods to test for the presence of jumps in high-frequency data have been introduced in recent years. Novotny (2010) propose following classification of market shocks filters: the model-independent price jump indicators, which do not require any specific form of underlying price process, and the model-dependent price jump indicators, which assume a specific form of the underlying price process The first group includes such methods as extreme returns, temperature, p-dependent realized volatility, the price jump index, and the wavelet filter, the second—integral and differential indicators based on the difference between the bi-power variance and standard deviation, and the bi-power statistics.

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