Research paperRetrodicting with the truncated Lévy flight☆
Introduction
Making predictions, especially about the future, is difficult, according to an old Danish proverb retold by many, from baseball player Yogi Berra to physicist Niels Bohr. Despite the statement’s comical redundancy, there is a point in also predicting the past (retrodicting) because we lack information about it. Helenus was said to be able to predict the past with great accuracy in The Iliad. We are dealing with such predictions about the past here.
We lack information when it comes to forecasting the future and the past. However, while the future is limitless, the past is not. We demonstrate how a power law can be used to connect bounded past events and unbounded future events.
To model data, we use a truncated Lévy flight and infer a power law between the truncation length and its standard deviation. The problem of induction plagues the prediction of the future in a truncated Lévy flight: how to successfully consider very extreme data until the next single big event occurs, rendering the current modeling useless. However, building on the power law finding [1], we argue that, while a truncated Lévy flight cannot predict future extreme events (as established in the literature [2]), it can still be used to model the past.
Although we consider a truncated Lévy flight, our approach is applicable to a broader class of symmetric distributions because we can do without an exact form for the probability density function: the distributional moments, with the exception of the mean, are allowed to vary over time.
We illustrate our point by analyzing intraday US dollar prices in 15 currencies traded on foreign exchange markets, totaling more than 32 million ticks.
Financial time series data from the past and future exhibit distinct stochastic characteristics. One of them is whether the data is bounded. Consider realized intraday financial returns at time , . The maximum daily return is known in this case and is a bounded variable with cutoff .
To make our point, we chose the benchmark truncated Lévy flight with a sharp truncation. Initially, the justification for the truncation was the finiteness of data [3], [4], [5] or the fact that all physical systems have boundary conditions [2]. The truncation process became more sophisticated [6], [7], [8], but we will stick to the sharp truncation form.
When it comes to forecasting the future, unrealized returns follow an unbounded distribution because bounds are still unknown. As a result, the truncated Lévy flight cannot explain or forecast extreme future returns. However, we can reconcile past and future financial returns as follows.
To begin, we propose a novel sharp truncation form for describing symmetrically distributed past data with standard deviation and cutoff . This method considers a weighted function of returns as well as in a probability model. We derive the power law for small values of , where is a damping coefficient and is a constant [1].
Then, we obtain the probability distribution of future (unbounded) absolute returns by averaging past truncated distributions to describe future returns. As a result, we get the standardized returns . Unlike the returns themselves, which are subject to volatility clusters, evolves according to the time-invariant threshold , allowing connections with classical extreme value models [9], [10], [11].
The remainder of the paper is organized as follows. The power law feature of two types of abruptly truncated symmetric distributions is introduced in Section 2. Our proposed model for the returns of the mean price value is described in Section 3. Section 4 uses tick-by-tick quote data from the dollar bid prices of 15 currencies traded on foreign exchange markets to illustrate our approach, and Section 5 concludes.
Section snippets
The abrupt truncation
Proposition 1 Consider a continuous random variable with the density function , , with and , and let be a sharp truncation defined by constraining to the range , with , and the density function represented as for ; and as otherwise, with If is the standard deviation of , then .
Proof The assumptions and are distributional properties related to the truncation hypothesis [3], [4], [5], where the assumption
A bounded probability model
Let be the price at day , where is denoted as in the discrete-time domain. Consider that the price update process runs over the expected price at time as well: where is the (local) time-varying mean. The rate is the return on the mean price value of a first-order stationary stochastic process with a zero mean and a time-varying variance . It is compatible with the no-arbitrage hypothesis, because we can write ,
Data and analysis
We use intraday spot exchange rate data of fifteen currencies against the US dollar transacted on foreign exchange (Forex) market to illustrate our method. There are 32,308,882 tick-by-tick bid prices with a millisecond timestamp kindly provided by Tick Data, LLC (Table 1).
The logarithms of the maximum daily absolute return and the estimated standard deviation are linearly correlated (Fig. 1), with the slopes shown in Table 2 for each line. This empirical result is consistent
Conclusion
We argue in this study that a truncated Lévy flight is still effective for predicting the past, even if it cannot forecast future extreme events. We address its induction problem and find that the power law connects the past and the future, where and are time-invariant parameters and the cutoff length and the standard deviation change through time.
This point is demonstrated in a new truncation model (Eq. (12)), but the power law (Eq. (20)) remains valid for other alternatives if
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
We thank Tick Data LLC (www.tickdata.com) for providing the data, which is available on Figshare ( https://doi.org/10.6084/m9.figshare.19808536.v1). FAP/DF, Brazil, DPI/DPP/UnB, Brazil, CNPq , and Capes, Brazil have all provided financial support.
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Cited by (3)
- ☆
Funding: CNPq, Brazil, Capes, Brazil, FAP/DF, Brazil, DPI/DPG/UnB, Brazil.
- 1
The authors contributed equally to this work.