Research paper
Retrodicting with the truncated Lévy flight

https://doi.org/10.1016/j.cnsns.2022.106900Get rights and content

Highlights

  • We derive a power law between truncation length and standard deviation.

  • We suggest standardized returns that evolve according to a time-invariant threshold.

  • We illustrate using intraday spot exchange rate data.

Abstract

There is a point in predicting the past (retrodicting) because we lack information about it. To address this issue, we consider a truncated Lévy flight to model data. We build on the finding that there is a power law between truncation length and standard deviation that connects the bounded past and unbounded future. Even if a truncated Lévy flight cannot predict future extreme events, we argue that it can still be used to model the past. Because we avoid the exact form of the probability density function while allowing its distributional moments, with the exception of the mean, to vary over time, our method is applicable to a wide range of symmetric distributions. We illustrate our point by using US dollar prices in 15 different currencies traded on foreign exchange markets.

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 t, rt. The maximum daily return is known in this case and rt is a bounded variable with cutoff t.

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 σt and cutoff t. This method considers a weighted function of returns as well as t in a probability model. We derive the power law t=ζσtβ for small values of t, where 0<β1 is a damping coefficient and ζ>0 is a constant [1].

Then, we obtain the probability distribution π of future (unbounded) absolute returns by averaging past truncated distributions to describe future returns. As a result, we get the standardized returns Zt=rt/σtβ. Unlike the returns themselves, which are subject to volatility clusters, Zt evolves according to the time-invariant threshold ζ, 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 Y with the density function g(y)=g(y), yR, with g(0)>0 and g(0)=0, and let r be a sharp truncation defined by constraining y to the range [,], with 0<1, and the density function represented as f(y)=g(y)2η(),for |y|; and as f(y)=0 otherwise, with η()=0g(y)dy.If σ is the standard deviation of r, then σ.

Proof

The assumptions g(y)=g(y) and g(0)=0 are distributional properties related to the truncation hypothesis [3], [4], [5], where the assumption g

A bounded probability model

Let Xt be the price at day t, where t is denoted as {1,2,} in the discrete-time domain. Consider that the price update process runs over the expected price at time t as well: Xt=(1+rt)mt,where mt=E[Xt] is the (local) time-varying mean. The rate rt is the return on the mean price value of a first-order stationary stochastic process with a zero mean and a time-varying variance σt2. It is compatible with the no-arbitrage hypothesis, because we can write lnXt=lnXtlnXt1=ln(1+rt)+lnmtrt+δt,

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 ˆt and the estimated standard deviation σˆt are linearly correlated (Fig. 1), with the slopes βˆ 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 t=ζσtβ connects the past and the future, where ζ and β are time-invariant parameters and the cutoff length and the standard deviation σ 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.

References (14)

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Funding: CNPq, Brazil, Capes, Brazil, FAP/DF, Brazil, DPI/DPG/UnB, Brazil.

1

The authors contributed equally to this work.

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