حرکت انفعالی در اختلالات دینامیکی به عنوان یک مدل برای قیمت بازار سهام

A new model for stock price fluctuations is proposed, based upon an analogy with the motion of tracers in Gaussian random fields, as used in turbulent dispersion models and in studies of transport in dynamically disordered media. Analytical and numerical results for this model in a special limiting case of a single-scale field show characteristics similar to those found in empirical studies of stock market data. Specifically, short-term returns have a non-Gaussian distribution, with super-diffusive volatility. Assuming a power-law decay of the time correlation of the disorder, the returns correlation decays rapidly but the correlation function of the absolute returns exhibits a slow power-law decay. The returns distribution converges towards Gaussian over long times. Some important characteristics of empirical data are not, however, reproduced by the model, notably the scaling of tails of the cumulative distribution function of returns. Implied volatilities for options pricing are found by numerical simulation.

Random fluctuations in stock market prices have long fascinated investors and mathematical modelers alike. Although the investors’ hopes of accurately predicting tomorrow's share price appear to be in vain, models which limit themselves to statistical characteristics such as distributions and correlations have had some success. Bachelier's classical model [1] treats the stock price S(t)S(t) as a random walk, leading to the conclusion that the distribution of prices is Gaussian. Samuelson [2] instead describes the log-price equation(1) x(t)=ln[S(t)/S(0)]x(t)=ln[S(t)/S(0)] Turn MathJax on as a random walk, and therefore concludes that the stock price S(t)S(t) should have a log-normal distribution. This model remains in common usage, despite the shortcomings listed below, not least because it permits the derivation of an equation for pricing options and other financial derivatives, e.g., the famous Black–Scholes equation [3]. Nevertheless, empirical evidence from stock market data indicates that the random walk model inadequately describes many important features of the stock price process. The following stylized facts are accepted as established by these studies [3] and [4]: (i) Short-term returns are non-Gaussian, with ‘fat tails’ and high central peaks. The center of the returns distribution is well fitted by Lévy distributions [5]. Recent studies indicate that the tails of the returns distribution decay as power-laws [6]. As the lag time increases, the returns distributions slowly converge towards Gaussian; recent analysis of the Standard & Poor 500 index (S&P500) estimates this convergence is seen only on lags longer than 4 days [6]. (ii) The correlation function of returns decays exponentially over a short timescale, consistent with market efficiency. However, the correlation function of the absolute value of the returns shows a much slower (power-law) decay [6]. (iii) The volatility (standard deviation of returns) grows like that of a diffusion (random walk) process, i.e., as the square root of the lag time, for lag times longer than about 10 min. However, higher frequency (shorter lag) returns demonstrate a super-diffusive volatility, which can be fitted to power-laws with exponents found to range between 0.67 and 0.77 [6] and [7]. (iv) The distribution of stock price returns exhibits a simple scaling: in Ref. [5] a power-law scaling of the peak of the returns distribution P(0)P(0) with lag time is shown to hold across many magnitudes of lag times. The exponent of the power-law is approximately -0.7-0.7. Most of the references cited here examine data from the Standard & Poor 500 index (S&P500), but other international markets are found to behave similarly [6]. The predictions of the random-walk model can be investigated using the random differential equation equation(2) View the MathML sourcedxdt=uw(t), Turn MathJax on which yields the log price x(t)x(t) from each realization of the random function uw(t)uw(t). The classical random walk follows from taking uw(t)uw(t) to be a Gaussian white-noise process with zero mean and (auto)correlation function [8]: equation(3) View the MathML source〈uw(t)uw(t′)〉=α2δ(t-t′), Turn MathJax on where δ(t)δ(t) is the Dirac delta function, and the angle brackets denote averaging over time or over an ensemble of realizations. White noise is the formal derivative of a Wiener process, which connects (2) to rigorous methods for stochastic differential equations [9]. Note that throughout this paper all stochastic processes are assumed (unless explicitly stated otherwise) to have zero mean and to be statistically stationary. Eq. (2) is easily solved, and leads to results which do not agree with the empirical facts (i)–(iv) listed above. Defining the return on the stock price S(t)S(t) at time t as the forward change in the logarithm of S(t)S(t) over the lag time ΔΔ: equation(4) View the MathML sourcerΔ(t)=x(t+Δ)-x(t)=lnS(t+Δ)-lnS(t), Turn MathJax on it is found that the white-noise case leads to probability distribution functions (PDFs) for rΔrΔ which are Gaussian at all lags ΔΔ, in contrast to (i). The lack of memory effects in white-noise leads to correlation functions which decay immediately to zero, unlike (ii). The volatility grows diffusively for all lag values, and so has no super-diffusive range, while the scaling of P(0)P(0) follows a power-law different from that found for actual prices, see (iv) above. A natural generalization of the white-noise model again takes a form similar to (2) equation(5) View the MathML sourcedxdt=uc(t), Turn MathJax on but with uc(t)uc(t) now being a non-white or colored process, incorporating some memory effects. For the particularly simple case of Gaussian noise, it can be described fully by its correlation function equation(6) View the MathML source〈uc(t)uc(t′)〉=α2R(t-t′). Turn MathJax on Here R(t)R(t) decays from a maximum of unity at t=0t=0 to zero as t→∞t→∞, and α2α2 is the variance of the process ucuc. Gaussian noise is attractive because it appears naturally as the limiting sum of many independent noise sources under the central limit theorem [8], and because of its analytical tractability. Indeed Eq. (5) is again easily solvable for Gaussian ucuc, but leads once more to Gaussian-distributed returns at all lag times, in contrast to (i). Moreover, the correlation of absolute returns does not decay more slowly that the returns correlation, as required by (ii). These failings of simple differential equation models have inspired many attempts to reproduce the empirical facts (i)–(iv) using various stochastic models. As a sample of just a few of these, we list the truncated Lévy flights model [10] and [11], ARCH and GARCH models [12] and [13], non-Gaussian Ornstein–Uhlenbeck processes [14], and a model based on a continuous superposition of jump processes [15] and [7]. The first three of these were reviewed in Ref. [16]. An ideal model should reproduce all the experimental facts (i)–(iv), by giving a simple picture of the stochastic process underlying the stock price time series, but no model has yet been found which fulfills all these criteria [17]. In this paper we examine the results of generalizing the random walk models (2) and (5) to motion in a random field: equation(7) View the MathML sourcedxdt=u(x,t). Turn MathJax on Here u(x,t)u(x,t) is a Gaussian random field with zero mean, which is fully described by its correlation function equation(8) 〈u(x,t)u(x′,t′)〉=α2Q(x-x′,t-t′)〈u(x,t)u(x′,t′)〉=α2Q(x-x′,t-t′) Turn MathJax on equation(9) View the MathML source=α2S(x-x′)R(t-t′). Turn MathJax on For simplicity we take the field to be homogeneous and stationary; the factorization in (9) of the correlation into separate time- and x-correlations is for ease of exposition only, and more complicated inter-dependent correlations can be considered. The averaging procedure in Eq. ( 8) is over an ensemble of realizations of the random field u, or using a uniform measure over all possible (x,t)(x,t) values. Eq. (7) implies that the log-price x(t)x(t) changes in a random fashion, but the rate of change is randomly dependent on both time and the current value of x. Note that random-walk models ( 2) and ( 5) may be recovered by taking S(x)≡1S(x)≡1 and choosing the time correlation R(t)R(t) appropriately. We will show that the x-dependence in the noise term yields qualitatively different results to the random walk models, and that many of the empirical observations (i)–(iv) of stock markets may be reproduced by ( 7). The x-dependent noise term in ( 7) is motivated by recent studies in turbulent dispersion [18] and transport in dynamically disordered media [19]. These investigations employ Eq. ( 7), but with a rather different interpretation: x(t)x(t) is the position vector (in 2 or 3 dimensions) of a passive tracer, and u(x,t)u(x,t) is the velocity vector field which transports the tracer. The tracer is called passive because it is assumed not to affect the velocity field by its presence, and so u(x,t)u(x,t) is a prescribed random field. An instructive example is the motion of a small buoy on the surface of the ocean [20]: the two-dimensional vector x(t)x(t) then describes the position of the buoy (by its longitude and latitude, say) while the tracer is moved by the ocean waves according to ( 7), with the Gaussian field u(x,t)u(x,t) providing an approximate description of the ocean wave field. The distribution of such tracers resulting from motion in a Gaussian field is known to be non-Gaussian, at least over short to intermediate timescales [18] and [21]. Crucial to understanding this effect is the difference between the Eulerian velocity u(x,t)u(x,t), i.e., the velocity measured at the fixed location x at time t, and the Lagrangian velocity equation(10) View the MathML sourcev(t)=u(x(t),t), Turn MathJax on which is the time series of velocity measurements made by the tracer itself as it moves through the random field [22]. When there is no x-dependence in the velocity field u, the Eulerian and Lagrangian velocities coincide, but otherwise a Gaussian Eulerian velocity field can yield non-Gaussian Lagrangian processes. The relationship between Eulerian and Lagrangian random processes is an active research area, especially in the case of a compressible (non-zero divergence) velocity field which is of interest here [23], [24] and [25]. Our model of the log-price uses a scalar “position” x(t)x(t) and “velocity” u(x,t)u(x,t) rather than the vector-valued analogues in the turbulent dispersion problems described above. Nevertheless, the concept of Eulerian and Lagrangian velocities carries over to the one-dimensional case, and so we will refer to x(t)x(t) as the “price tracer”, with its “velocity” u(x,t)u(x,t) related by Eq. (7). The fact that the Eulerian velocity field u(x,t)u(x,t) is Gaussian allows us to obtain analytical results, and may be justified as a consequence of the central limit theorem applied to the sum of many independent agents trading in the stock. The remainder of this paper is structured as follows. In Section 2 the consequences of the x-dependence of the noise in Eq. ( 7) are examined through some simple examples, and a special limiting case (single-scale field) is highlighted. Results from the theory of Gaussian disorder are used to derive the x-dependence of the noise from a simple model of trader behavior. The statistics of the Lagrangian velocity ( 10) in the single-scale case are found analytically in Sections 3 and 4, and are related to the statistics of the returns rΔrΔ. Numerical simulations of Eq. ( 7) are considered in Section 5, and a simple method for generating Monte Carlo time series in the single-scale case is found. Results from such simulations are presented in Section 6, and compared to the stylized facts (i)–(iv), as presented in Refs. [6] and [5]. The implied volatility for pricing of options is calculated, leading to volatility ‘smiles’. Section 7 comprises of discussion of the results and directions for future work.

We have proposed a new model for the fluctuations of stock prices, in which the rate of change of the log price is randomly dependent upon both time and the current price, see Eq. (7). This is a very general modelling concept, with the attractive feature that Gaussian Eulerian fields may give non-Gaussian Lagrangian statistics in the observable data, i.e., the time series of returns. Moreover, random field models can be derived from some simple assumptions on the behavior of trading agents (Section 2.1). As noted in Section 5, Monte Carlo simulations of Eq. (7) can be used to find the model predictions for any random field. In this paper we focus on a special case, the ε→0ε→0 limit of a single-scale random field, for which exact analytical results are obtainable, and for which numerical simulations are particularly efficient. Our main analytical results are the expression (48) for the Lagrangian velocity correlation L(t)L(t) in terms of the given Eulerian field correlation R(t)R(t), and the quadrature formulas (40) and (52) giving the correlation function and volatility of the returns. Numerical simulations of stock price time series are efficiently performed using Eq. (62), and allow us to examine features not amenable to exact analysis. We find that several of the important empirical stylized facts (i)–(iv) listed in the Introduction are reproduced by our model, using the correlation function (61) for computational convenience. The parameter values k and T are chosen by comparing the model's volatility to the data in Fig. 3(c) of Ref. [6]. With reference to the stylized facts listed in Section 1, the results of the single-scale random field model may be summarized as follows. (i) Model returns are non-Gaussian, with fat tails and the center of the distribution well-fitted by the Lévy distribution used in Ref. [5]. The tails of the cumulative distributions of model returns do not, however, have the power-law scaling found in Ref. [6]. The normalized model returns distributions exhibit a slow return toward Gaussian, on timescales an order of magnitude longer than the characteristic time of R(t)R(t). However, these timescales are an order of magnitude smaller those found for the convergence to Gaussian in the S&P500 returns in Ref. [6]. (ii) The correlation function of model returns decays quickly (similar to exponentially) over a short timescale. Following the fast decay, the correlation becomes negative, with magnitude decaying more slowly to zero. Although the fast decay for lags under 10 min is very similar that found for the S&P500 in Ref. [6], the empirical correlation function does not reach negative values, rather remaining at a constant “noise level”. Whether this noise level could be masking negative correlation values as predicted by the model is a question requiring further extensive analysis of market data. (iii) The volatility of the model returns grows diffusively for lag times longer than 10 min. The super-diffusive volatility at higher frequencies (shorter lags) is actually due to a logarithmic correction to power-law growth (53), but for the range of lags examined in Refs. [6] and [7] it matches well to a super-diffusive power-law. Higher frequency data is needed to determine which form best fits the actual stock market volatility. (iv) Our model does not have a simple scaling of P(0)P(0) with lag, in contrast to the case of S&P500 prices as demonstrated in Fig. 1 of Ref. [5]. In summary, the single-scale random field model yields time series which have many (but not all) of the important properties of empirical data. The model gives an appealingly intuitive picture of the causes of non-Gaussian behavior in markets, and provides a simple algorithm for generating time series with many market-like features. We anticipate that this algorithm will prove particularly useful in Monte Carlo simulations to determine prices of options and derivatives, as demonstrated in Section 6.1. The concept of modelling stock prices by motion in a random field is quite general—our restriction in this paper to the single-scale case is purely because of the existence of analytical results in this case. It would be very interesting to determine the sensitivity of the results obtained here to the form of the random field. As shown in Fig. 2, the small-εε limit of non-single-scale fields again leads to motion along the u=0u=0 contours, but may also include fast jumps in stock prices. We anticipate that the results presented here will be largely applicable to the contour-following motion in more general fields, but further work is required to describe the statistics of the jumps in stock price. Further work is also in progress to investigate the dependence of all results on finite values of εε, using numerical simulations of Eq. (7).