Asset prices in financial markets, following Madan and Schoutens (2016b), must change of necessity to support the price with a return for the investor. Furthermore, the changes must be surprises, if all available information has already entered the price. As a consequence the changes must occur at surprise times. The simplest model for such times are the arrival times of a Poisson process. The price process is then rendered to be a pure jump process with no continuous component, either deterministic or random. The process for the logarithm of the price is therefore described by the structure for the arrival rates of jumps for all jump sizes *x*,−*∞*<*x*<*∞*. This arrival rate function may be denoted by *k*(*x*,*t*) where the dependence on *t* is in all generality, adapted to all information available at time *t*.

It was noted in Madan and Schoutens (2016b) that when the local motion is described by the generalized class of limit laws (Khintchine 1938; Lévy 1937; Sato 1999), of which the Gaussian law is an example, that then, in the non-Gaussian limit cases, the integral of the arrival rate function must be infinite. Such processes were termed infinite activity processes in Carr et al. (2002). The random variables at a finite horizon with arrival rate functions, that only depend on the jump size *x*, are then self-decomposable laws (Sato 1999). The models investigated in the paper are infinite activity processes.

It is then possible for the small jumps to be so frequent as they get smaller that the sum of all the small positive jumps is infinite and the sum of all the small negative jumps is negative infinity. Such processes are called processes of infinite variation. On the contrary, for finite variation processes the sum of all the small positive and negative jumps are separately finite, in any interval. More generally one may take the sum of all positive and negative jumps and their squares to be finite, and not just the small ones. These considerations require that

$$\int_{-\infty }^{\infty }xk(x,t)dx<\infty ;\text{}\int_{-\infty }^{\infty }x^{2}k(x,t)dx<\infty. $$

The processes being considered for the logarithm of the price are then infinite activity processes of finite variation and quadratic variation.

The use of an infinite activity process is motivated by recognizing that though the number of price moves in reality will be finite it is often quite large for daily data. Such observations inspire the use of limit laws. The limit laws are a special case of infinitely divisible laws termed self decomposable laws with a special structure to their arrival rate functions (Sato 1999). The arrival rate functions when scaled by the absolute jump size must be decreasing functions of the absolute jump size. In particular they are all infinite activity processes.

Finite quadratic variation is a consequence of working with semimartingales that is a maintained hypothesis for arbitrage free price dynamics (Delbaen and Schcharemayer 1994). Finite variation is a simplification that allows one to describe the price process as the difference of two increasing processes, one for the price up ticks and the other for the down ticks. In addition there is considerable evidence (see for a recent example Madan 2016c) that such processes are empirically adequate for describing the physical as well as the risk neutral process.

The stock price process *S*(*t*), with *X*(*t*)= ln(*S*(*t*)) then satisfies

$$\begin{array}{@{}rcl@{}} X(t) &=&X(0)+\sum\limits_{s\leq t}\Delta X(s) \\ \Delta X(s) &=&X(s)-X(s_{-}). \end{array} $$

Additionally, associated with such a jump process is an integer valued random measure *μ*(*d*
*x*,*d*
*t*) that counts the jumps occurring in measurable subsets of space and time. One may then write

$$\begin{array}{@{}rcl@{}} X(t) &=&\left(x\ast \mu \right)_{t} \\ &=&\int_{0}^{t}\int_{-\infty }^{\infty }x\mu (dx,ds). \end{array} $$

The counting measure *μ* has a compensator *ν*(*d*
*x*,*d*
*t*) announcing the arrival rate for jumps at all times that is supposed to be defined by the arrival rate function *k*(*x*,*t*) where

$$\nu (dx,dt)=k(x,t)dxdt. $$

All information about the log price process is then embedded in the arrival rate function and all modeling and estimation efforts are directed to the specification and estimation of arrival rate functions.

In particular there is no room for modeling drifts with or without mean reversion as the price process has no purely deterministic time component. Furthermore, there is also no room for a diffusive volatility given that there is no continuous martingale component in the price process. At the instantaneous level one may only speak of variation and quadratic variation but not directly of drift and volatility. Madan and Schoutens (2016b) shows that the limit of the continuously compounded drift over a horizon *h*, denoted by say *m*
_{
t
}(*h*) tends on division by *h*, to the exponential variation. Specifically, it may be observed that

$${\lim}_{h\rightarrow 0}\frac{m_{t}(h)}{h}=\int_{-\infty }^{\infty }\left(e^{x}-1\right) k(x,t)dx. $$

The concept of reward compensating for risk defined in terms of arrival rate functions is then the exponential variation.

Madan and Schoutens (2016b) model the risk of holding stock positions by an assessment of their necessary fluctuations. The view taken is that prices must move to afford positions with a return and the risk is then that of how far up and down they may go. A temporally conservative evaluation of the magnitude of motion in both directions is constructed from the upper and lower prices prevailing in two price economies that not only exclude arbitrage but also eliminate highly acceptable trades.

The acceptable trades may be defined following Artzner et al. (1999) by a convex cone containing the nonnegative random variables. Every such cone may equivalently be defined by the class of random variables with a nonnegative expectation under a convex collection of test probability measures. The upper and lower prices are then suprema and infima of expectations under all test probabilities (see for example Cherny and Madan (2010), also Madan and Schoutens 2016a). Modeling the test probabilities as those delivering event probabilities bounded above by a concave distribution function evaluated at the physical event probability, leads to upper and lower prices as distorted expectations. The upper/lower distorted expectations (Kusuoka 2001) evaluate expectations using the distorted distribution function obtained by composing a convex/concave distribution function with the physical distribution function of the random variable in question.

Risk exposures are then measured by the gap between the upper price and the expectation and the expectation less the lower price. Parametric distortions are employed with parameters calibrated to enforce the absence of acceptable opportunities in the options market for the S&P 500 index. The convex and concave distorted physical distribution functions then sandwich the risk neutral distribution of the S&P 500 index. In the continuous time limit probability distortions may be replaced by measure distortions (Madan et al. 2016) of arrival rate functions to construct upper and lower prices and their associated risk exposures. Such option calibrated risk exposures are then observed to be compensated by observed exponential variations across a wide variety of stock days. The result is an alternative asset pricing theory based on risk premia for two sided price fluctuations in opposition to classical covariation principles.

Efficient frontiers for risk and reward in terms of multidimensional arrival rate functions are presented in Madan (2016a). Optimal portfolios are here based on maximizing the lower price of the portfolio. This lower price may also be written as the exponential variation, seen as the reward, less the risk measured by the upper price for the centered and negated risk exposure. The continuous time centering is accomplished by subtracting the exponential variation.

Questions about the presence of local mean reversion or momentum in price processes are then to be addressed and answered by modeling and estimating arrival rate functions. Such considerations lead us to write arrival rate functions that depend on both the space variable *X* and the jump size *x*. The data set on which the models will be estimated are short horizon returns from one to, say, five or ten days and hence time homogeneous models with arrival rate functions that do not specifically depend on calendar time are considered. Pure jump Markov processes with arrival rate functions depending on the space variable are called Hunt processes (Hunt 1966). They were formulated and estimated risk neutrally in Madan (2016b) using data on option prices. By way of contrast, the objective of this paper is on their estimation as a physical process from time series data. Mean reversion and momentum are important aspects of price dynamics influencing responses to abnormal price movements. With strong mean reversion one would sell on an extraordinary uptick while with momentum the appropriate action would be to buy on such upticks. Furthermore the structure of mean reversion and momentum can be different with respect to upward and downward moves and this is commented on in greater detail later. For examples of other applications of Hunt processes in the literature we cite Cont and Minca (2013) and Cousin et al. (2012).

Local mean reversion and momentum may be evaluated in the first instance by determining the effects on drift of down and up moves. Positive drifts associated with down moves and vice versa for the up moves represent mean reversion. The opposite result constitutes momentum. Beyond the drift one may address the impact on conditional probabilities of up moves of a given size conditional on such a move. If the conditional probability falls after such an up move then one has mean reversion upwards while if it rises then there is momentum upwards. Similar evaluations may be made for mean reversion and momentum downwards. This second approach was employed in Madan (2016b) to address mean reversion and momentum of risk neutral martingales where these aspects are absent in risk neutral drifts by construction.

The outline of the rest of the paper is as follows. “Spatially inhomogeneous variance gamma processes” section introduces the spatially inhomogeneous variance gamma process to be estimated. “Estimation on time series data” section presents two estimation procedures based on matching digital moments and maximum likelihood. Results are presented for a variety of asset price time series in “Digital moment estimation results” section. “Drift structure of SIVG” section discusses the structure of mean reversion and momentum in the implied drift structure. “Mean reversion and momentum in the process” section reports on the structure of momentum and mean reversion for the process in aggregate, combining the drift and the martingale components. “Risk compensation” section reports on the implicit risk reward relationships. “The S&P 500 index during the financial crisis” section presents results for the S&P 500 index for the period of the financial crisis covering the years 2007 to 2009. “Conclusion” section concludes this paper.