Leverage effect is a well-known stylized fact of financial data. It refers to the negative correlation between price returns and volatility increments: when the price of an asset is increasing, its volatility drops, while when it decreases, the volatility tends to become larger. The name “leverage” comes from the following interpretation of this phenomenon: When an asset price declines, the associated company becomes automatically more leveraged since the ratio of its debt with respect to the equity value becomes larger. Hence the risk of the asset, namely its volatility, should become more important. Another economic interpretation of the leverage effect, inverting causality, is that the forecast of an increase of the volatility should be compensated by a higher rate of return, which can only be obtained through a decrease in the asset value.
Some statistical methods enabling us to use high frequency data have been built to measure volatility. In financial engineering, it has become clear in the late eighties that it is necessary to introduce leverage effect in derivatives pricing frameworks in order to accurately reproduce the behavior of the implied volatility surface. This led to the rise of famous stochastic volatility models, where the Brownian motion driving the volatility is (negatively) correlated with that driving the price for stochastic volatility models.
Traditional explanations for leverage effect are based on “macroscopic” arguments from financial economics. Could microscopic interactions between agents naturally lead to leverage effect at larger time scales? We would like to know whether part of the foundations for leverage effect could be microstructural. To do so, our idea is to consider a very simple agent-based model, encoding well-documented and understood behaviors of market participants at the microscopic scale. Then we aim at showing that in the long run, this model leads to a price dynamic exhibiting leverage effect. This would demonstrate that typical strategies of market participants at the high frequency level naturally induce leverage effect.
One could argue that transactions take place at the finest frequencies and prices are revealed through order book type mechanisms. Therefore, it is an obvious fact that leverage effect arises from high frequency properties. However, under certain market conditions, typical high frequency behaviors, having probably no connection with the financial economics concepts, may give rise to some leverage effect at the low frequency scales. It is important to emphasize that leverage effect should be fully explained by high frequency features.
Another important stylized fact of financial data is the rough nature of the volatility process. Indeed, for a very wide range of assets, historical volatility time-series exhibit a behavior which is much rougher than that of a Brownian motion. More precisely, the dynamics of the log-volatility are typically very well modeled by a fractional Brownian motion with Hurst parameter around 0.1, that is a process with Hölder regularity of order 0.1. Furthermore, using a fractional Brownian motion with small Hurst index also enables to reproduce very accurately the features of the volatility surface.
The fact that for basically all reasonably liquid assets, volatility is rough, with the same order of magnitude for the roughness parameter, is of course very intriguing. Tick-by-tick price model is based on a bi-dimensional Hawkes process, which is a bivariate point process (Nt+, Nt−)t≥0 taking values in (R+)2 and with intensity (λ+t, λ−t) of the form
Here μ+ and μ− are positive constants and the functions (φi)i=1,…4 are non-negative with associated matrix called kernel matrix. Hawkes processes are said to be self-exciting, in the sense that the instantaneous jump probability depends on the location of the past events. Hawkes processes are nowadays of standard use in finance, not only in the field of microstructure but also in risk management or contagion modeling. The Hawkes process generates behavior that mimics financial data in a pretty impressive way. And back-fitting, yields coorespndingly good results. Some key problems remain the same whether you use a simple Brownian motion model or this marvelous technical apparatus.
In short, back-fitting only goes so far.
The essentially random nature of living systems can lead to entirely different outcomes if said randomness had occurred at some other point in time or magnitude. Due to randomness, entirely different groups would likely succeed and fail every time the “clock” was turned back to time zero, and the system allowed to unfold all over again. Goldman Sachs would not be the “vampire squid”. The London whale would never have been. This will boggle the mind if you let it.
Extraction of unvarying physical laws governing a living system from data is in many cases is NP-hard. There are far many varieties of actors and variety of interactions for the exercise to be tractable.
Given the possibility of their extraction, the nature of the components of a living system are not fixed and subject to unvarying physical laws – not even probability laws.
The conscious behavior of some actors in a financial market can change the rules of the game, some of those rules some of the time, or complete rewire the system form the bottom-up. This is really just an extension of the former point.
Natural mutations over time lead to markets reworking their laws over time through an evolutionary process, with never a thought of doing so.
Thus, in this approach, Nt+ corresponds to the number of upward jumps of the asset in the time interval [0,t] and Nt− to the number of downward jumps. Hence, the instantaneous probability to get an upward (downward) jump depends on the arrival times of the past upward and downward jumps. Furthermore, by construction, the price process lives on a discrete grid, which is obviously a crucial feature of high frequency prices in practice.
This simple tick-by-tick price model enables to encode very easily the following important stylized facts of modern electronic markets in the context of high frequency trading:
- Markets are highly endogenous, meaning that most of the orders have no real economic motivation but are rather sent by algorithms in reaction to other orders.
- Mechanisms preventing statistical arbitrages take place on high frequency markets. Indeed, at the high frequency scale, building strategies which are on average profitable is hardly possible.
- There is some asymmetry in the liquidity on the bid and ask sides of the order book. This simply means that buying and selling are not symmetric actions. Indeed, consider for example a market maker, with an inventory which is typically positive. She is likely to raise the price by less following a buy order than to lower the price following the same size sell order. This is because its inventory becomes smaller after a buy order, which is a good thing for her, whereas it increases after a sell order.
A significant proportion of transactions is due to large orders, called metaorders, which are not executed at once but split in time by trading algorithms.
In a Hawkes process framework, the first of these properties corresponds to the case of so-called nearly unstable Hawkes processes, that is Hawkes processes for which the stability condition is almost saturated. This means the spectral radius of the kernel matrix integral is smaller than but close to unity. The second and third ones impose a specific structure on the kernel matrix and the fourth one leads to functions φi with heavy tails.