Section: Application Domains

Economy and finance

Basel III and Solvency 2 regulations

As amply demonstrated above, economy is a field where the performativity of mathematical models is particularly noticeable. This has become even more so in recent years in finance because international regulations have fundamentally changed since the Basel II Accords. Among other evolutions, Basel II and III explicitly impose that computations of capital requirements be model-based. The same is true of the Solvency 2 directive, a European regulation aiming in particular at evaluating the amount of capital that insurance companies must hold to reduce the risk of insolvency, much in the spirit in the Basel Accords.

This paradigm shift in risk management has been the source of strong debates among both practitioners and academics, who question whether such model-based regulations are indeed more efficient.

A common feeling in the industry is that regulations will sometimes give a false impression of security: risk managers tend to think that a financial company that would fulfil all the criteria of, say, the Basel III Accords on capital adequacy, is not necessarily on the safe side. This is so mainly because many risks, and most significantly systemic or system-wide risks, are not properly modelled, and also because it is easy to manipulate to some extent various risk measures, such as Value at Risk (VaR).

In parallel, a fast growing body of academic research provides various arguments explaining why current regulations are not well fitted to address risk management in an adequate way, and may even, in certain cases, worsen the situation. In other words, they have a divergent performativity effect.

Our first angle to tackle the performativity of these regulations is to question the Gaussian assumption that is implicitly made in designing them. More precisely, we have already shown in [11], [12] that, in some situations, and because of this assumption, prudential rules are themselves the source of a systemic risk. In [12], it was explained how a wrong model of price dynamics coupled to the regulatory VaR constraint tends to systematically increase Tail Conditional Expectation. [11] details how trying to minimize VaR under Gaussian beliefs for the dynamics of returns when actual movements are stable non-Gaussian results in fact in maximization of VaR. Along with the concept of endogenous risk put forward in [44], this body of work provides a mathematical description of how models perform financial reality: this is a perfect example of divergent performativity, since, because of a wrong model, (mandatory) actions are taken that make financial markets even less similar to the model. More technically, assume the simplest model of returns movements, that is, Brownian motion. Brownian motion is the symmetric stable motion characterized by the stability index $\alpha =2$ and a given scale parameter $\sigma$ (recall that a stable motion is a process with independent and identically distributed increments, where each increment follows a stable law $S_{\alpha }(\sigma ,\beta ,\mu )$. The parameter $\alpha \in (0,2]$ characterizes the jump intensity - the smallest $\alpha$, the largest the jump intensity, with no jumps when $\alpha =2$, that is, for Brownian motion -, $\sigma$ is the scale parameter - proportional to the variance when $\alpha =2$ -, $\beta$ is the skewness parameter and $\mu$ the location one.). Under reasonable assumptions, minimizing VaR in a Brownian market amounts to minimizing the variance. However, in a stable market where $\alpha <2$, which therefore is subject to jumps, minimizing VaR requires to maximize $\alpha$ while choosing an intermediate value of $\sigma$. Furthermore, actions taken under a Brownian belief will tend not only to minimize $\sigma$ but also $\alpha$: therefore, implementing VaR-based regulations founded on the wrong Brownian model tends to decrease $\alpha$, making the market even “more” non-Brownian. This is exactly the definition of divergent performativity.

The work in [11], [12] is only one possible mechanism of performativity, although maybe the simplest one. Starting from this, one may progress in two directions: propose regulations that will avoid at least the particular kind of performativity just described, and study more complex models and their performative effects.

As for the first direction, assuming a stable non-Brownian market, we need to understand what kind of constraints would lead to actions favouring an increase rather than a decrease of $\alpha$. Our first idea is to explore counter-cyclical measures, as current regulations are often blamed for their pro-cyclical effect. In a nutshell, pro-cyclicity is entailed by the fact that, in market downs, actors will be forced by regulations to reduce their exposure, thus amplifying downwards movements. We plan to investigate how this translates into modifications of the $(\alpha ,\sigma )$ couple, and check whether basing regulations on the time evolution of this couple would be efficient. For instance, one might imagine measuring $(\alpha ,\sigma )$ as a function of time, and let financial companies increase or decrease their solvency capital requirements based on the coupled evolution.

As for the second direction, we remark that, since regulations tend to endogenously modify both volatility and jump intensity, it seems natural to define and study processes where the local regularity varies in time, possibly in relation with the value of the process. We have introduced such classes of processes in recent years. We plan to deepen their study in the light of their possible adequacy for the mathematical modelling of performativity. We briefly describe now the first actions we will take in this respect.

Multistable and self-stabilizing processes for financial modelling

It is widely accepted that the dynamics of most financial instruments display jumps and there is a huge literature dealing with jump processes in all areas of financial engineering [32]. In order to get a better understanding of these dynamics, we have developed in recent years various instances of multistable processes. These processes were introduced in [4] and further studied e.g. in [8]. Their main feature is that their local intensity of jumps varies in time. In view of their application, we plan to study the following points:

• Recognizing that the local characteristics (intensity of jumps and scale) vary in time implies that evolution equations these parameters must be proposed for these parameters. We have started to develop Hull and White-like models, where auxiliary EDS are satisfied by both scale and the intensity of jumps. This will hopefully allow one to model in a satisfactorily manner implicit volatility surfaces.

• Robust statistical estimation of $\alpha \left(t\right)$ (or of the couple $\left(\alpha \right(t),h(t\left)\right)$ in the case of the so-called linear multifractional multistable motion) is necessary. Some results are presented in [45], but other methods should be studied.

• Self-regulating processes are processes where the local regularity is a function of the amplitude. They were introduced in [1] and further studied e.g. in [3]. It seems natural to follow the same approach and define “self-stabilizing processes” as processes where the local index of stability is a function of the amplitude. Certain tools used for defining some SRP, namely the fixed point theorem, could be adapted, with the difference that the underlying space will not be the one of continuous functions, but the one of càdlàg functions. As a consequence, the Prohorov metric may have to be considered instead of the sup-norm. We have some preliminary results in this direction, which also include the definition of Markovian self-stabilizing processes. Statistical issues (that is, the estimation of the “self-stabilizing” function) need also be addressed.

Multifractional and self-regulating processes for financial modelling

Besides multistable motions, we will also continue to investigate the use of multifractional Brownian motion in financial modelling. Previous works [29] have shown the potential of this approach, in particular for reproducing certain features of the volatility process [51], and we plan to pursue this line of study. More precisely, we will investigate the following matters:

• The instance of self-regulating processes built so far [1] are not progressive, in the sense that paths are constructed globally rather than in a chronological manner. For this reason, they do not provide adequate models for time series encountered in economy and finance. We will put some effort in trying to construct progressive self-regulating processes. Our first attempts will be based on pathwise stochastic integrals as well as on Skohorod integrals.

• Once progressive self-regulating processes have been built and their basic probabilistic properties been investigated, the second step will consist in constructing estimators for the self-regulating function (that is, the function relating amplitude and regularity). This is of course essential for applications.

• We will finally investigate precisely which economical or financial times series display self-regulation, and examine the performative effect of current regulations when such models are in force.

Performativity of monetary policies

It seems clear that, besides prudential regulations, monetary policies such as quantitative easing used by central banks in Europe, Japan and the USA have a strong impact on economy (In a nutshell, quantitative easing is an unconventional monetary policy by which central banks create new money to buy financial assets in view of stimulating the economy.). There is already a huge literature studying this impact. From a broader perspective, many actions taken by financial authorities are designed in a conceptual frame where volatility is all there is to risk. We believe that incorporating at least another dimension related to jumps is essential for proper control. In this respect, we plan to analyse in a quantitative way what is the impact on the stability of markets of the various measures taken by central banks in recent years, such as Zero Interest Rates Policies, Large Scale Assets Purchases, Forward Guidance or Long Term Refinancing Operations, when one takes into account the jump dimension of risk. Such measures have led to typically very low volatility on the markets. But, as C. Borio of BIS recently stated [30], “history teaches us that low volatility and risk premia are not the signs of smaller risk, but rather than investors are ready to take large risks. The less investors fear risk, the more dangerous the situation is”. In other words, recent monetary policies seem to have lowered volatility at the expense of increasing the intensity of jumps. This view is supported by a number of studies in recent years by the BIS. For instance, [26] argues that the accommodative monetary policy have pushed volatility to low levels in various ways: directly by reducing the amplitude of interest rate movements and by removing to a large extent uncertainty about interest rate changes; and indirectly because an environment of low yields on high- quality benchmark bonds favours risk-taking. Investors then tend to have a lower perception of risk, and thus be inclined to take riskier positions.

Studying such a performative effect is typically in the focus ofAnja. Our first attempts in this direction will be again to use stable or multistable processes in place of the Brownian motion as a source of randomness. The obvious approach is to rewrite current models with this modification. This will however require to define several new notions adapted to this situation. More precisely, most computations in classical models crucially depend on the fact that all the quantities involved are square integrable, a property not available when one deals with (multi-)stable processes. As a consequence, correlations, for instance, are not well-defined; this is a problem as they serve as a fundamental tool in such studies. One possible way out would be to use CGMY or other tempered stable processes instead of stable ones, since this would bring us back in the realm of $L^2$ random variables. The price to pay is that we lose stability, meaning that aggregate behaviours are more difficult to assess. A more ambitious but potentially more fruitful approach is to to start again from the modified classical models but to extend their study in a stable frame so as to be able to compute joint distributions.

Another, very different path, is to use the mathematical theory of causality to tackle these questions [49]. We will recall in the next section some facts about causality. Recent studies have tried to tackle the question of determining the causal structure among economic quantities. For instance, results in [33] suggest that per capita real balances and real per capita private gross domestic product are both causes of real per capita consumption expenditures and that real per capita consumption expenditures and real per capita private gross domestic product in turn cause real per capita gross private domestic fixed investment in a four-variables vector autoregressive model of US macro-economic data for the period January 1949 to April 2002. We plan to use both constraint-based methods and Bayesian approaches to study the causal structure in a graph where the nodes are the various quantities manipulated by quantitative easing policies. As always, one of the main problems will be to define the set of sufficient variables.