current research: quantitative analysis in football (soccer)

I have recently developed a strong interest in quantitative analysis of sports, especially soccer; Top tier newspapers, such as The New York Times, Le Monde and El Pais, have recently published my pieces on the FIFA World Cup, FIFA rankings, the UEFA Champions League, and the UEFA Euro 2016.

Rethinking the FIFA World Cup final draw: The official rules produce unbalanced groups, are unfair to some teams, and do not produce equally likely outcomes. I suggest a new tractable draw procedure that produces eight balanced and geographically diverse groups, is fair to all teams, and gives equally likely outcomes.

The new rule that I suggest for a fairer draw is described in the following articles:
- The World Cup Draw Is Unfair. Here's a Better Way, The New York Times, June 4, 2014. Available online here.
- La FIFA doit aussi revoir le tirage au sort de sa Coupe du monde, Le Monde, June 4, 2014. Available online here.
- Repenser le tirage au sort de la Coupe du monde, So Foot, June 4, 2014. Available online here.
- El sistema del sorteo de grupos del Mundial es injusto. Cambiémoslo, El Pais, June 16, 2014. Available online here.
- Interview in La Tête Au Carré on the French national public radio France Inter available online here (May 30, 2014; 600,000+ listeners).
- This is an article in the French newspaper Le Monde promoting my work (May 14, 2014).
- An academic article was published in Journal of Quantitative Analysis of Sports, 11(3):169-182, 2015. An extented version is available online here.
- I was invited to present my research at MathSport International 2015, EURO 2015, and at Harvard University for the New England Symposium on Statistics in Sports (NESSIS).

Flaws of the FIFA Rankings system: The FIFA Rankings calculation method is flawed in many ways. Home advantage is ignored, as well as goal difference. Loosing against team #1 or team #200 makes no difference. Not playing friendlies can be very beneficial. The system penalizes the hosts of major tournaments. This is explained in the following articles:
- A Better Way to Rank Soccer Teams in a Fairer World Cup, The New York Times, June 13, 2014. Available online here.
- Pourquoi la France va dégringoler au classement FIFA, So Foot, June 23, 2015. Available online here.

UEFA Champions League: How to Solve the Seeding Problem: Starting in 2015, UEFA has put in place a new seeding rule for the Champions League, in order to protect more domestic champions. This is done by placing the domestic champions of the best leagues in Pot 1. I suggest another way, which takes into account the relative strength of European domestic leagues.

The new rule that I suggest for a fairer draw is described in the following articles:
- Champions League: How to Solve the Seeding Problem, The New York Times, January 21, 2015. Available online here.
- Ligue des champions : comment améliorer le tirage au sort, Le Monde, February 24, 2015. Available online here.
- Ligue des champions : comment résoudre le problème des têtes de série, So Foot, February 24, 2015. Available online here.

What a fairer UEFA Euro 2016 could look like: For the first time in 2016, the UEFA Euro gathered 24 men's national teams. Since the 24 teams are divided into 6 groups of 4, it is not straightforward to build a fair format for the knockout stage. We critically examine a number of flaws in the current knockout bracket of the UEFA Euro 2016 and we suggest two fairer procedures that eliminate group advantage and lack of win incentive and still satisfy the balance and group constraints. The suggested procedures apply to any tournament consisting of a round robin stage made of 6 groups of 4, followed by a knockout stage.

The new formats that I suggest for a fairer 24-team UEFA Euro are described in the following articles:
- What a Fairer UEFA Euro 2016 Could Look Like, SSRN, January 12, 2016. Available online here.
- Euro 2016 : comment le tableau final favorise la France, par Julien Guyon, mathématicien, Le Monde, December 12, 2015 (in French). Available online here.
- Euro 2016 : un autre tableau final est possible, Le Monde, June 24, 2016 (in French). Available online here.

current research: quantitative finance

My current research in quantitative finance is mainly about nonlinear option pricing, the associated numerical probabilistic methods, and volatility and correlation modeling.
My contributions to numerical methods for solving nonlinear option pricing problems include:
- Model-free bounds for VIX futures given S&P 500 smiles using optimal transportation theory
- Cross-dependent volatility models
- Path-dependent volatility models
- Local correlation models
- Monte Carlo methods for chooser options; lower and upper bounds (optimal stopping problems, dual formulation)
- Monte Carlo pricing in the uncertain volatility model (backward SDEs)
- Links between spot volatilities and implied volatilities (heat kernel expansion)
- The ''particle method'' calibration of local stochastic volatility models, stochastic rates and local volatility models, stochastic rates and local stochastic volatility models, local volatility and local correlation models, path-dependent volatility models (PDEs, non-linear SDEs, Malliavin calculus)
- Pricing of reinsurance deals with uncertainty on lapse and mortality (PDEs, Monte Carlo, backward SDEs)

past research: quantitative finance

Past research topics in quantitative finance include:
- Stochastic volatility models: static and dynamic properties
- FX options vanilla smile
- Constant Maturity Swaps and the smile of swaptions
- Time-extrapolations for numerical schemes for SDEs

PhD research: probability theory and statistics applied to finance and biology

- Approximation in density of stochastic processes. Application to option pricing and hedging. I have derived functional expansions for the density of the Euler scheme approximating the solution of a regular stochastic differential equation. Such expansions allow infinite differentiation with respect to both space variables, the starting point x and the current point y. They involve functional coefficients and remainders that have gaussian tails. I have also studied the way these coefficients and remainders explode when time tends to zero. I have given applications to option pricing and hedging, proving numerical convergence rates for prices, deltas and gammas on a lognormal stock market. I have proved that applying the Euler scheme of order n to the logarithm of the underlying leads to approximations of the price, the deltas and the gammas which converge to the true price, deltas and gammas with speed 1/n. I have also showed that the principal part of the error explodes as the maturity t tends to 0 as t^{-1/2} for the prices, t^{-1} for the deltas and t^{-3/2} for the gammas. As for the O(1/n^2) remainder, the rates are respectively t^{-2}, t^{-5/2} and t^{-3}.

- Stochastic volatility models. I have studied ergodic stochastic volatility models, as presented in the book "Derivatives in Financial Markets with Stochastic Volatility" by J.-P. Fouque, G. Papanicolaou and K. R. Sircar (Cambridge University Press). Besides the theoretical study of such models, I have performed simulations in the case when the mean reversion parameter is not large. Such models generate a wide range of volatility smiles.

- Statistics for unbalanced tree data. Limit theorems for bifurcating Markov chains. Application to the detection of cellular aging. Together with J.-F. Delmas, we have studied dependent data on a binary tree. Think of the descent of an initial individual, where each individual in one generation gives rise to two offspring, one of type 0 and one of type 1, in the next generation. Suppose we are interested in a particular characteristic of these individuals. We assume that this characteristic is stochastic and depends on the ancestors' only through the mother's. The dependency structure may be described by a transition probability P(x,dydz) which gives the probability that the couple of daughters' characteristics is around (y,z) given that the mother's characteristic is x. Note that y, the characteristic of the daughter of type 0, and z, that of the daughter of type 1, may be conditionally dependent given x, and their respective conditional ldistributions may differ (that is why we speak of unbalanced data tree). We then speak of bifurcating Markov chains. We have derived laws of large numbers and central limit theorems for such stochastic processes. We have applied these results to detect cellular aging in Escherichia Coli, using the data of the Laboratoire de Genetique moleculaire, evolutive et medicale (INSERM U571, Faculte de Medecine Necker, Paris) and a bifurcating autoregressive model.