By Aimé Lachapelle & Marie-Therese Wolfram (Cahier de la Chaire n°40)

In this paper we present a new class of pedestrian crowd models based on the mean field games theory introduced by Lasry and Lions in 2006. This macroscopic approach is based on a microscopic model, that considers smart pedestrians who rationally interact and anticipate the future. This leads to a forward-backward structure in time. We focus on two-population interactions and validate the modeling with simple examples such as self-organization behavior as for instance lane formation. Two complementary classes of problems are addressed, namely the case of crowd aversion and the one of congestion. In both cases we describe the model, build a numerical solver (respectively based on optimization formulation and partial differential equations), and finally provide some numerical tests involving complex group behaviors such as symmetry breaking and lane formation.

Keywords: Mean field games, interacting populations, Nash equilibrium, rational expectations, flow of pedestrians, lane formation, numerical approximation

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By Jean-Michel Zakoïan (Cahier de la Chaire n°39)

This paper considers Dynamic Conditional Correlations (DCC) GARCH models in which the time-varying coefficients, including the conditional correlation matrix, are functions of the realizations of an exogenous stochastic process. Time series generated by this model are in general nonstationary. Necessary and sufficient conditions are given for the existence of non-explosive solutions, and for the existence of second-order moments of these solutions. Potential applications concern the modeling of the volatility of a vector of energy prices, the model coefficients depending on the weather conditions.

Keywords: Dynamic conditional correlation, Existence of nonexplosive solutions, Multivariate GARCH, Nonstationary processes, Time-varying models

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By René Aïd, Luciano Campi & Nicolas Langrené (Cahier de la Chaire n°38)

We develop a structural risk-neutral model for energy market modifying along several directions the approach introduced in [Aïd et al., 2009]. In particular a scarcity function is introduced to allow important deviations of the spot price from the marginal fuel price, producing price spikes. We focus on pricing and hedging electricity derivatives. The hedging instruments are forward contracts on fuels and electricity. The presence of production capacities and electricity demand makes such a market incomplete. We follow a local risk minimization approach to price and hedge energy derivatives. Despite the richness of information included in the spot model, we obtain closed-form formulae for futures prices and semi-explicit formulae for spread options and European options on electricity forward contracts. An analysis of the electricity price risk premium is provided showing the contribution of demand and capacity to the futures prices. We show that when far from delivery, electricity futures behave like a basket of futures on fuels.

Keywords: Electricity spot and forward prices, fuels, capacity, electricity demand, scarcity function, local risk minimization, minimal martingale measure, power derivatives, spread options, extended incomplete Goodwin-Staton integral

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By Delphine Lautier & Franck Raynaud (Cahier de la Chaire n°37)

This article presents an empirical study of thirteen derivative markets for commodity and financial assets. This paper goes beyond statistical analysis by including the maturity as a variable for futures contracts’s daily returns, from 1998 to 2010 and for delivery dates up to 120 months. We observe that the mean and variance of the commodities follow a scaling behavior in the maturity dimension with an exponent characteristic of the Samuelson effect. The comparison of the tails of the probability distribution according to the expiration dates shows that there is a segmentation in the fat tails exponent term structure above the Lévy stable region. Finally, we compute the average tail exponent for each maturity and we observe two regimes of extreme events for derivative markets, reminding of a phase diagram with a sharp transition at the 18th delivery month.

Keywords: Econophysics, Derivatives, Term structures, Tail exponents, PACS:

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By Gilles-Edouard Espinosa & Nizar Touzi (Cahier de la Chaire n°36)

Let X be a mean reverting scalar process, X the corresponding running maximum, T0 the first time X hits the level zero and ` a loss function, mainly increasing and convex. We consider the following optimal stopping problem (…) over all stopping times with values in [0; T0]. Under mild conditions, we prove that an optimal stopping time exists and is defined by (…) where the boundary  is explicitly characterized as the concatenation of the solutions of two equations. We investigate some examples such as the Ornstein-Uhlenbeck process, the CIR-Feller process, as well as the standard and drifted Brownian motions. Finally, we perform an empirical examination of the efficiency of this strategy on real financial data.

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By Dylan Possamai, H. Mete Soner & Nizar Touzi (Cahier de la Chaire n°35)

We consider a financial market with liquidity cost as in Cetin, Jarrow and Protter (2004) where the supply function  depends on a parameter  with  corresponding to the perfect liquid situation. Using the PDE characterization of Cetin, Soner and Touzi (2007) of the super-hedging cost of an option written on such a stock, we provide a Taylor expansion of the super-hedging cost in powers of . In particular, we explicitly compute the first term in the expansion for a European Call option and give bounds for the order of the expansion for a European Digital Option.

Key words: Super-replication, liquidity, viscosity solutions, asymptotic expansions

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By H. Mete Soner, Nizar Touzi & Jianfeng Zhang (Cahier de la Chaire n°34)

We provide an existence and uniqueness theory for an extension of backward SDEs to the second order. While standard Backward SDEs are naturally connected to semilinear PDEs, our second order extension is connected to fully nonlinear PDEs, as suggested in Cheridito et al. (2007). In particular, we provide a fully nonlinear extension of the Feynman-Kac formula. Unlike Cheridito et al. (2007), the alternative formulation of this paper insists that the equation must hold under a non-dominated family of mutually singular probability measures. The key argument is a stochastic representation, suggested by the optimal control interpretation, and analyzed in the accompanying paper (Soner et al. 2009).

Keywords: Backward SDEs, non-dominated family of mutually singular measures, viscosity solutions for second order PDEs

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By Bruno Bouchard & Nizar Touzi (Cahier de la Chaire n°33)

We prove a weak version of the dynamic programming principle for standard stochastic control problems and mixed control-stopping problems, which avoids the technical difficulties related to the measurable selection argument. In the Markov case, our result is tailor-made for the derivation of the dynamic programming equation in the sense of viscosity solutions.

Keywords: Optimal control, Dynamic programming, discontinuous viscosity solutions

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By Arash Fahim, Nizar Touzi & Xavier Warin (Cahier de la Chaire n°32)

We consider the probabilistic numerical scheme for fully nonlinear PDEs suggested in Cheridito et al (2007), and show that it can be introduced naturally as a combination of Monte Carlo and finite differences scheme without appealing to the theory of backward stochastic differential equations. Our first main result provides the convergence of the discrete-time approximation and derives a bound on the discretization error in terms of the time step. An explicit implementable scheme requires to approximate the conditional expectation operators involved in the discretization. This induces a further Monte Carlo error. Our second main result is to prove the convergence of the latter approximation scheme, and to derive an upper bound on the approximation error. Numerical experiments are performed for the approximation of the solution of the mean curvature flow equation in  dimensions two and three, and for two and five-dimensional (plus time) fully-nonlinear Hamilton-Jacobi-Bellman equations arising in the theory of portfolio optimization in financial mathematics.

Keywords: Viscosity Solutions, monotone schemes, Monte Carlo approximation, second order backward stochastic dierential equations.

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by Christina Atanasova & Gilles Chemla (Cahier de la Chaire n°31)

This paper provides new evidence that familiarity bias affects the portfolios ofinstitutional investors. Using a sample of large US defined-benefit pension plans for the period1992 to 2002, we show that the corporate focus of the sponsoring firm has an impact on theinvestment policy of the pension plan. Pension plans sponsored by firms with a high proportionof foreign sales are more likely to invest in international assets, plans sponsored by firms thatare active in research and development are more likely to invest in private equity, and planswith sponsors that have more fixed assets are more likely to invest in real estate andmortgages. Comparing to existing explanations of why plans tilt their portfolios towards thesponsor’s focus, familiarity bias is the most compelling one. The worse performance of pensionplans with such portfolio allocation bias is consistent with pension managers being overconfidentabout familiar assets.

Keywords: Institutional investment, defined benefit pension plans, familiarity bias

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By D. Crisan, K. Manolarakis & N. Touzi (Cahier de la Chaire n°30)

We propose a generic framework for the analysis of Monte Carlo simulation schemes of backward SDEs. The general results are used to re-visit the convergence of the algorithm suggested by Bouchard and Touzi (2004). By keeping the higher order terms in the expansion of the Skorohod integrals resulting from the Malliavin integration by parts in Bouchard and Touzi (2004), we introduce a variant of the latter algorithm which allows for a significant reduction of the numerical complexity. We prove the convergence of this improved Malliavin based algorithm, and derive a bound on the induced error. In particular, we show that the price to pay for our simplification is to use a more accurate localizing function.

Keywords: BSDEs, Weak approximations, Monte Carlo methods, Malliavin calculus

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By Bruno Bouchard, Romuald Elie & Nizar Touzi (Cahier de la Chaire n°29)

We consider the problem of finding the minimal initial data of a controlled process which
guarantees to reach a controlled target with a given probability of success or, more generally,
with a given level of expected loss. By suitably increasing the state space and the controls,
we show that this problem can be converted into a stochastic target problem, i.e. nd the
minimal initial data of a controlled process which guarantees to reach a controlled target with
probability one. Unlike the existing literature on stochastic target problems, our increased
controls are valued in an unbounded set. In this paper, we provide a new derivation of the
dynamic programming equation for general stochastic target problems with unbounded controls,
together with the appropriate boundary conditions. These results are applied to the problem
of quantile hedging in nancial mathematics, and are shown to recover the explicit solution of
Follmer and Leukert.

Keywords: Stochastic target problem, discontinuous viscosity solutions, quantile hedging

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By Gilles Chemla & Christopher A. Hennessy (Cahier de la Chaire n°28)

Privately informed owners securitizing assets signal positive information by retaining sufficient interest. Signaling provides social bene…fits, allowing uninformed investors to insure without fearing adverse selection. Instead of signaling, owners of high value assets may prefer a pooling equilibrium in which they securitize more of the asset, relying on speculators to gather information and bring prices closer to fundamentals. This induces suboptimal risk sharing, since uninformed investors face adverse selection. We analyze privately optimal securitization and the choice between signaling and reliance on speculative markets. In the model, prices are set competitively, with an endogenously informed speculator trading against uninformed hedgers placing rational orders. If a structuring exists providing sufficient speculator gains, her effort is high, mispricing is low, and all/most of the asset is securitized in a pooling equilibrium. Here risky debt and levered equity are optimal, with optimal face value trading off higher unit profi…ts for the speculator against lower hedging demand. Hedgers imperfectly insure, buying only the concave claim, the only source of speculator pro…ts. If risk-aversion is low and/or endowment shocks are small, hedging demand is low, leading to low speculator effort. Here high types sell only safe debt in a separating equilibrium with perfect risk sharing. The owner’s incentive to choose the separating equilibrium is weak when risk-aversion is high and/or endowment shocks are large, precisely when efficient risk sharing has high social value.

Keywords: securitization, debt, equity, speculator, hedging demand, separating equilibrium, pooling equilibrium

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By Ivar Ekeland (Cahier de la Chaire n°27)

According to the Stern Report on climate change, the course of the next fifty years is set: present policies will impact only in the very long term, fifty to two hundred years from now. There is no market for interest rates, so far into the future, and economists must find other ways to set interest rates in a coherent way. This paper reviews some of the methods which have been used. We start with the classical Ramsey model of economic growth, which remains a central reference in the current debate, and we study the determinants of the interest rate in that framework. We then adapt the model (and the results) to take into account various concerns, namely (a) the existence of the environment as a distinct, non-producible good, (b) uncertainty on the parameters or on the model (c) intergenerational equity.

Keywords: Climate change, Long-term policy-making, Interest rates, Growth models.

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by Laure Durand-Viel & Bertrand Villeneuve (Cahier de la Chaire n°26)

This article analyzes the impact of incomplete contracts’ length on investment in a bilateral relationship. The seller has the power to set the contract terms whereas the buyer decides on the investment level, which acts as a cap on future demand. Two-part tariffs succeed at implementing the optimal investment and consumption even if commitment is limited, and the contract’s duration is irrelevant. Interestingly, this efficient solution is rendered possible by subsidies on consumption during the contract. In other terms, duration matters hugely for the contract details (the timing of transfers), not for its performance. Under certain circumstances that we discuss, linear pricing may have to be used, which leads to suboptimal investment. We show that longer contracts are less efficient, meaning that a degree of completeness (pricing width) may be strictly complementary to another one (contract length). The buyer’s surplus increases with respect to the contract duration, whereas the seller loses more in profit than the social surplus decreases. A longer contract actually protects expropriable investors rather than investment itself.

Keywords: Long-term Contracts, Incomplete Contracting, Infrastructure Investment.

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by Nazim Regnard and Jean-Michel Zakoïan (Cahier de la Chaire n°25)

A novel GARCH(1,1) model, with coefficients function of the realizations of an exogenous process, is considered for the volatility of daily gas prices. A distinctive feature of the model is that it produces non-stationary solutions. The probability properties, and the convergence and asymptotic normality of the Quasi-Maximum Likelihood Estimator (QMLE) have been derived by Regnard and Zakoian (2009). The prediction properties of the model are considered. We derive a strongly consistent estimator of the asymptotic variance of the QMLE. An application to daily gas spot prices from the Zeebruge market is presented. Apart from conditional heteroskedasticity, an empirical finding is the existence of distinct volatility regimes depending on the temperature level.

Keywords: GARCH, Nonstationary models, Periodic models, Quasi-maximum likelihood estimation, Time-varying coefficients.

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by René Aïd, Luciano Campi, Adrien Nguyen Huu and Nizar Touzi (Cahier de la Chaire n°24)

The objective of this paper is to present a model for electricity spot prices and the corresponding forward contracts, which relies on the underlying market of fuels, thus avoiding the electricity nonstorability restriction. The structural aspect of our model comes from the fact that the electricity spot prices depend on the dynamics of the electricity demand at the maturity T, and on the random available capacity of each production means. Our model explains, in a stylized fact, how the prices of dierent fuels together with the demand combine to produce electricity prices. This modeling methodology allows one to transfer to electricity prices the risk-neutral probabilities of the market of fuels and under the hypothesis of independence between demand and outages on one hand, and prices of fuels on the other hand, it provides a regression-type relation between electricity forward prices and forward prices of fuels. Moreover, the model produces, by nature, the well-known peaks observed on electricity market data. In our model, spikes occur when the producer has to switch from one technology to the lowest cost available one. Numerical tests performed on a very crude approximation of the French electricity market using only two fuels (gas and oil) provide an illustration of the potential interest of this model.

Keywords: energy markets, electricity prices, fuel prices, risk-neutral probability, no-arbitrage pricing, forward contract

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