## Section: New Results

### Financial Mathematics

#### Indifference pricing for carbon emission allowances

In collaboration with N. Maïzi (CMA - Mines Paristech) and O. Pourtallier (INRIA -COPRIN Team), M. Bossy, O. Davidau and I. Hammad studied the indifference pricing for carbon emission allowances, as a short term model value of carbon (see also Section 7.2 ). The indifference pricing methodology describes the way an industrial agent on the emission allowances market chooses his production strategy. An utility function represents the preferences of the producer and its risk aversion. The outputs of its production have stochastic prices on the market, so that the optimal production strategy arises as the solution of a stochastic control problem.

In that case, the resulting stochastic control problem is degenerate, but its well-posedness could be proved: the value function is the unique solution of a Hamilton-Jacobi-Bellman equation. This value function may then be computed through finite difference methods. The indifference price of the producer for an amount of allowances could be then computed in a simplified model.

The participants are currently working on the sensitivity analysis of the indifference price function with respect to some allowance market parameters. Forward Backward Stochastic Differential Equations formulations are considered as representation tools for the viscosity solution of the HJB equation. The objective is to establish a representation formula for the value function which allows us to obtain tractable dependencies on the derivatives with respect to the interesting parameters.

#### Mean field games

M. Bossy, O. Davidau and I. Hammad have formulated a mean field type model associated with an optimization problem of power consumption. They model an agent who wants to use electricity in a world price of electricity which depends on the probability distribution of individual consumptions, modelled by a controlled diffusion.

This model is the limit model of a mean field game in the sense of Lasry and Lions  ,  . The agents interact only through the functional costs. This simple model served as benchmark tests for the development of numerical methods for the mean field game equations.

J. Huang and D. Talay have studied a mean-field game which models the time evolution of an asset price whose mean instantaneous return is controlled. Two different numerical methods have been developed and tested successfully.

#### Artificial boundary conditions for nonlinear PDEs in finance

M. Bossy and D. Talay continued their collaboration with M. Cissé (ENSAE-Sénégal) after his Ph.D. thesis under their supervision. They studied the problem of artificial boundary conditions for nonlinear PDEs such as variational inequalities characterizing prices of American options.

They established a representation formula for the space derivative of the viscosity solution of the variational inequalities with Neumann boundary condition. The formula relies on generalized RBSDEs coupled with a reflected forward stochastic differential equations.

Motivated by applications in Finance, where the space derivative of the solution of such PDEs allows one to construct hedging strategies of American options, they estimated the localization error on this derivative inside the domain in terms of the misspecification of the Neumann boundary condition.

#### Mathematical modelling for technical analysis techniques

After her summer internship in the team, D. Ibrahim started her Ph.D. under supervision of E. Tanré and D. Talay on mathematical modelling for technical analysis techniques in finance. She is studying rigorously the “Bollinger Bandes” indicator for an investor using a risky asset whose instantaneous rate of volatility changes at an unknown random time.

#### Super-replication of barrier options

This work is part of the contract with NATIXIS (see Section  7.10 ).

N. Champagnat, S. Maroso and E. Tanré have studied the problem of super-replication of up-and-out barrier options under Gamma constraints. Extending the results of Cheridito, Soner and Touzi  , they identified the Hamilton-Jacobi-Bellman equation solved by the minimal super-replication cost.

#### Impulse control with delay

This work is part of the contract with NATIXIS (see Section  7.10 ).

N. Champagnat, S. Maroso, D. Talay, and E. Tanré have continued their theoretical and numerical study of the problem of hedging of barrier options with strategies having a minimal delay between successive portfolio reallocations. They have obtained the Hamilton-Jacobi-Bellman equation solved by the minimal quadratic risk and described several discretization algorithms (explicit and implicit-explicit). They studied numerically their performances by comparing the value function and the optimal strategy with those obtained under two sets of strategies: first, the set of strategies with fixed, discrete portfolio reallocation times (which is the strategy commonly used by practitioners), and second, the set of strategies without constraints (which corresponds to the Black and Scholes hedging of the option).

#### Modelling of financial techniques: resistance and support levels

One of the concepts that appears frequently in technical analysis is resistance. We will refer to resistance level as a given price level when market pressures seem to force a repeated maximum. Although the study of some historical data shows the eventual appearance of resistance levels, most of the mathematical models used for stock prices, including the well known Black and Scholes model, do not exhibit this kind of phenomenon.

B. Bérard-Bergery (Univ. of Nancy), C. Garcia, B. Guan (Interns at INRIA, Erasmus Mundus program), C. Profeta (Univ. of Nancy) and E. Tanré have developed a mathematical model derived from the Black and Scholes model to include resistance phenomena and a trading strategy for optimizing the logarithm of the returns.

They also developed numerical tests based on Monte Carlo estimations of the wealths obtained with different strategies: the classical Black and Scholes strategy, a simple technical analysis strategy and the optimal strategy for the resistance model. These tests show that the optimal strategy with resistance is robust according to variations of the parameters of the model and produces greater wealth than the other strategies.

#### Rate of convergence in the Robbins-Monro algorithm

Under the supervision of D. Talay, J. Huang continued the study on the convergence of the Robbins-Monro (RM) algorithm. He defended his thesis  in July.

By using the smoothing inequality for characteristic functions, they provided a Berry-Esseen type rate of convergence in the central limit theorem for martingales. This result is applied to precise the convergence rate of the RM algorithm, thus establishing a Berry-Esseen bound for the RM algorithm n = n-1 + nh( n-1) + n n,

where h is assumed to be a smooth function with bounded derivatives and n is a martingale difference.

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