## Section: New Results

### Financial Mathematics

Participants : Mireille Bossy, Mamadou Cissé, Nicolas Champagnat, Anthony Floryszczak, Junbo Huang, Thanh-Hoang Mai, Stephania Maroso, Denis Talay, Etienne Tanré, Pierre Vallois.

#### Modelling of financial techniques

In collaboration with R. Gibson (Zürich University), C. Blanchet and S. Rubenthaler (Université de Nice – Sophia-Antipolis), B. de Saporta (Université Bordeaux 4), D. Talay and E. Tanré elaborate an appropriate mathematical framework to develop the analysis of the financial performances of financial techniques which are often used by the traders. This research is funded by NCCR FINRISK (Switzerland) and is a part of its project “Conceptual Issues in Financial Risk Management”.

In the financial industry, there are three main approaches to investment: the fundamental approach, where strategies are based on fundamental economic principles, the technical analysis approach, where strategies are based on past prices behavior, and the mathematical approach where strategies are based on mathematical models and studies. The main advantage of technical analysis is that it avoids model specification, and thus calibration problems, misspecification risks, etc. On the other hand, technical analysis methods have limited theoretical justifications, and therefore none can assert that they are risk-less, or even efficient.

Consider an unstable financial economy. It is impossible to specify and calibrate models which can capture all the sources of instability during a long time interval. Thus it is natural to compare the performances obtained by using erroneously calibrated mathematical models and the performances obtained by technical analysis techniques. To our knowledge, this question has not been investigated in the literature.

We deal with the following model for a financial market, in which two assets are traded continuously. The first one (the bond) is an asset without systematic risk. The second one (the stock) is subject to systematic risk and its instantaneous rate of return changes at independent exponentially distributed random times. The trader does not observe the times of change.

This year we extended our earlier results [46] , [47] by taking transaction costs into account. At any time, the trader is allowed to invest all his wealth in the bond or in the stock. This leads to a non-classical stochastic control problem. We have proved that the value function of this control problem is continuous, satisfies a dynamic programming principle, and is the unique viscosity solution of a Hamilton–Jacobi–Bellman (HJB) equation. We developed an algorithm to compute the value function and a sub-optimal trading strategy, and compared its performance to the performance of the moving average strategy. When transactions costs are high, it is difficult for the technical analyst to outperform miscalibrated mathematical strategies. We are now working on the numerical analysis of the HJB equation. We aim to precise the rate of discretization methods.

#### Optimal dynamic cross pricing of CO2 market

Company strategists in the energy sector find themselves faced with new constraints, such as emission quotas, that can be exchanged through a financial market.

In collaboration with Nadia Maïzi (CMA ENSMP Sophia Antipolis) and Odile Pourtallier (CORPIN INRIA Sophia Antipolis – Méditerranée), M. Bossy and A. Floryszczak propose an approach in order to assess the impact of CO2 tax and quota levels on CO2 market prices. We model the expectation of the yield of an industrial with and without CO2 market, in order to derive the indifference price. This price is established through an optimal dynamic approach leading to the resolution of an Hamilton-Jacobi-Bellman equation. First results, based on simplified tax functions, have been presented in [Oops!] .

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

Under M. Bossy and D. Talay's supervision, M. Cissé studied the problem of artificial boundary conditions for nonlinear PDEs. The motivation of this research concerns American option pricing and the numerical resolution of the variational inequality characterizing prices of American options.

First, we have extended the theorem on existence and uniqueness of solutions of the reflected backward stochastic differential equations (RBSDE) with fixed terminal time to the case of bounded random terminal time. We obtained a probabilistic interpretation of the previous solution as a viscosity solution of variational inequalities with Dirichlet boundary conditions. We obtained a general expression of the localization error in terms of the boundary conditions.

Secondly, we were interested in variational inequalities with Neumann boundary conditions. They can be interpreted as a generalized RBSDEs coupled with a reflected forward stochastic differential equations (RSDE). We used the derivative in the sense of distributions with respect to the initial data of the RSDE and the representation of the gradient of Ma and Zhang [59] to establish a representation theorem for the space derivative of the viscosity solution of the variational inequalities. We apply this result to get a priori estimates on the error induced in numerical simulations by artificial boundary conditions for the PDEs related to American options pricing. The differentiability of the reflected forward SDE in the sense of distributions with respect to the initial data is deduced from results by Bouleau and Hirsch [49] on the absolute continuity of probability measures.

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

In his thesis under D. Talay's supervision, J. Huang continues the work on the numerical analysis of HJB equations.

As Karatzas have mentioned in [56] , some portfolio optimization problems corresponding to the given HJB equations can also be solved by the martingale method. Then we can expect the martingale method may give a good approximation of boundary conditions for the aforementioned HJB equations. The martingale method is, however, (roughly speaking) a special application of the approach of Lagrange multipliers, which is the basic tool in nonlinear constrained optimization. The Lagrange multiplier often acts as the root of some constraint equation. This implies that in order to approximate the boundary conditions of the HJB equation by using the martingale method one has to solve the constraint equation and find the corresponding root. To this end, we invoke a particular stochastic algorithm firstly proposed by Robbins and Monro and then developed by several other authors.

Generally speaking, for a given equation h( ) = 0 , the Robbins-Monro algorithm is like

where
_{n} is the step size,
y_{n} is stochastic and so is
_{n} . Under some assumptions, people have proved that
is asymptotically normal, where
^{*} is the sought-for root. We, however, are interested in the Berry-Esseen bound for the aforementioned term because this kind of bound usually provides a more accurate and neat estimate of the error term in the normal approximation. Thorough analyzes show that
is a sum of two terms: one is a sum of martingale increments and the other is a random variable converging to zero almost surely. Thus our work resides in solving two problems: one is to give the Berry-Esseen bound for a sum of martingale increments and the other is to specify the
convergence rate to zero of the second variable mentioned above.

#### Liquidity risk

P. Protter (Cornell University) and D. Talay are addressing the following question. We have the possibility of trading in a risky asset (which we refer to as a stock) with both liquidity and transaction costs. We further assume that the stock price follows a diffusion, and that the stock is highly liquid. We limit our trading strategies to those which change our holdings only by jumps (i.e., discrete trading strategies), and we begin with $0 and 0 shares, and we end with a liquidated portfolio (that is, we no longer hold any shares of the stock), on or before a predetermined ending time T . The question then is, given the structure of the liquidity and transaction costs, what is the optimal trading strategy which will maximize our gains? This amounts to maximize the value of our risk free savings account. This problem can be solved in this context if it is formulated as a non classical problem in stochastic optimal control. We prove existence and uniqueness results for the related Hamilton–Jacobi–Bellman equation.

#### Portfolio optimization in incomplete markets

N. Champagnat, T.-H. Mai, S. Maroso, D. Talay and E. Tanré are working on various aspects of portfolio management in incomplete markets within a contract with Natixis (see Section 7.6 ).

#### Optimal stopping problems

Keywords : Optimal stopping, h-transform.

In collaboration with P. Patie (University of Bern, Switzerland), M. Cissé and E. Tanré solve explicitly the optimal stopping problem with random discounting and an additive functional as cost of observations for a regular linear diffusion. This generalizes a result by Beibel and Lerche [40] , [41] . The approach relies on a combination of Doob's h -transform, time-changes and martingales techniques. Our results combined with Patie's results [62] allows us to treat a few examples, one of them being the evaluation of a perpetual American type option with payoff defined by

where the process X is the instantaneous interest rate modeled by the Vasicek or Ornstein-Uhlenbeck process.

#### Memory-based persistent counting random walk and applications

In [Oops!] , P. Vallois and C. Tapiero (ESSEC) have considered a memory-based persistent counting random walk, based on a Markov memory of the last event. The usefulness to some problems in insurance, finance and risk analysis are discussed.