Section: New Results

Financial Mathematics

Participants : Mireille Bossy, Paul Charton, Dalia Ibrahim, Denis Talay, Etienne Tanré.

• Mireille Bossy, in collaboration with H. Quinteros (Univ. Chile) worked on the rate of convergence of non Lipschitz diffusion processes discretized with the symetrized Milstein scheme. Under the same kind of hypotheses than in [41] on the symetrized Euler scheme, they obtained the expected improvement of the strong rate of convergence, when the diffusion coefficient is of the form $\sigma \left(x\right)=x^{\alpha }$, with $\alpha \in [1/2,1[$.

  A preprint is being written.

• P. Charton continued his PhD. under the supervision of M. Deaconu and A. Lejay. He studied some storage strategies for wind farms.

• Mathematical modelling for technical analysis techniques Since November 2009, D. Ibrahim has been working on her PhD. thesis on Mathematical modeling of technical analysis in finance, under supervision of D. Talay and E. Tanré. The aim of her work is to study the performances of a technical analysis tool designed to detect changes in the volatility term: The Bollinger Bands. She studied the performances of this indicator in a modified Black-Scholes model such that the volatility is equal to ${\sigma }_0$ up to a random time $\tau$, independent of the Brownian motion governing the prices. After $\tau$, the volatility is equal to ${\sigma }_1$. She proved that Bollinger Bandwidth indicator can detect the time change (at which the volatility changes its value), in the case of small and large volatilities. She has also exhibited a mathematical optimal allocation strategy, by decomposing the initial allocation problem into an allocation problem before the change time $\tau$ and an allocation problem after $\tau$, in order to circumvent some technical problems brought from the change of volatility.

  This work is part of the contract with FINRISK.

• In collaboration with C. Michel (CA-CIB) and V. Reutenauer (Citi), D. Talay and E. Tanré worked on the

  • the study of the liquidity risk in the interest rate options market;

  • the minimization of the hedging error in interest rates Gaussian models by means of strategies designed in an effective way by using stochastic optimization algorithms.

• P. Protter (Columbia University) and D. Talay continue to work on bubbles time evolution models, which leads them to try to extend Feller's results on explosion times for stochastic differetential equations.