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Abstract. The latter author, together with collaborators, proposed a numerical scheme to calculate the price of barrier options in [3,4,5]. The scheme is based on a symmetrization of diffusion process. The present paper aims to give a mathematical credit to the use of the numerical scheme for Heston or SABR type stochastic volatility models. This will be done by showing a fairly general result on the symmetrization (in multi-dimension/multi-reflections). Further applications (to time-inhomogeneous diffusions/ to time dependent boundaries/to curved boundaries) are also discussed.

In this paper a stochastic equation on compact groups in discrete negative time is studied. The diagonal group action on the extreme points of solutions is proved to be transitive by means of the coupling method. This result is applied to generalize Yor's work which is closely related to Tsirelson's stochastic differential equation and to give criteria for existence of a strong solution and for uniqueness in law.

We consider a heat kernel approach for the development of stochastic pricing kernels. The kernels are constructed by positive propagators, which are driven by time-inhomogeneous Markov processes. We multiply such a propagator with a positive, time-dependent and decreasing weight function, and integrate the product over time. The result is a so-called weighted heat kernel that by construction is a supermartingale with respect to the filtration generated by the timeinhomogeneous Markov processes. As an application, we show how this framework naturally fits the information-based asset pricing framework where time-inhomogeneous Markov processes are utilized to model partial information about random economic factors. We present examples of pricing kernel models which lead to analytical formulae for bond prices along with explicit expressions for the associated interest rate and market price of risk. Furthermore, we also address the pricing of fixed-income derivatives within this framework.

The aim of this paper is to provide a mathematical contribution on the semi-static hedge of timing risk associated to positions in American-style options under a multi-dimensional market model. Barrier options are considered in the paper and semi-static hedges are studied and discussed for a fairly large class of underlying price dynamics. Timing risk is identified with the uncertainty associated to the time at which the payoff payment of the barrier option is due. Starting from the work by [9], where the authors show that the timing risk can be hedged via static positions in plain vanilla options, the present paper extends the static hedge formula proposed in [9] by giving sufficient conditions to decompose a generalized timing risk into an integral of knock-in options in a multi-dimensional market model. A dedicated study of the semi-static hedge is then conducted by defining the corresponding strategy based on positions in barrier options. The proposed methodology allows to construct not only first order hedges but also higher order semi-static hedges, that can be interpreted as asymptotic expansions of the hedging error. The convergence of these higher order semi-static hedges to an exact hedge is shown. An illustration of the main theoretical results is provided for i) a symmetric case, ii) a one dimensional case, where the first order and second order hedging errors are derived in analytic closed form. The materiality of the hedging benefit gain of going from order one to order two by re-iterating the timing risk hedging strategy is discussed through numerical evidences by showing that order two can bring to more than 90% reduction of the hedging 'cost' w.r.t. order one (depending on the specific barrier option characteristics).