We introduce sufficient conditions for the solution of a multi-dimensional, Markovian BSDE to have a density. We show that a system of BSDEs possesses a density if its corresponding semilinear PDE exhibits certain regularity properties, which we verify in the case of several examples.

# Past Mathematical and Computational Finance Seminar

Backward SDEs have proven to be a useful tool in mathematical finance. Their applications include the solution to various pricing and equilibrium problems in complete and incomplete markets, the estimation of value adjustments in the presence of funding costs, and the solution to many utility/risk optimisation type of problems.

In this work, we prove an explicit error expansion for the approximation of BSDEs. We focus our work on studying the cubature method of solution. To profit fully from these expansions in this case, e.g. to design high order approximation methods, we need in addition to control the complexity growth of the base algorithm. In our work, this is achieved by using a sparse grid representation. We present several numerical results that confirm the efficiency of our new method. Based on joint work with J.F. Chassagneux.

We show how to adapt methods originally developed in

model-independent finance / martingale optimal transport to give a

geometric description of optimal stopping times tau of Brownian Motion

subject to the constraint that the distribution of tau is a given

distribution. The methods work for a large class of cost processes.

(At a minimum we need the cost process to be adapted. Continuity

assumptions can be used to guarantee existence of solutions.) We find

that for many of the cost processes one can come up with, the solution

is given by the first hitting time of a barrier in a suitable phase

space. As a by-product we thus recover Anulova's classical solution of

the inverse first passage time problem.

We introduce a novel stochastic volatility model where the squared volatility of the asset return follows a Jacobi process. It contains the Heston model as a limit case. We show that the the joint distribution of any finite sequence of log returns admits a Gram--Charlier A expansion in closed-form. We use this to derive closed-form series representations for option prices whose payoff is a function of the underlying asset price trajectory at finitely many time points. This includes European call, put, and digital options, forward start options, and forward start options on the underlying return. We derive sharp analytical and numerical bounds on the series truncation errors. We illustrate the performance by numerical examples, which show that our approach offers a viable alternative to Fourier transform techniques. This is joint work with Damien Ackerer and Damir Filipovic.