A parsimonious stochastic correlation framework to model the joint dynamics of assets
Chris Bardgett (University of Zurich, Switzerland)

Tuesday June 3, 14:00-14:30 | session 2.8 | Stochastic Analysis | room 1+2

We introduce a novel and flexible stochastic correlation framework for asset returns. The system of stochastic differential equations satisfied by the correlation processes is proven to admit a unique strong solution and the correlation matrix is shown to be positive semidefinite over time. We investigate the case of correlated returns each of them following a stochastic volatility model and argue that our setup presents two main advantages compared to existing ones. First, the stochastic dependence structure is specified independently of the asset's individual dynamics, which makes it possible to estimate separately each asset's dynamics and their dependence structure. Second, our framework is parsimonious in the number of stochastic factors which is proportional to the number $n$ of assets, as opposed to quadratic in $n$. Finally, in an numerical experiment we examine the impact of stochastic correlations on the steepness of the implied volatility smile of index options. To avoid the curse of dimensionality when pricing basket options, we propose two solutions. The first one is to use standard Monte Carlo techniques. The second one is to solve the high dimensional partial differential equation that option prices satisfy using the Quantized Tensor Train representation for large matrices entering in the Finite Difference discretization. This low parametric format for high dimensional tensors makes it possible for the storage cost and computational complexity to grow linearly with the number of assets.

Conic martingales
Frederic Vrins (ING Bank and Louvain School of Management (UCL), Belgium)

Tuesday June 3, 14:30-15:00 | session 2.8 | Stochastic Analysis | room 1+2

Mathematical finance extensively relies on martingales. In some cases, they need to meet constraints. For instance, they can be required to evolve in a compact set; discounted zero-coupon bond prices are risk-neutral martingales in $[0,1]$ if interest rates are positive. So are survival probabilities $\mathbb{E}_t[1_{\{\tau>T\}}]$. These martingales are usually tackled via latent processes, but some constraints are typically relaxed in order to fit the initial term-structure (eg positivity in Hull-White or CIR++ models for default intensities). In that context, tools for direct modelling of such bounded processes could be valuable. Surprisingly, however, such approaches haven't received much attention so far. Filling this gap is the goal of this paper.
We first review the conditions for a pure diffusion SDE to admit a (strong) solution, and for the latter to be a martingale. We then proceed with bounded (or conic) martingales. Martingales in $[0,1]$ can be obtained by solving pure diffusion SDEs whose diffusion coefficient has a specific form, eg $x(1-x)$ or the logistic curve. Unfortunately, explicit solutions to those are unavailable. On the other hand, martingales in $[a,b]$ can be obtained by mapping latent processes through smooth bijections $F$ with image $[a,b]$, provided that the associated drift is chosen according to the mapping's score. We study several mappings, and a unique solution is found in one case. Its variance, variance of increments and distribution are derived, a useful feature for tractability purposes. The solution's asymptotic distribution is proven to be typically singular (Bernoulli type). This is natural for floored/capped martingales. In particular, any quantile of the exponential martingale distribution tend to zero at some point. Interestingly, this proves that the maximum variance is attainable asymptotically. The existence of a unique solution is not always easy to establish for bounded martingale SDEs. We show how to choose the mapping for turning the SDE of a bounded martingale with separable diffusion coefficient of the form $h(t)g(x)$ to that of a drifted process with autonomous diffusion coefficient. We conclude with an example where the method is applied to survival probability modelling. In this setup, calibration to initial survival probability curve is trivial. Potential applications cover pricing of CDS options or CVA on CDS under weak collateral regimes, among others.

Simulated Likelihood for Discretely Observed Jump-Diffusions
Gustavo Schwenkler (Boston University School of Management, USA)
Joint work with Kay Giesecke

Tuesday June 3, 15:00-15:30 | session 2.8 | Stochastic Analysis | room 1+2

This paper develops, analyzes, and tests likelihood estimators for the parameters of a discretely observed one-dimensional jump-diffusion process whose drift, volatility, jump intensity, and jump magnitude are allowed to be arbitrary parametric functions of the state. The observation intervals need not be short. The estimators are based on a novel representation of the transition density of the process, which facilitates the construction of an unbiased Monte Carlo approximation. Under conditions, the estimators are consistent and asymptotically normal. The estimators do not suffer from the second-order bias that alternative discretization-based estimators exhibit. Numerical results illustrate our approach.

Discrete time approximation of fully nonlinear HJB equations via BSDEs with nonpositive jumps
Idris Kharroubi (University Paris Dauphine, France)
Joint work with Nicolas Langrené and Huyên Pham

Tuesday June 3, 15:30-16:00 | session 2.8 | Stochastic Analysis | room 1+2

We propose a new probabilistic numerical scheme for fully nonlinear equation of Hamilton-Jacobi-Bellman (HJB) type associated to stochastic control problem, which is based on the Feynman-Kac representation by means of control randomization and backward stochastic differential equation with nonpositive jumps. We study a discrete time approximation for the minimal solution to this class of BSDE when the time step goes to zero, which provides both an approximation for the value function and for an optimal control in feedback form. We obtained a convergence rate without any ellipticity condition on the controlled diffusion coefficient.