Abstract: Pricing and hedging exotic options using local stochastic volatility models drew a serious attention within the last decade, and nowadays became almost a standard approach to this problem. In this paper we show how this framework could be extended by adding to the model stochastic interest rates and correlated jumps in all three components. We also propose a new fully implicit modification of the popular Hundsdorfer and Verwer and Modified Craig-Sneyd finite-difference schemes which provides second order approximation in space and time, is unconditionally stable and preserves positivity of the solution, while still has a linear complexity in the number of grid nodes.

Abstract: We will present a method to numerically price options by solving the Fokker-Planck equation for the conditional probability density p(s,t|s_0,t_0). This enables the pricing of several contracts with pay-offs ϕ(s,K,T) (with strike-price K and time of maturity T) by integrating p(s,T|s_0,t_0) multiplied by ϕ(s,K,T) and discount to today's price. From a numerical perspective the initial condition for the Fokker-Planck equation is particularly challenging since it is a Dirac delta function. In [1] a closed-form expansion for the conditional probability density was introduced that is valid for small time-steps. We use this for the computation of p(s,t_0+∆t|s_0,t_0) the first time-step. For the remaining time-steps we discretize the Fokker-Planck equation using BDF-2 in time and Radial Basis Function (RBF) approximation in space with Gaussian RBFs. Finally, the computation of the option prices from the obtained p(s,T|s_0,t_0) can be done analytically for many pay-off functions ϕ(s,K,T), due to the Gaussian RBFs. We will demonstrate the good qualities of our proposed method for European call options and barrier options. [1] Y. Aït-Sahalia, Maximum-likelihood estimation of discretely-sampled diffusions: A closed-form approximation approach, Econometrica, 70: 223–262, 2002.

Abstract: Tail dependence refers to clustering of extreme events. In the context of financial risk management, the clustering of high-severity risks has a devastating effect on the well-being of firms and is thus of pivotal importance in risk analysis. When it comes to quantifying the extent of tail dependence, it is generally agreed that measures of tail dependence must be independent of the marginal distributions of the risks but rather solely copula-dependent. Indeed, all classical measures of tail dependence are such, but they investigate the amount of tail dependence along the main diagonal of copulas, which has often little in common with the concentration of extremes in the copulas' domain of definition. In this paper we urge that the classical measures of tail dependence may underestimate the level of tail dependence in copulas. For the Gaussian copula, however, we prove that the classical measures are maximal. As, in spite of the numerous criticisms, the Gaussian copula remains ubiquitous in a great variety of practical applications, our ndings must be a welcome news for risk professionals.

Abstract: A radial basis function generated finite difference (RBF-FD) method has been considered for solving multidimensional PDEs arising in pricing of financial contracts, mainly basket options. Being mesh-free while yielding a sparse differentiation matrix, this method aims to exploit the best properties from, both, finite difference (FD) methods and radial basis function (RBF) methods. Moreover, the RBF-FD method is expected to be advantageous for high-dimensional problems compared to: Monte Carlo (MC) methods which converge slowly, global RBF methods since they produce dense matrices, and FD methods because they require regular grids. The method was succesfully tested in solving the standard Black-Scholes-Merton equation for pricing European and American options with discrete or continuous dividends in 1D. Then, it is developed further in order to price European call basket and spread options in 2D on adapted domains, and some groundwork has been done in solving 3D problems as well. The method features a non-uniform node placement in space, as well as a variable spatial stencil size, in order to improve the accuracy in the regions with known low regularity. Performance of the method and the error profiles have been studied with respect to discretization in space, size and form of stencils, and RBF shape parameter. The results highlight RBF-FD as a competitive, sparse method, capable of achieving high accuracy with a small number of nodes in space.

Abstract: Credit risk models largely bifurcate into two classes – the structural models and the reduced form models. Attempts have been made to reconcile the two approaches via restricting information by adjusting filtrations, but they are technically complicated. Here we propose a reconciliation inspired by actuarial science’s approach to survival analysis. Extending the work of Chen, we model the hazard rate curve itself as a stochastic process. This puts default models in a form resembling the HJM model for interest rates, yielding a unifying framework for default modeling. All credit models can be put in this form, and default dependent derivatives can be directly priced in this framework. Predictability of default has a simple interpretation in this framework. The framework enables us to disentangle predictability and the distribution of the default time from calibration decisions such as whether to use market prices or balance sheet information. It also allows us a simple way to define new default models.