Applying default probabilities in an exponential barrier structural model

This paper shows that the use of a time-dependant barrier in a structural model improve its flexibility because it allows to incorporate, as input, the probability of default. The main result achieved is the assessment that the default barrier is, indeed,characterized by a non flat structure. JEL Classification: G13, G33default structural models, barrier options with exponential boundaries, implied default probability

Strong order 1/2 convergence of full truncation Euler approximations to the Cox-Ingersoll-Ross process

We study convergence properties of the full truncation Euler scheme for the Cox-Ingersoll-Ross process in the regime where the boundary point zero is inaccessible. Under some conditions on the model parameters (precisely, when the Feller ratio is greater than three), we establish the strong order 1/2 convergence in $L^{p}$ of the scheme to the exact solution. This is consistent with the optimal rate of strong convergence for Euler approximations of stochastic differential equations with globally Lipschitz coefficients, despite the fact that the diffusion coefficient in the Cox-Ingersoll-Ross model is not Lipschitz.Comment: 16 pages, 1 figur

High Performance and Low Power Monte Carlo Methods to Option Pricing Models via High Level Design and Synthesis

This article compares the performance and energy consumption of GPUs and FPGAs via implementing financial market models. The case studies used in this comparison are the Black-Scholes model and the Heston model for option pricing problems, which are analyzed numerically by Monte Carlo method. The algorithms are computationally intensive but not memory-intensive and thus well suited for FPGA implementation. High-level synthesis was performed starting from parallel models written in OpenCL and then various micro-architectures were explored and optimized on FPGAs. The final implementations of both models to several options on FPGAs achieved the best parallel acceleration systems, in terms of both performance-per-operation and energy-per-operation, compared not only to the kernels on advanced GPUs but also to the RTL implementations found in the literatures

Uniform approximation of the Cox-Ingersoll-Ross process

The Doss-Sussmann (DS) approach is used for uniform simulation of the Cox-Ingersoll-Ross (CIR) process. The DS formalism allows to express trajectories of the CIR process through solutions of some ordinary differential equation (ODE) depending on realizations of a Wiener process involved. By simulating the first-passage times of the increments of the Wiener process to the boundary of an interval and solving the ODE, we uniformly approximate the trajectories of the CIR process. In this respect special attention is payed to simulation of trajectories near zero. From a conceptual point of view the proposed method gives a better quality of approximation (from a path-wise point of view) than standard, or even exact simulation of the SDE at some discrete time grid.Comment: 24 page

Discretizing the Heston Model: An Analysis of the Weak Convergence Rate

In this manuscript we analyze the weak convergence rate of a discretization scheme for the Heston model. Under mild assumptions on the smoothness of the payoff and on the Feller index of the volatility process, respectively, we establish a weak convergence rate of order one. Moreover, under almost minimal assumptions we obtain weak convergence without a rate. These results are accompanied by several numerical examples. Our error analysis relies on a classical technique from Talay & Tubaro, a recent regularity estimate for the Heston PDE by Feehan & Pop and Malliavin calculus

Random Time Forward Starting Options

We introduce a natural generalization of the forward-starting options, first discussed by M. Rubinstein. The main feature of the contract presented here is that the strike-determination time is not fixed ex-ante, but allowed to be random, usually related to the occurrence of some event, either of financial nature or not. We will call these options {\bf Random Time Forward Starting (RTFS)}. We show that, under an appropriate "martingale preserving" hypothesis, we can exhibit arbitrage free prices, which can be explicitly computed in many classical market models, at least under independence between the random time and the assets' prices. Practical implementations of the pricing methodologies are also provided. Finally a credit value adjustment formula for these OTC options is computed for the unilateral counterparty credit risk.Comment: 19 pages, 1 figur