Pricing European and American Options under Heston Model using Discontinuous Galerkin Finite Elements

This paper deals with pricing of European and American options, when the underlying asset price follows Heston model, via the interior penalty discontinuous Galerkin finite element method (dGFEM). The advantages of dGFEM space discretization with Rannacher smoothing as time integrator with nonsmooth initial and boundary conditions are illustrated for European vanilla options, digital call and American put options. The convection dominated Heston model for vanishing volatility is efficiently solved utilizing the adaptive dGFEM. For fast solution of the linear complementary problem of the American options, a projected successive over relaxation (PSOR) method is developed with the norm preconditioned dGFEM. We show the efficiency and accuracy of dGFEM for option pricing by conducting comparison analysis with other methods and numerical experiments

Radial basis functions with partition of unity method for American options with stochastic volatility

In this article, we price American options under Heston's stochastic volatility model using a radial basis function (RBF) with partition of unity method (PUM) applied to a linear complementary formulation of the free boundary partial differential equation problem. RBF-PUMs are local meshfree methods that are accurate and flexible with respect to the problem geometry and that produce algebraic problems with sparse matrices which have a moderate condition number. Next, a Crank-Nicolson time discretisation is combined with the operator splitting method to get a fully discrete problem. To better control the computational cost and the accuracy, adaptivity is used in the spatial discretisation. Numerical experiments illustrate the accuracy and efficiency of the proposed algorithm

Option pricing under stochastic volatility on a quantum computer

We develop quantum algorithms for pricing Asian and barrier options under the Heston model, a popular stochastic volatility model, and estimate their costs, in terms of T-count, T-depth and number of logical qubits, on instances under typical market conditions. These algorithms are based on combining well-established numerical methods for stochastic differential equations and quantum amplitude estimation technique. In particular, we empirically show that, despite its simplicity, weak Euler method achieves the same level of accuracy as the better-known strong Euler method in this task. Furthermore, by eliminating the expensive procedure of preparing Gaussian states, the quantum algorithm based on weak Euler scheme achieves drastically better efficiency than the one based on strong Euler scheme. Our resource analysis suggests that option pricing under stochastic volatility is a promising application of quantum computers, and that our algorithms render the hardware requirement for reaching practical quantum advantage in financial applications less stringent than prior art.Comment: 45 pages, 9 figures. The circuits in Appendices B and C have been improved, which makes the algorithm based on strong Euler scheme more efficient than befor

Radial Basis Functions with Partition of Unity Method for American Options with Stochastic Volatility

In this article, we price American options under Heston's stochastic volatility model using a radial basis function (RBF) with partition of unity method (PUM) applied to a linear complementary formulation of the free boundary partial differential equation problem. RBF-PUMs are local meshfree methods that are accurate and flexible with respect to the problem geometry and that produce algebraic problems with sparse matrices which have a moderate condition number. Next, a Crank-Nicolson time discretisation is combined with the operator splitting method to get a fully discrete problem. To better control the computational cost and the accuracy, adaptivity is used in the spatial discretisation. Numerical experiments illustrate the accuracy and efficiency of the proposed algorithm

Efficient valuation of exotic derivatives with path-dependence and early exercise features

The main objective of this thesis is to provide effective means for the valuation of popular financial derivative contracts with path-dependence and/or early-exercisable provisions. Starting from the risk-neutral valuation formula, the approach we propose is to sequentially compute convolutions of the value function of the contract at a monitoring date with the transition density between two dates, to provide the value function at the previous monitoring date, until the present date. A rigorous computational algorithm for the convolutions is then developed based on transformations to the Fourier domain. In the first part of the thesis, we deal with arithmetic Asian options, which, due to the growing popularity they enjoy in the financial marketplace, have been researched signicantly over the last two decades. Although few remarkable approaches have been proposed so far, these are restricted to the market assumptions imposed by the standard Black-Scholes-Merton paradigm. Others, although in theory applicable to Lévy models, are shown to suffer a non-monotone convergence when implemented numerically. To solve the Asian option pricing problem, we initially propose a flexible framework for independently distributed log-returns on the underlying asset. This allows us to generalize firstly in calculating the price sensitivities. Secondly, we consider an extension to non-Lévy stochastic volatility models. We highlight the benefits of the new scheme and, where relevant, benchmark its performance against an analytical approximation, control variate Monte Carlo strategies and existing forward convolution algorithms for the recovery of the density of the underlying average price. In the second part of the thesis, we carry out an analysis on the rapidly growing market of convertible bonds (CBs). Despite the vast amount of research which has been undertaken yet. This is due to the need for proper modelling of the CBs composite payout structure and the multi factor modelling arising in the CB valuation. Given the dimensional capacity of the convolution algorithm, we are now able to introduce a new jump diffusion structural approach in the CB literature, towards more realistic modelling of the default risk, and further include correlated stochastic interest rates. This aims at fixing dimensionality and convergence limitations which previously have been restricting the range of applicability of popular grid- based, lattice and Monte Carlo methods. The convolution scheme further permits flexible handling of real-world CB specications; this allows us to properly model the call policy and investigate its impact on the computed CB prices. We illustrate the performance of the numerical scheme and highlight the effects originated by the inclusion of jumps

Evolutionary Algorithms and Computational Methods for Derivatives Pricing

This work aims to provide novel computational solutions to the problem of derivative pricing. To achieve this, a novel hybrid evolutionary algorithm (EA) based on particle swarm optimisation (PSO) and differential evolution (DE) is introduced and applied, along with various other state-of-the-art variants of PSO and DE, to the problem of calibrating the Heston stochastic volatility model. It is found that state-of-the-art DEs provide excellent calibration performance, and that previous use of rudimentary DEs in the literature undervalued the use of these methods. The use of neural networks with EAs for approximating the solution to derivatives pricing models is next investigated. A set of neural networks are trained from Monte Carlo (MC) simulation data to approximate the closed form solution for European, Asian and American style options. The results are comparable to MC pricing, but with offline evaluation of the price using the neural networks being orders of magnitudes faster and computationally more efficient. Finally, the use of custom hardware for numerical pricing of derivatives is introduced. The solver presented here provides an energy efficient data-flow implementation for pricing derivatives, which has the potential to be incorporated into larger high-speed/low energy trading systems

THE IMPACT OF MERGERS AND ACQUISITIONS ON THE ECONOMIC PERFORMANCES

This paper aims to measure the extent to which banking systems from Central and Eastern Europe have been characterized by banking consolidation through mergers and acquisitions and the extent to which this process led to the concentration of the banking activity. Also, we measure the impact of mergers and acquisitions on the economic performance of the targeted banks. The conclusions show that the banking system from Central and Eastern Europe was subject to numerous acquisitions from Western banks that chose to enter aggressively into these markets with high growth potential. The effect of the acquisitions on the economic indicators is not obvious in the first years because of the strategy aiming larger market shares.mergers; acquisitions; consolidation; financial repression; economic performances.