Efficient Numerical Methods for Pricing American Options under Lévy Models

Two new numerical methods for the valuation of American and Bermudan options are proposed, which admit a large class of asset price models for the underlying. In particular, the methods can be applied with Lévy models that admit jumps in the asset price. These models provide a more realistic description of market prices and lead to better calibration results than the well-known Black-Scholes model. The proposed methods are not based on the indirect approach via partial differential equations, but directly compute option prices as risk-neutral expectation values. The expectation values are approximated by numerical quadrature methods. While this approach is initially limited to European options, the proposed combination with interpolation methods also allows for pricing of Bermudan and American options. Two different interpolation methods are used. These are cubic splines on the one hand and a mesh-free interpolation by radial basis functions on the other hand. The resulting valuation methods allow for an adaptive space discretization and error control. Their numerical properties are analyzed and, finally, the methods are validated and tested against various single-asset and multi-asset options under different market models

Deep combinatorial optimisation for optimal stopping time problems : application to swing options pricing

A new method for stochastic control based on neural networks and using randomisation of discrete random variables is proposed and applied to optimal stopping time problems. The method models directly the policy and does not need the derivation of a dynamic programming principle nor a backward stochastic differential equation. Unlike continuous optimization where automatic differentiation is used directly, we propose a likelihood ratio method for gradient computation. Numerical tests are done on the pricing of American and swing options. The proposed algorithm succeeds in pricing high dimensional American and swing options in a reasonable computation time, which is not possible with classical algorithms

Speed-up credit exposure calculations for pricing and risk management

We introduce a new method to calculate the credit exposure of European and path-dependent options. The proposed method is able to calculate accurate expected exposure and potential future exposure profiles under the risk-neutral and the real-world measure. Key advantage of is that it delivers an accuracy comparable to a full re-evaluation and at the same time it is faster than a regression-based method. Core of the approach is solving a dynamic programming problem by function approximation. This yields a closed form approximation along the paths together with the option's delta and gamma. The simple structure allows for highly efficient evaluation of the exposures, even for a large number of simulated paths. The approach is flexible in the model choice, payoff profiles and asset classes. We validate the accuracy of the method numerically for three different equity products and a Bermudan interest rate swaption. Benchmarking against the popular least-squares Monte Carlo approach shows that our method is able to deliver a higher accuracy in a faster runtime.Comment: arXiv admin note: substantial text overlap with arXiv:1905.0023

Markov-functional and stochastic volatility modelling

In this thesis, we study two practical problems in applied mathematical fi nance. The first topic discusses the issue of pricing and hedging Bermudan swaptions within a one factor Markov-functional model. We focus on the implications for hedging of the choice of instantaneous volatility for the one-dimensional driving Markov process of the model. We find that there is a strong evidence in favour of what we term \parametrization by time" as opposed to \parametrization by expiry". We further propose a new parametrization by time for the driving process which takes as inputs into the model the market correlations of relevant swap rates. We show that the new driving process enables a very effective vega-delta hedge with a much more stable gamma profile for the hedging portfolio compared with the existing ones. The second part of the thesis mainly addresses the topic of pricing European options within the popular stochastic volatility SABR model and its extension with mean reversion. We investigate some effcient approximations for these models to be used in real time. We first derive a probabilistic approximation for three different versions of the SABR model: Normal, Log-Normal and a displaced diffusion version for the general constant elasticity of variance case. Specifically, we focus on capturing the terminal distribution of the underlying process (conditional on the terminal volatility) to arrive at the implied volatilities of the corresponding European options for all strikes and maturities. Our resulting method allows us to work with a variety of parameters which cover long dated options and highly stress market condition. This is a different feature from other current approaches which rely on the assumption of very small total volatility and usually fail for longer than 10 years maturity or large volatility of volatility. A similar study is done for the extension of the SABR model with mean reversion (SABR-MR). We first compare the SABR model with this extended model in terms of forward volatility to point out the fundamental difference in the dynamics of the two models. This is done through a numerical example of pricing forward start options. We then derive an effcient probabilistic approximation for the SABRMR model to price European options in a similar fashion to the one for the SABR model. The numerical results are shown to be still satisfactory for a wide range of market conditions

Function approximation for option pricing and risk management Methods, theory and applications.

PhD Thesis.This thesis investigates the application of function approximation techniques for computationally demanding problems in nance. We focus on the use of Chebyshev interpolation and its multivariate extensions. The main contribution of this thesis is the development of a new pricing method for path-dependent options. In each step of the dynamic programming time-stepping we approximate the value function with Chebyshev polynomials. A key advantage of this approach is that it allows us to shift all modeldependent computations into a pre-computation step. For each time step the method delivers a closed form approximation of the price function along with the options' delta and gamma. We provide a theoretical error analysis and nd conditions that imply explicit error bounds. Numerical experiments con rm the fast convergence of prices and sensitivities. We use the new method to calculate credit exposures of European and path-dependent options for pricing and risk management. The simple structure of the Chebyshev interpolation allows for a highly e cient evaluation of the exposures. We validate the accuracy of the computed exposure pro les numerically for di erent equity products and a Bermudan swaption. Benchmarking against the least-squares Monte Carlo approach shows that our method delivers a higher accuracy in a faster runtime. We extend the method to e ciently price early-exercise options depending on several risk-factors. As an example, we consider the pricing of callable bonds in a hybrid twofactor model. We develop an e cient and stable calibration routine for the model based on our new pricing method. Moreover, we consider the pricing of early-exercise basket options in a multivariate Black-Scholes model. We propose a numerical smoothing in the dynamic programming time-stepping using the smoothing property of a Gaussian kernel. An extensive numerical convergence analysis con rms the e ciency

Fourier Neural Network Approximation of Transition Densities in Finance

This paper introduces FourNet, a novel single-layer feed-forward neural network (FFNN) method designed to approximate transition densities for which closed-form expressions of their Fourier transforms, i.e. characteristic functions, are available. A unique feature of FourNet lies in its use of a Gaussian activation function, enabling exact Fourier and inverse Fourier transformations and drawing analogies with the Gaussian mixture model. We mathematically establish FourNet's capacity to approximate transition densities in the $L_2$-sense arbitrarily well with finite number of neurons. The parameters of FourNet are learned by minimizing a loss function derived from the known characteristic function and the Fourier transform of the FFNN, complemented by a strategic sampling approach to enhance training. Through a rigorous and comprehensive error analysis, we derive informative bounds for the $L_2$ estimation error and the potential (pointwise) loss of nonnegativity in the estimated densities. FourNet's accuracy and versatility are demonstrated through a wide range of dynamics common in quantitative finance, including L\'{e}vy processes and the Heston stochastic volatility models-including those augmented with the self-exciting Queue-Hawkes jump process.Comment: 27 pages, 5 figure

Irregular grid methods for pricing high-dimensional American options

This thesis proposes and studies numerical methods for pricing high-dimensional American options; important examples being basket options, Bermudan swaptions and real options. Four new methods are presented and analysed, both in terms of their application to various test problems, and in terms of their theoretical stability and convergence properties. A method using matrix roots (Chapter 2) and a method using local consistency conditions (Chapter 4) are found to be stable and to give accurate solutions, in up to ten dimensions for the latter case. A method which uses local quadratic functions to approximate the value function (Chapter 3) is found to be vulnerable to instabilities in two dimensions, and thus not suitable for high-dimensional problems. A proof of convergence related to these methods is provided in Chapter 6. Finally, a method based on interpolation of the value function (Chapter 5) is found to be effective in pricing Bermudan swaptions.