No-Arbitrage Deep Calibration for Volatility Smile and Skewness

Volatility smile and skewness are two key properties of option prices that are represented by the implied volatility (IV) surface. However, IV surface calibration through nonlinear interpolation is a complex problem due to several factors, including limited input data, low liquidity, and noise. Additionally, the calibrated surface must obey the fundamental financial principle of the absence of arbitrage, which can be modeled by various differential inequalities over the partial derivatives of the option price with respect to the expiration time and the strike price. To address these challenges, we have introduced a Derivative-Constrained Neural Network (DCNN), which is an enhancement of a multilayer perceptron (MLP) that incorporates derivatives in the output function. DCNN allows us to generate a smooth surface and incorporate the no-arbitrage condition thanks to the derivative terms in the loss function. In numerical experiments, we apply the stochastic volatility model with smile and skewness parameters and simulate it with different settings to examine the stability of the calibrated model under different conditions. The results show that DCNNs improve the interpolation of the implied volatility surface with smile and skewness by integrating the computation of the derivatives, which are necessary and sufficient no-arbitrage conditions. The developed algorithm also offers practitioners an effective tool for understanding expected market dynamics and managing risk associated with volatility smile and skewness.Comment: 9 pages, 7 figure

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

The mathematical modelling and numerical solution of options pricing problems

Accurate and efficient numerical solutions have been described for a selection of financial options pricing problems. The methods are based on finite difference discretisation coupled with optimal solvers of the resulting discrete systems. Regular Cartesian meshes have been combined with orthogonal co-ordinate transformations chosen for numerical accuracy rather than reduction of the differential operator to constant coefficient form. They allow detailed resolution in the regions of interest where accuracy is most desired, and grid coarsening where there is least interest. These transformations are shown to be effective in producing accurate solutions on modest computational grids. The spatial discretisation strategy is chosen to meet accuracy requirements as sell as to produce coefficient matrices with favourable sparsity and stability properties. In the case of single factor European options, a modified Crank-Nicolson, second order accurate finite difference scheme is presented, which uses adaptive upwind differences when the mesh Peclet conditions are violated. The resulting tridiagonal system of equations is solved using a direct solver. A careful study of grid refinement displays convergence towards the true solution and demonstrates a high level of accuracy can be obtained with this approach. Laplace inversion methods are also implemented as an alternative solution approach for the one-factor European option. Results are compared to those produced by the direct solver algorithm and are shown to be favourable. It is shown how Semi-Lagrange time-integration can solve the path-dependent Asian pricing problem, by integrating out the average price term and simplifying the finite difference equations into a parameterised Black-Scholes form. The implicit equations that result are unconditionally stable, second order accurate and can be solved using standard tridiagonal solvers. The Semi-Lagrange method is shown to be easily used in conjunction with co-ordinate transformations applied in both spatial directions. A variable time-stepping scheme is implemented in the algorithm. Early exercise is also easily incorporated, the resulting linear complementarity problem can be solved using a projection or penalty method (the penalty method is shown to be slightly more efficient). Second order accuracy has been confirmed for Asian options that must be held to maturity. A comparison with published results for continuous-average-rate put and call options, with and without early exercise, shows that the method achieves basis point accuracy and that Richardson extrapolation can also be applied

The ANOVA decomposition and generalized sparse grid methods for the high-dimensional backward Kolmogorov equation

In this thesis, we discuss numerical methods for the solution of the high-dimensional backward Kolmogorov equation, which arises in the pricing of options on multi-dimensional jump-diffusion processes. First, we apply the ANOVA decomposition and approximate the high-dimensional problem by a sum of lower-dimensional ones, which we then discretize by a θ-scheme and generalized sparse grids in time and space, respectively. We solve the resultant systems of linear equations by iterative methods, which requires both preconditioning and fast matrix-vector multiplication algorithms. We make use of a Linear Program and an algebraic formula to compute optimal diagonal scaling parameters. Furthermore, we employ the OptiCom as non-linear preconditioner. We generalize the unidirectional principle to non-local operators and develop a new matrix-vector multiplication algorithm for the OptiCom. As application we focus on the Kou model. Using a new recurrence formula, the computational complexity of the operator application remains linear in the number of degrees of freedom. The combination of the above-mentioned methods allows us to efficiently approximate the solution of the backward Kolmogorov equation for a ten-dimensional Kou model.Die ANOVA-Zerlegung und verallgemeinerte dünne Gitter für die hochdimensionale Kolmogorov-Rückwärtsgleichung In der vorliegenden Arbeit betrachten wir numerische Verfahren zur Lösung der hochdimensionalen Kolmogorov-Rückwärtsgleichung, die beispielsweise bei der Bewertung von Optionen auf mehrdimensionalen Sprung-Diffusionsprozessen auftritt. Zuerst wenden wir eine ANOVA-Zerlegung an und approximieren das hochdimensionale Problem mit einer Summe von niederdimensionalen Problemen, die wir mit einem θ-Verfahren in der Zeit und mit verallgemeinerten dünnen Gittern im Ort diskretisieren. Wir lösen die entstehenden linearen Gleichungssysteme mit iterativen Verfahren, wofür eine Vorkonditionierung als auch schnelle Matrix-Vektor-Multiplikationsalgorithmen nötig sind. Wir entwickeln ein Lineares Programm und eine algebraische Formel, um optimale Diagonalskalierungen zu finden. Des Weiteren setzen wir die OptiCom als nicht-lineares Vorkonditionierungsverfahren ein. Wir verallgemeinern das unidirektionale Prinzip auf nicht-lokale Operatoren und entwickeln einen für die OptiCom optimierten Matrix-Vektor-Multiplikationsalgorithmus. Als Anwendungsbeispiel betrachten wir das Kou-Modell. Mit einer neuen Rekurrenzformel bleibt die Gesamtkomplexität der Operatoranwendung linear in der Anzahl der Freiheitsgrade. Unter Einbeziehung aller genannten Methoden ist es nun möglich, die Lösung der Kolmogorov-Rückwärtsgleichung für ein zehndimensionales Kou-Modell effizient zu approximieren

High dimensional American options

Pricing single asset American options is a hard problem in mathematical finance. There are no closed form solutions available (apart from in the case of the perpetual option), so many approximations and numerical techniques have been developed. Pricing multi–asset (high dimensional) American options is still more difficult. We extend the method proposed theoretically by Glasserman and Yu (2004) by employing regression basis functions that are martingales under geometric Brownian motion. This results in more accurate Monte Carlo simulations, and computationally cheap lower and upper bounds to the American option price. We have implemented these models in QuantLib, the open–source derivatives pricing library. The code for many of the models discussed in this thesis can be downloaded from quantlib.org as part of a practical pricing and risk management library. We propose a new type of multi–asset option, the “Radial Barrier Option” for which we find analytic solutions. This is a barrier style option that pays out when a barrier, which is a function of the assets and their correlations, is hit. This is a useful benchmark test case for Monte Carlo simulations and may be of use in approximating multi–asset American options. We use Laplace transforms in this analysis which can be applied to give analytic results for the hitting times of Bessel processes. We investigate the asymptotic solution of the single asset Black–Scholes–Merton equation in the case of low volatility. This analysis explains the success of some American option approximations, and has the potential to be extended to basket options

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal