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Calibrating the time-dependent Implied Volatility Surface (IVS) using sparse market data is an essential challenge in computational finance, particularly for real-time applications. This task requires not only fitting market data but also satisfying a specified partial differential equation (PDE) and no-arbitrage conditions modelled by differential inequalities. This paper proposes a novel Physics-Informed Neural Networks (PINNs) approach called Whack-a-mole Online Learning (WamOL) to address this multi-objective optimisation problem. WamOL integrates self-adaptive and auto-balancing processes for each loss term, efficiently reweighting objective functions to ensure smooth surface fitting while adhering to PDE and no-arbitrage constraints and updating for intraday predictions. In our experiments, WamOL demonstrates superior performance in calibrating intraday IVS from uneven and sparse market data, effectively capturing the dynamic evolution of option prices and associated risk profiles. This approach offers an efficient solution for intraday IVS calibration, extending PINNs applications and providing a method for real-time financial modelling., Comment: 9 pages, 7 figures, 3 tables

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 objective 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 train the model using prices generated with the SABR model to produce smile and skewness parameters. We carry out 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: 25 pages, 13 figures

We present new numerical schemes for pricing perpetual Bermudan and American options as well as $\alpha$-quantile options. This includes a new direct calculation of the optimal exercise barrier for early-exercise options. Our approach is based on the Spitzer identities for general L\'evy processes and on the Wiener-Hopf method. Our direct calculation of the price of $\alpha$-quantile options combines for the first time the Dassios-Port-Wendel identity and the Spitzer identities for the extrema of processes. Our results show that the new pricing methods provide excellent error convergence with respect to computational time when implemented with a range of L\'evy processes., Comment: 29 pages, 14 figures, 5 tables

We present numerical methods based on the fast Fourier transform (FFT) to solve convolution integral equations on a semi-infinite interval (Wiener-Hopf equation) or on a finite interval (Fredholm equation). We extend and improve a FFT-based method for the Wiener-Hopf equation due to Henery, expressing it in terms of the Hilbert transform, and computing the latter in a more sophisticated way with sinc functions. We then generalise our method to the Fredholm equation reformulating it as two coupled Wiener-Hopf equations and solving them iteratively. We provide numerical tests and open-source code., Comment: 30 pages, 19 figures

In this note, we discuss the impact of the COVID-19 outbreak from the perspective of the market-structure. We observe that the US market-structure has dramatically changed during the past four weeks and that the level of change has followed the number of infected cases reported in the USA. Presently, market-structure resembles most closely the structure during the middle of the 2008 crisis but there are signs that it may be starting to evolve into a new structure altogether. This is the first article of a series where we will be analyzing and discussing market-structure as it evolves to a state of further instability or, more optimistically, stabilization and recovery., Comment: 1 page, figures

We present a numerical scheme to calculate fluctuation identities for exponential L\'evy processes in the continuous monitoring case. This includes the Spitzer identities for touching a single upper or lower barrier, and the more difficult case of the two-barriers exit problem. These identities are given in the Fourier-Laplace domain and require numerical inverse transforms. Thus we cover a gap in the literature that has mainly studied the discrete monitoring case; indeed, there are no existing numerical methods that deal with the continuous case. As a motivating application we price continuously monitored barrier options with the underlying asset modelled by an exponential L\'evy process. We perform a detailed error analysis of the method and develop error bounds to show how the performance is limited by the truncation error of the sinc-based fast Hilbert transform used for the Wiener-Hopf factorisation. By comparing the results for our new technique with those for the discretely monitored case (which is in the Fourier-$z$ domain) as the monitoring time step approaches zero, we show that the error convergence with continuous monitoring represents a limit for the discretely monitored scheme., Comment: 30 pages, 13 figures

We show how spectral filters can improve the convergence of numerical schemes which use discrete Hilbert transforms based on a sinc function expansion, and thus ultimately on the fast Fourier transform. This is relevant, for example, for the computation of fluctuation identities, which give the distribution of the maximum or the minimum of a random path, or the joint distribution at maturity with the extrema staying below or above barriers. We use as examples the methods by Feng and Linetsky (2008) and Fusai, Germano and Marazzina (2016) to price discretely monitored barrier options where the underlying asset price is modelled by an exponential L\'evy process. Both methods show exponential convergence with respect to the number of grid points in most cases, but are limited to polynomial convergence under certain conditions. We relate these rates of convergence to the Gibbs phenomenon for Fourier transforms and achieve improved results with spectral filtering., Comment: 27 pages, 12 figures, submitted