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One of the most discussed problems in the financial world is stock option pricing. The Black-Scholes Equation is a Parabolic Partial Differential Equation which provides an option pricing model. The present work proposes an approach based on Neural Networks to solve the Black-Scholes Equations. Real-world data from the stock options market were used as the initial boundary to solve the Black-Scholes Equation. In particular, times series of call options prices of Brazilian companies Petrobras and Vale were employed. The results indicate that the network can learn to solve the Black-Sholes Equation for a specific real-world stock options time series. The experimental results showed that the Neural network option pricing based on the Black-Sholes Equation solution can reach an option pricing forecasting more accurate than the traditional Black-Sholes analytical solutions. The experimental results making it possible to use this methodology to make short-term call option price forecasts in options markets., Comment: 15 pages and 8 figures

We consider the Fourier-Laplace transforms of a broad class of polynomial Ornstein-Uhlenbeck (OU) volatility models, including the well-known Stein-Stein, Sch\"obel-Zhu, one-factor Bergomi, and the recently introduced Quintic OU models motivated by the SPX-VIX joint calibration problem. We show the connection between the joint Fourier-Laplace functional of the log-price and the integrated variance, and the solution of an infinite dimensional Riccati equation. Next, under some non-vanishing conditions of the Fourier-Laplace transforms, we establish an existence result for such Riccati equation and we provide a discretized approximation of the joint characteristic functional that is exponentially entire. On the practical side, we develop a numerical scheme to solve the stiff infinite dimensional Riccati equations and demonstrate the efficiency and accuracy of the scheme for pricing SPX options and volatility swaps using Fourier and Laplace inversions, with specific examples of the Quintic OU and the one-factor Bergomi models and their calibration to real market data., Comment: 41 pages, 8 figures

This paper proposes a probabilistic machine learning method to price catastrophe (CAT) bonds in the primary market. The proposed method combines machine-learning-based predictive models with Conformal Prediction, an innovative algorithm that generates distribution-free probabilistic forecasts for CAT bond prices. Using primary market CAT bond transaction records between January 1999 and March 2021, the proposed method is found to be more robust and yields more accurate predictions of the bond spreads than traditional regression-based methods. Furthermore, the proposed method generates more informative prediction intervals than linear regression and identifies important nonlinear relationships between various risk factors and bond spreads, suggesting that linear regressions could misestimate the bond spreads. Overall, this paper demonstrates the potential of machine learning methods in improving the pricing of CAT bonds.

This paper introduces StockGPT, an autoregressive ``number'' model trained and tested on 70 million daily U.S. stock returns over nearly 100 years. Treating each return series as a sequence of tokens, StockGPT automatically learns the hidden patterns predictive of future returns via its attention mechanism. On a held-out test sample from 2001 to 2023, a daily rebalanced long-short portfolio formed from StockGPT predictions earns an annual return of 119% with a Sharpe ratio of 6.5. The StockGPT-based portfolio completely spans momentum and long-/short-term reversals, eliminating the need for manually crafted price-based strategies, and also encompasses most leading stock market factors. This highlights the immense promise of generative AI in surpassing human in making complex financial investment decisions., Comment: 19 pages, 3 figures, 7 tables

I explore the relationship between investor emotions expressed on social media and asset prices. The field has seen a proliferation of models aimed at extracting firm-level sentiment from social media data, though the behavior of these models often remains uncertain. Against this backdrop, my study employs EmTract, an open-source emotion model, to test whether the emotional responses identified on social media platforms align with expectations derived from controlled laboratory settings. This step is crucial in validating the reliability of digital platforms in reflecting genuine investor sentiment. My findings reveal that firm-specific investor emotions behave similarly to lab experiments and can forecast daily asset price movements. These impacts are larger when liquidity is lower or short interest is higher. My findings on the persistent influence of sadness on subsequent returns, along with the insignificance of the one-dimensional valence metric, underscores the importance of dissecting emotional states. This approach allows for a deeper and more accurate understanding of the intricate ways in which investor sentiments drive market movements., Comment: arXiv admin note: text overlap with arXiv:2112.03868

We propose a unifying framework for the pricing of debt securities under general time-inhomogeneous short-rate diffusion processes. The pricing of bonds, bond options, callable/putable bonds, and convertible bonds (CBs) are covered. Using continuous-time Markov chain (CTMC) approximation, we obtain closed-form matrix expressions to approximate the price of bonds and bond options under general one-dimensional short-rate processes. A simple and efficient algorithm is also developed to price callable/putable debts. The availability of a closed-form expression for the price of zero-coupon bonds allows for the perfect fit of the approximated model to the current market term structure of interest rates, regardless of the complexity of the underlying diffusion process selected. We further consider the pricing of CBs under general bi-dimensional time-inhomogeneous diffusion processes to model equity and short-rate dynamics. Credit risk is also incorporated into the model using the approach of Tsiveriotis and Fernandes (1998). Based on a two-layer CTMC method, an efficient algorithm is developed to approximate the price of convertible bonds. When conversion is only allowed at maturity, a closed-form matrix expression is obtained. Numerical experiments show the accuracy and efficiency of the method across a wide range of model parameters and short-rate models.

We present closed analytical approximations for the pricing of basket options, also applicable to Asian options with discrete averaging under the Black-Scholes model with time-dependent parameters. The formulae are obtained by using a stochastic Taylor expansion around a log-normal proxy model and are found to be highly accurate for Asian options in practice as well as for vanilla options with discrete dividends.

This paper is a supplement to our recent paper ``Alternative models for FX, arbitrage opportunities and efficient pricing of double barrier options in L\'evy models". We introduce the class of regime-switching L\'evy models with memory, which take into account the evolution of the stochastic parameters in the past. This generalization of the class of L\'evy models modulated by Markov chains is similar in spirit to rough volatility models. It is flexible and suitable for application of the machine-learning tools. We formulate the modification of the numerical method in ``Alternative models for FX, arbitrage opportunities and efficient pricing of double barrier options in L\'evy models", which has the same number of the main time-consuming blocks as the method for Markovian regime-switching models.

This paper presents a Monte-Carlo-based artificial neural network framework for pricing Bermudan options, offering several notable advantages. These advantages encompass the efficient static hedging of the target Bermudan option and the effective generation of exposure profiles for risk management. We also introduce a novel optimisation algorithm designed to expedite the convergence of the neural network framework proposed by Lokeshwar et al. (2022) supported by a comprehensive error convergence analysis. We conduct an extensive comparative analysis of the Present Value (PV) distribution under Markovian and no-arbitrage assumptions. We compare the proposed neural network model in conjunction with the approach initially introduced by Longstaff and Schwartz (2001) and benchmark it against the COS model, the pricing model pioneered by Fang and Oosterlee (2009), across all Bermudan exercise time points. Additionally, we evaluate exposure profiles, including Expected Exposure and Potential Future Exposure, generated by our proposed model and the Longstaff-Schwartz model, comparing them against the COS model. We also derive exposure profiles at finer non-standard grid points or risk horizons using the proposed approach, juxtaposed with the Longstaff Schwartz method with linear interpolation and benchmark against the COS method. In addition, we explore the effectiveness of various interpolation schemes within the context of the Longstaff-Schwartz method for generating exposures at finer grid horizons., Comment: 34 pages

This study proposes a new exact simulation scheme of the Ornstein-Uhlenbeck driven stochastic volatility model. With the Karhunen-Lo\`eve expansions, the stochastic volatility path following the Ornstein-Uhlenbeck process is expressed as a sine series, and the time integrals of volatility and variance are analytically derived as the sums of independent normal random variates. The new method is several hundred times faster than Li and Wu [Eur. J. Oper. Res., 2019, 275(2), 768-779] that relies on computationally expensive numerical transform inversion. The simulation algorithm is further improved with the conditional Monte-Carlo method and the martingale-preserving control variate on the spot price.

L\'evy-driven Ornstein-Uhlenbeck (OU) processes represent an intriguing class of stochastic processes that have garnered interest in the energy sector for their ability to capture typical features of market dynamics. However, in the current state-of-the-art, Monte Carlo simulations of these processes are not straightforward for two main reasons: i) algorithms are available only for some particular processes within this class; ii) they are often computationally expensive. In this paper, we introduce a new simulation technique designed to address both challenges. It relies on the numerical inversion of the characteristic function, offering a general methodology applicable to all L\'evy-driven OU processes. Moreover, leveraging FFT, the proposed methodology ensures fast and accurate simulations, providing a solid basis for the widespread adoption of these processes in the energy sector. Lastly, the algorithm allows an optimal control of the numerical error. We apply the technique to the pricing of energy derivatives, comparing the results with existing benchmarks. Our findings indicate that the proposed methodology is at least one order of magnitude faster than existing algorithms, all while maintaining an equivalent level of accuracy.

We explore a stochastic model that enables capturing external influences in two specific ways. The model allows for the expression of uncertainty in the parametrisation of the stochastic dynamics and incorporates patterns to account for different behaviours across various times or regimes. To establish our framework, we initially construct a model with random parameters, where the switching between regimes can be dictated either by random variables or deterministically. Such a model is highly interpretable. We further ensure mathematical consistency by demonstrating that the framework can be elegantly expressed through local volatility models taking the form of standard jump diffusions. Additionally, we consider a Markov-modulated approach for the switching between regimes characterised by random parameters. For all considered models, we derive characteristic functions, providing a versatile tool with wide-ranging applications. In a numerical experiment, we apply the framework to the financial problem of option pricing. The impact of parameter uncertainty is analysed in a two-regime model, where the asset process switches between periods of high and low volatility imbued with high and low uncertainty, respectively.

An extensive empirical study of the class of Volterra Bergomi models using SPX options data between 2011 and 2022 reveals the following fact-check on two fundamental claims echoed in the rough volatility literature: Do rough volatility models with Hurst index $H \in (0,1/2)$ really capture well SPX implied volatility surface with very few parameters? No, rough volatility models are inconsistent with the global shape of SPX smiles. They suffer from severe structural limitations imposed by the roughness component, with the Hurst parameter $H \in (0,1/2)$ controlling the smile in a poor way. In particular, the SPX at-the-money skew is incompatible with the power-law shape generated by rough volatility models. The skew of rough volatility models increases too fast on the short end, and decays too slow on the longer end where "negative" $H$ is sometimes needed. Do rough volatility models really outperform consistently their classical Markovian counterparts? No, for short maturities they underperform their one-factor Markovian counterpart with the same number of parameters. For longer maturities, they do not systematically outperform the one-factor model and significantly underperform when compared to an under-parametrized two-factor Markovian model with only one additional calibratable parameter. On the positive side: our study identifies a (non-rough) path-dependent Bergomi model and an under-parametrized two-factor Markovian Bergomi model that consistently outperform their rough counterpart in capturing SPX smiles between one week and three years with only 3 to 4 calibratable parameters. \end{abstract}

This paper presents a new framework for player valuation in European football by fusing principles from financial mathematics and network theory. The valuation model leverages a "passing matrix" to encapsulate player interactions on the field, utilizing centrality measures to quantify individual influence. Unlike traditional approaches, this model is both metric-driven and cohort-free, providing a dynamic and individualized framework for ascertaining a player's fair market value. The methodology is empirically validated through a case study in European football, employing real-world match and financial data. The paper advances the disciplines of sports analytics and financial mathematics by offering a cross-disciplinary mechanism for player valuation, and also links together two well-known econometric methods in marginal revenue product and expected present valuation., Comment: 15 pages, 4 figures, 2 tables

Credit Valuation Adjustment is a balance sheet item which is nowadays subject to active risk management by specialized traders. However, one of the most important risk factors, which is the vector of default intensities of the counterparty, affects in a non-differentiable way the most general Monte Carlo estimator of the adjustment, through simulation of default times. Thus the computation of first and second order (pure and mixed) sensitivities involving these inputs cannot rely on direct path-wise differentiation, while any approach involving finite differences shows very high statistical noise. We present ad hoc analytical estimators which overcome these issues while offering very low runtime overheads over the baseline computation of the price adjustment. We also discuss the conversion of the so-obtained sensitivities to model parameters (e.g. default intensities) into sensitivities to market quotes (e.g. Credit Default Swap spreads)., Comment: 22 pages, 6 figures

We study an optimal stopping problem with an unbounded, time-dependent and discontinuous reward function. This problem is motivated by the pricing of a variable annuity (VA) contract with guaranteed minimum maturity benefit, under the assumption that the policyholder's surrender behaviour maximizes the contract's risk-neutral value. We consider a general fee and surrender charge function, and give a condition under which optimal stopping always occurs at maturity. Using an alternative representation for the value function of the optimization problem, we study its analytical properties and the resulting surrender (or exercise) region. In particular, we show that the non-emptiness and the shape of the surrender region are fully characterized by the fee and the surrender charge functions, which provides a powerful tool for understanding the link between fees and surrender functions and how they affect early surrender and the optimal surrender boundary. When the fee and surrender charge only depend on time, we develop three different representations of the value function; two are analogous to their American option counterpart, and one is new to the actuarial and American option pricing literature. Our results allow for the development of new algorithms for the valuation of variable annuity contracts. We provide three such algorithms, based on continuous-time Markov chain approximations. The efficiency of these three algorithms is studied numerically and compared to other commonly used approaches.

In this paper similar to [P. Carr, A. Itkin, 2019] we construct another Markovian approximation of the rough Heston-like volatility model - the ADO-Heston model. The characteristic function (CF) of the model is derived under both risk-neutral and real measures which is an unsteady three-dimensional PDE with some coefficients being functions of the time $t$ and the Hurst exponent $H$. To replicate known behavior of the market implied skew we proceed with a wise choice of the market price of risk, and then find a closed form expression for the CF of the log-price and the ATM implied skew. Based on the provided example, we claim that the ADO-Heston model (which is a pure diffusion model but with a stochastic mean-reversion speed of the variance process, or a Markovian approximation of the rough Heston model) is able (approximately) to reproduce the known behavior of the vanilla implied skew at small $T$. We conclude that the behavior of our implied volatility skew curve ${\cal S}(T) \propto a(H) T^{b\cdot (H-1/2)}, \, b = const$, is not exactly same as in rough volatility models since $b \ne 1$, but seems to be close enough for all practical values of $T$. Thus, the proposed Markovian model is able to replicate some properties of the corresponding rough volatility model. Similar analysis is provided for the forward starting options where we found that the ATM implied skew for the forward starting options can blow-up for any $s > t$ when $T \to s$. This result, however, contradicts to the observation of [E. Alos, D.G. Lorite, 2021] that Markovian approximation is not able to catch this behavior, so remains the question on which one is closer to reality., Comment: 23 pages, 3 figures, 3 tables

Stochastic volatility models, where the volatility is a stochastic process, can capture most of the essential stylized facts of implied volatility surfaces and give more realistic dynamics of the volatility smile/skew. However, they come with the significant issue that they take too long to calibrate. Alternative calibration methods based on Deep Learning (DL) techniques have been recently used to build fast and accurate solutions to the calibration problem. Huge and Savine developed a Differential Machine Learning (DML) approach, where Machine Learning models are trained on samples of not only features and labels but also differentials of labels to features. The present work aims to apply the DML technique to price vanilla European options (i.e. the calibration instruments), more specifically, puts when the underlying asset follows a Heston model and then calibrate the model on the trained network. DML allows for fast training and accurate pricing. The trained neural network dramatically reduces Heston calibration's computation time. In this work, we also introduce different regularisation techniques, and we apply them notably in the case of the DML. We compare their performance in reducing overfitting and improving the generalisation error. The DML performance is also compared to the classical DL (without differentiation) one in the case of Feed-Forward Neural Networks. We show that the DML outperforms the DL. The complete code for our experiments is provided in the GitHub repository: https://github.com/asridi/DML-Calibration-Heston-Model

In this research work, we propose a high-order time adapted scheme for pricing a coupled system of fixed-free boundary constant elasticity of variance (CEV) model on both equidistant and locally refined space-grid. The performance of our method is substantially enhanced to improve irregularities in the model which are both inherent and induced. Furthermore, the system of coupled PDEs is strongly nonlinear and involves several time-dependent coefficients that include the first-order derivative of the early exercise boundary. These coefficients are approximated from a fourth-order analytical approximation which is derived using a regularized square-root function. The semi-discrete equation for the option value and delta sensitivity is obtained from a non-uniform fourth-order compact finite difference scheme. Fifth-order 5(4) Dormand-Prince time integration method is used to solve the coupled system of discrete equations. Enhancing the performance of our proposed method with local mesh refinement and adaptive strategies enables us to obtain highly accurate solution with very coarse space grids, hence reducing computational runtime substantially. We further verify the performance of our methodology as compared with some of the well-known and better-performing existing methods.

In this study, we introduce a physical model inspired by statistical physics for predicting price volatility and expected returns by leveraging Level 3 order book data. By drawing parallels between orders in the limit order book and particles in a physical system, we establish unique measures for the system's kinetic energy and momentum as a way to comprehend and evaluate the state of limit order book. Our model goes beyond examining merely the top layers of the order book by introducing the concept of 'active depth', a computationally-efficient approach for identifying order book levels that have impact on price dynamics. We empirically demonstrate that our model outperforms the benchmarks of traditional approaches and machine learning algorithm. Our model provides a nuanced comprehension of market microstructure and produces more accurate forecasts on volatility and expected returns. By incorporating principles of statistical physics, this research offers valuable insights on understanding the behaviours of market participants and order book dynamics.

In this paper we propose a semi-analytic approach to pricing American options for time-dependent jump-diffusions models with exponential jumps The idea of the method is to further generalize our approach developed for pricing barrier, [Itkin et al., 2021], and American, [Carr and Itkin, 2021; Itkin and Muravey, 2023], options in various time-dependent one factor and even stochastic volatility models. Our approach i) allows arbitrary dependencies of the model parameters on time; ii) reduces solution of the pricing problem for American options to a simpler problem of solving a system of an algebraic nonlinear equation for the exercise boundary and a linear Fredholm-Volterra equation for the the option price; iii) the options Greeks solve a similar Fredholm-Volterra linear equation obtained by just differentiating Eq. (25) by the required parameter. Once done, the American option price is presented in close form., Comment: 23 pages, 1 table, 2 figures

Ghost, or fictitious points allow to capture boundary conditions that are not located on the finite difference grid discretization. We explore in this paper the impact of ghost points on the stability of the explicit Euler finite difference scheme in the context of the diffusion equation. In particular, we consider the case of a one-touch option under the Black-Scholes model. The observations and results are however valid for a much wider range of financial contracts and models.

We introduce a Path Shadowing Monte-Carlo method, which provides prediction of future paths, given any generative model. At any given date, it averages future quantities over generated price paths whose past history matches, or `shadows', the actual (observed) history. We test our approach using paths generated from a maximum entropy model of financial prices, based on a recently proposed multi-scale analogue of the standard skewness and kurtosis called `Scattering Spectra'. This model promotes diversity of generated paths while reproducing the main statistical properties of financial prices, including stylized facts on volatility roughness. Our method yields state-of-the-art predictions for future realized volatility and allows one to determine conditional option smiles for the S\&P500 that outperform both the current version of the Path-Dependent Volatility model and the option market itself.

Introduced in the late 90s, the passport option gives its holder the right to trade in a market and receive any positive gain in the resulting traded account at maturity. Pricing the option amounts to solving a stochastic control problem that for $d>1$ risky assets remains an open problem. Even in a correlated Black-Scholes (BS) market with $d=2$ risky assets, no optimal trading strategy has been derived in closed form. In this paper, we derive a discrete-time solution for multi-dimensional BS markets with uncorrelated assets. Moreover, inspired by the success of deep reinforcement learning in, e.g., board games, we propose two machine learning-powered approaches to pricing general options on a portfolio value in general markets. These approaches prove to be successful for pricing the passport option in one-dimensional and multi-dimensional uncorrelated BS markets.

Semi-analytical pricing of American options in a time-dependent Ornstein-Uhlenbeck model was presented in [Carr, Itkin, 2020]. It was shown that to obtain these prices one needs to solve (numerically) a nonlinear Volterra integral equation of the second kind to find the exercise boundary (which is a function of the time only). Once this is done, the option prices follow. It was also shown that computationally this method is as efficient as the forward finite difference solver while providing better accuracy and stability. Later this approach called "the Generalized Integral transform" method has been significantly extended by the authors (also, in cooperation with Peter Carr and Alex Lipton) to various time-dependent one factor, and stochastic volatility models as applied to pricing barrier options. However, for American options, despite possible, this was not explicitly reported anywhere. In this paper our goal is to fill this gap and also discuss which numerical method (including those in machine learning) could be efficient to solve the corresponding Volterra integral equations., Comment: 38 pages, 5 figures, 1 table

We propose a neural network-based approach to calibrating stochastic volatility models, which combines the pioneering grid approach by Horvath et al. (2021) with the pointwise two-stage calibration of Bayer et al. (2018) and Liu et al. (2019). Our methodology inherits robustness from the former while not suffering from the need for interpolation/extrapolation techniques, a clear advantage ensured by the pointwise approach. The crucial point to the entire procedure is the generation of implied volatility surfaces on random grids, which one dispenses to the network in the training phase. We support the validity of our calibration technique with several empirical and Monte Carlo experiments for the rough Bergomi and Heston models under a simple but effective parametrization of the forward variance curve. The approach paves the way for valuable applications in financial engineering - for instance, pricing under local stochastic volatility models - and extensions to the fast-growing field of path-dependent volatility models., Comment: 38 pages, 22 figures, and 9 tables

This paper generalizes the local variance gamma model of Carr and Nadtochiy, to a piecewise quadratic local variance function. The formulation encompasses the piecewise linear Bachelier and piecewise linear Black local variance gamma models. The quadratic local variance function results in an arbitrage-free interpolation of class $\mathcal{C}^3$. The increased smoothness over the piecewise-constant and piecewise-linear representation allows to reduce the number of knots when interpolating raw market quotes, thus providing an interesting alternative to regularization while reducing the computational cost., Comment: arXiv admin note: text overlap with arXiv:2004.08650

The paper studies intraday price movement of stocks that is considered as an image classification problem. Using a CNN-based model we make a compelling case for the high-level relationship between the first hour of trading and the close. The algorithm managed to adequately separate between the two opposing classes and investing according to the algorithm's predictions outperformed all alternative constructs but the theoretical maximum. To support the thesis, we ran several additional tests. The findings in the paper highlight the suitability of computer vision techniques for studying financial markets and in particular prediction of stock price movements.

This note improves the lower and upper bounds of the Black-Scholes implied volatility (IV) in Tehranchi (SIAM J. Financial Math., 7 (2016), p. 893). The proposed tighter bounds are systematically based on the bounds of the option delta. While Tehranchi used the bounds to prove IV asymptotics, we apply the result to the accurate numerical root-finding of IV. We alternatively formulate the Newton-Raphson method on the log price and demonstrate that the iteration always converges rapidly for all price ranges if the new lower bound found in this study is used as an initial guess.

Recent technological developments have changed the fundamental ways stock markets function, bringing regulatory instances to assess the benefits of these developments. In parallel, the ongoing machine learning revolution and its multiple applications to trading can now be used to design a next generation of financial models, and thereby explore the systemic complexity of financial stock markets in new ways. We here follow on a previous groundwork, where we designed and calibrated a novel agent-based model stock market simulator, where each agent autonomously learns to trade by reinforcement learning. In this Paper, we now study the predictions of this model from a regulator's perspective. In particular, we focus on how the market quality is impacted by smaller order book tick sizes, increasingly larger metaorders, and higher trading frequencies, respectively. Under our model assumptions, we find that the market quality benefits from the latter, but not from the other two trends.

Silkswap is an automated market maker model designed for efficient stablecoin trading with minimal price impact. The original purpose of Silkswap is to facilitate the trading of fiat-pegged stablecoins with the stablecoin Silk, but it can be applied to any pair of stablecoins. The Silkswap invariant is a hybrid function that generates an asymmetric price impact curve. We present the derivation of the Silkswap model and its mathematical properties. We also compare different numerical methods used to solve the invariant equation. Finally, we compare our model with the well-known Curve Finance model.

The stochastic-alpha-beta-rho (SABR) model has been widely adopted in options trading. In particular, the normal ($\beta=0$) SABR model is a popular model choice for interest rates because it allows negative asset values. The option price and delta under the SABR model are typically obtained via asymptotic implied volatility approximation, but these are often inaccurate and arbitrageable. Using a recently discovered price transition law, we propose a Gaussian quadrature integration scheme for price options under the normal SABR model. The compound Gaussian quadrature sum over only 49 points can calculate a very accurate price and delta that are arbitrage-free.

In this work, the Fourier-cosine series (COS) method has been combined with the Boundary Element Method (BEM) for a fast evaluation of barrier option prices. After a description of its use in the Black and Scholes (BS) model, the focus of the paper is on the application of the proposed methodology to the barrier option evaluation in the Heston model, where its contribution is fundamental to improve computational efficiency and to make BEM appealing among Finance practitioners as a valid alternative to Monte Carlo (MC) or other more traditional approaches. An error analysis is provided on the number of terms used in the Fourier-cosine series expansion, where the error bound estimation is based on the characteristic function of the log-asset price process.

A novel approach called Moate Simulation is presented to provide an accurate numerical evolution of probability distribution functions represented on grids arising from stochastic differential processes where initial conditions are specified. Where the variables of stochastic differential equations may be transformed via It\^o-Doeblin calculus into stochastic differentials with a constant diffusion term, the probability distribution function for these variables can be simulated in discrete time steps. The drift is applied directly to a volume element of the distribution while the stochastic diffusion term is applied through the use of convolution techniques such as Fast or Discrete Fourier Transforms. This allows for highly accurate distributions to be efficiently simulated to a given time horizon and may be employed in one, two or higher dimensional expectation integrals, e.g. for pricing of financial derivatives. The Moate Simulation approach forms a more accurate and considerably faster alternative to Monte Carlo Simulation for many applications while retaining the opportunity to alter the distribution in mid-simulation., Comment: 17 pages, 1 pdf figure plus internal tikz figures. JEL Classifications: C15, C63, G12, G13, G17

We propose a deep learning method for solving the American options model with a free boundary feature. To extract the free boundary known as the early exercise boundary from our proposed method, we introduce the Landau transformation. For efficient implementation of our proposed method, we further construct a dual solution framework consisting of a novel auxiliary function and free boundary equations. The auxiliary function is formulated to include the feed forward deep neural network (DNN) output and further mimic the far boundary behaviour, smooth pasting condition, and remaining boundary conditions due to the second-order space derivative and first-order time derivative. Because the early exercise boundary and its derivative are not a priori known, the boundary values mimicked by the auxiliary function are in approximate form. Concurrently, we then establish equations that approximate the early exercise boundary and its derivative directly from the DNN output based on some linear relationships at the left boundary. Furthermore, the option Greeks are obtained from the derivatives of this auxiliary function. We test our implementation with several examples and compare them with the existing numerical methods. All indicators show that our proposed deep learning method presents an efficient and alternative way of pricing options with early exercise features.

We focus on extending existing short-rate models, enabling control of the generated implied volatility while preserving analyticity. We achieve this goal by applying the Randomized Affine Diffusion (RAnD) method to the class of short-rate processes under the Heath-Jarrow-Morton framework. Under arbitrage-free conditions, the model parameters can be exogenously stochastic, thus facilitating additional degrees of freedom that enhance the calibration procedure. We show that with the randomized short-rate models, the shapes of implied volatility can be controlled and significantly improve the quality of the model calibration, even for standard 1D variants. In particular, we illustrate that randomization applied to the Hull-White model leads to dynamics of the local volatility type, with the prices for standard volatility-sensitive derivatives explicitly available. The randomized Hull-White (rHW) model offers an almost perfect calibration fit to the swaption implied volatilities., Comment: 22 Pages, 9 Figures, 2 Tables

The aim of this work is to propose an extension of the Deep BSDE solver by Han, E, Jentzen (2017) to the case of FBSDEs with jumps. As in the aforementioned solver, starting from a discretized version of the BSDE and parametrizing the (high dimensional) control processes by means of a family of ANNs, the BSDE is viewed as model-based reinforcement learning problem and the ANN parameters are fitted so as to minimize a prescribed loss function. We take into account both finite and infinite jump activity by introducing, in the latter case, an approximation with finitely many jumps of the forward process., Comment: 31 pages

We present a framework for constructing multivariate risk measures that is inspired from univariate Optimized Certainty Equivalent (OCE) risk measures. We show that this new class of risk measures verifies the desirable properties such as convexity, monotonocity and cash invariance. We also address numerical aspects of their computations using stochastic algorithms instead of using Monte Carlo or Fourier methods that do not provide any error of the estimation.

Partial differential equations sometimes have critical points where the solution or some of its derivatives are discontinuous. The simplest example is a discontinuity in the initial condition. It is well known that those decrease the accuracy of finite difference methods. A common remedy is to stretch the grid, such that many more grid points are present near the critical points, and fewer where the solution is deemed smooth. An alternative solution is to insert points such that the discontinuities fall in the middle of two grid points. This paper compares the accuracy of both approaches in the context of the pricing of financial derivative contracts in the Black-Scholes model and proposes a new fast and simple stretching function.

We propose a risk neutral approach to forecast the cashflows of music catalogs, based on historical revenue data. We use a discounted cashflows formula to produce reasonable ranges of multipliers for these assets, based on the age of the catalog, the last-twelve-months revenue and the duration of the contract. We compare the multipliers implied by the cashflows of top, median and bottom performing songs on the Royalty Exchange platform. We find that ask prices are close to the multipliers justified by median song cashflows. The best bids are near the multipliers justified by the bottom decile of song cashflows.

The class of Affine (Jump) Diffusion (AD) has, due to its closed form characteristic function (ChF), gained tremendous popularity among practitioners and researchers. However, there is clear evidence that a linearity constraint is insufficient for precise and consistent option pricing. Any non-affine model must pass the strict requirement of quick calibration -- which is often challenging. We focus here on Randomized AD (RAnD) models, i.e., we allow for exogenous stochasticity of the model parameters. Randomization of a pricing model occurs outside the affine model and, therefore, forms a generalization that relaxes the affinity constraints. The method is generic and can apply to any model parameter. It relies on the existence of moments of the so-called randomizer- a random variable for the stochastic parameter. The RAnD model allows flexibility while benefiting from fast calibration and well-established, large-step Monte Carlo simulation, often available for AD processes. The article will discuss theoretical and practical aspects of the RAnD method, like derivations of the corresponding ChF, simulation, and computations of sensitivities. We will also illustrate the advantages of the randomized stochastic volatility models in the consistent pricing of options on the S&P 500 and VIX., Comment: 7424 words, 24 figures

The main goal of this paper is to use the enlargement of ltration framework for pricing zerocoupon CAT bonds. For this purpose, we develop two models where the trigger event time is perfectly covered by an increasing sequence of stopping times with respect to a reference ltration. Hence, depending on the nature of these stopping times the trigger event time can be either accessible or totally inaccessible. When some of these stopping times are not predictable, the trigger event time is totally inaccessible, and very nice mathematical computations can be derived. When the stopping times are predictable, the trigger event time is accessible, and this case would be a meaningful choice for Model 1 from a practical point of view since features like seasonality are already captured by some quantities such as the stochastic intensity of the Poisson process. We compute the main tools for pricing the zero-coupon CAT bond and show that our constructions are more general than some existing models in the literature. We obtain some closed-form prices of zero-coupon CAT bonds in Model 2 so we give a numerical illustrative example for this latter., Comment: Journal of Mathematical Finance, Scientific Research, In press

We present a stochastic local volatility model for derivative contracts on commodity futures. The aim of the model is to be able to recover the prices of derivative claims both on futures contracts and on indices on futures strategies. Numerical examples for calibration and pricing are provided for the S&P GSCI Crude Oil excess-return index., Comment: 22 pages, 3 Figures

We consider the pricing of variable annuities (VAs) with general fee structures under popular stochastic volatility models such as Heston, Hull-White, Scott, $\alpha$-Hypergeometric, $3/2$, and $4/2$ models. In particular, we analyze the impact of different VIX-linked fee structures on the optimal surrender strategy of a VA contract with guaranteed minimum maturity benefit (GMMB). Under the assumption that the VA contract can be surrendered before maturity, the pricing of a VA contract corresponds to an optimal stopping problem with an unbounded, time-dependent, and discontinuous payoff function. We develop efficient algorithms for the pricing of VA contracts using a two-layer continuous-time Markov chain approximation for the fund value process. When the contract is kept until maturity and under a general fee structure, we show that the value of the contract can be approximated by a closed-form matrix expression. We also provide a quick and simple way to determine the value of early surrenders via a recursive algorithm and give an easy procedure to approximate the optimal surrender surface. We show numerically that the optimal surrender strategy is more robust to changes in the volatility of the account value when the fee is linked to the VIX index.

We propose a stable sixth-order compact finite difference scheme with a dynamic fifth-order staggered boundary scheme and 3(2) R-K Bogacki and Shampine adaptive time stepping for pricing American style options. To locate, fix and compute the free-boundary simultaneously with option and delta sensitivity, we introduce a Landau transformation. Furthermore, we remove the convective term in the pricing model which could further introduce errors. Hence, an efficient sixth-order compact scheme can easily be implemented. The main challenge in coupling the sixth order compact scheme in discrete form is to efficiently account for the near-boundary scheme. In this work, we introduce novel fifth- and sixth-order Dirichlet near-boundary schemes suitable for solving our model. The optimal exercise boundary and other boundary values are approximated using a high-order analytical approximation obtained from a novel fifth-order staggered boundary scheme. Furthermore, we investigate the smoothness of the first and second derivatives of the optimal exercise boundary which is obtained from this high-order analytical approximation. Coupled with the 3(2) RK-Bogacki and Shampine time integration method, the interior values are then approximated using the sixth order compact operator. The expected convergence rate is obtained, and our present numerical scheme is very fast and gives highly accurate approximations with very coarse grids.

Pricing a multi-asset derivative is an important problem in financial engineering, both theoretically and practically. Although it is suitable to numerically solve partial differential equations to calculate the prices of certain types of derivatives, the computational complexity increases exponentially as the number of underlying assets increases in some classical methods, such as the finite difference method. Therefore, there are efforts to reduce the computational complexity by using quantum computation. However, when solving with naive quantum algorithms, the target derivative price is embedded in the amplitude of one basis of the quantum state, and so an exponential complexity is required to obtain the solution. To avoid the bottleneck, the previous study~[Miyamoto and Kubo, IEEE Transactions on Quantum Engineering, \textbf{3}, 1--25 (2022)] utilizes the fact that the present price of a derivative can be obtained by its discounted expected value at any future point in time and shows that the quantum algorithm can reduce the complexity. In this paper, to make the algorithm feasible to run on a small quantum computer, we use variational quantum simulation to solve the Black-Scholes equation and compute the derivative price from the inner product between the solution and a probability distribution. This avoids the measurement bottleneck of the naive approach and would provide quantum speedup even in noisy quantum computers. We also conduct numerical experiments to validate our method. Our method will be an important breakthrough in derivative pricing using small-scale quantum computers., Comment: 18 pages, 4 figures

This paper studies equity basket options -- i.e., multi-dimensional derivatives whose payoffs depend on the value of a weighted sum of the underlying stocks -- and develops a new and innovative approach to ensure consistency between options on individual stocks and on the index comprising them. Specifically, we show how to resolve a well-known problem that when individual constituent distributions of an equity index are inferred from the single-stock option markets and combined in a multi-dimensional local/stochastic volatility model, the resulting basket option prices will not generate a skew matching that of the options on the equity index corresponding to the basket. To address this ``insufficient skewness'', we proceed in two steps. First, we propose an ``effective'' local volatility model by mapping the general multi-dimensional basket onto a collection of marginal distributions. Second, we build a multivariate dependence structure between all the marginal distributions assuming a jump-diffusion model for the effective projection parameters, and show how to calibrate the basket to the index smile. Numerical tests and calibration exercises demonstrate an excellent fit for a basket of as many as 30 stocks with fast calculation time., Comment: 23 pages, 21 figures

Driven by the good results obtained in computer vision, deep generative methods for time series have been the subject of particular attention in recent years, particularly from the financial industry. In this article, we focus on commodity markets and test four state-of-the-art generative methods, namely Time Series Generative Adversarial Network (GAN) Yoon et al. [2019], Causal Optimal Transport GAN Xu et al. [2020], Signature GAN Ni et al. [2020] and the conditional Euler generator Remlinger et al. [2021], are adapted and tested on commodity time series. A first series of experiments deals with the joint generation of historical time series on commodities. A second set deals with deep hedging of commodity options trained on he generated time series. This use case illustrates a purely data-driven approach to risk hedging., Comment: 15 pages

We propose a new jump-diffusion process, the Heston-Queue-Hawkes (HQH) model, combining the well-known Heston model and the recently introduced Queue-Hawkes (Q-Hawkes) jump process. Like the Hawkes process, the HQH model can capture the effects of self-excitation and contagion. However, since the characteristic function of the HQH process is known in closed-form, Fourier-based fast pricing algorithms, like the COS method, can be fully exploited with this model. Furthermore, we show that by using partial integrals of the characteristic function, which are also explicitly known for the HQH process, we can reduce the dimensionality of the COS method, and so its numerical complexity. Numerical results for European and Bermudan options show that the HQH model offers a wider range of volatility smiles compared to the Bates model, while its computational burden is considerably smaller than that of the Heston-Hawkes (HH) process.

This article aims to propose and apply a machine learning method to analyze the direction of returns from Exchange Traded Funds (ETFs) using the historical return data of its components, helping to make investment strategy decisions through a trading algorithm. In methodological terms, regression and classification models were applied, using standard datasets from Brazilian and American markets, in addition to algorithmic error metrics. In terms of research results, they were analyzed and compared to those of the Na\"ive forecast and the returns obtained by the buy & hold technique in the same period of time. In terms of risk and return, the models mostly performed better than the control metrics, with emphasis on the linear regression model and the classification models by logistic regression, support vector machine (using the LinearSVC model), Gaussian Naive Bayes and K-Nearest Neighbors, where in certain datasets the returns exceeded by two times and the Sharpe ratio by up to four times those of the buy & hold control model., Comment: Co-author did not agree with publishing here