Learning Parameter Dependence for Fourier-Based Option Pricing with Tensor Trains

Stochastic volatility of underlying assets has been shown to affect significantly the price of many financial derivatives. In particular, a fast mean-reverting factor of the stochastic volatility plays a major role in the pricing of options. This paper deals with the interest rate model dependence of the stochastic volatility impact on defaultable interest rate derivatives. We obtain an asymptotic formula of the price of defaultable bonds and bond options based on a quadratic term structure model and investigate the stochastic volatility and default risk effects and compare the results with those of the Vasicek model.

Options pricing is a critical problem and one of the fundamental building blocks in mathematical finance. Monte Carlo (MC) simulation is the most widely used solution for options pricing. In most cases options pricing must be performed in real time. Although today’s multi-core CPUs can provide a high computing power, the options pricing on today’s multi-core CPUs is far from responding in real time. In this paper, we use modern graphics processors (GPUs), which provide a much higher computing power than multi-core CPUs, to perform MC-based options pricing. The challenge is that GPUs only provide a small on-chip scratchpad while MC-based options pricing requires a large memory space and accesses the memory space irregularly. To make MC-based options pricing efficient on GPUs, we propose a recycling approach which compacts the data to shrink the memory space, and a crossing-path layout which reorganizes the data to make memory accesses GPU-friendly. We use real market data and benchmarks to evaluate our optimization approaches. The experimental result shows that our MC-based options pricing on the latest GPU is 43-fold faster than the latest multi-core CPU for single precision computation (with Intel C++ Compiler 11.1); compared with GCC 4.2.4, the speedup is as high as 145.

Option pricing, a fundamental problem in finance, often requires solving non-linear partial differential equations (PDEs). When dealing with multi-asset options, such as rainbow options, these PDEs become high-dimensional, leading to challenges posed by the curse of dimensionality. While deep learning-based PDE solvers have recently emerged as scalable solutions to this high-dimensional problem, their empirical and quantitative accuracy remains not well-understood, hindering their real-world applicability. In this study, we aimed to offer actionable insights into the utility of Deep PDE solvers for practical option pricing implementation. Through comparative experiments, we assessed the empirical performance of these solvers in high-dimensional contexts. Our investigation identified three primary sources of errors in Deep PDE solvers: (i) errors inherent in the specifications of the target option and underlying assets, (ii) errors originating from the asset model simulation methods, and (iii) errors stemming from the neural network training. Through ablation studies, we evaluated the individual impact of each error source. Our results indicate that the Deep BSDE method (DBSDE) is superior in performance and exhibits robustness against variations in option specifications. In contrast, some other methods are overly sensitive to option specifications, such as time to expiration. We also find that the performance of these methods improves inversely proportional to the square root of batch size and the number of time steps. This observation can aid in estimating computational resources for achieving desired accuracies with Deep PDE solvers.

In an efficient market, the no‐arbitrage condition implies that the price difference between any two assets must be the market value of all differences in their cash flows. We use this logic to deduce the price of the prepayment option embedded in fixed‐rate Government National Mortgage Association (GNMA) mortgage‐backed securities. The option price equals the difference between an observed GNMA price and the cost of a synthetic, nonprepayable GNMA constructed from the least expensive portfolio of Treasury securities that exactly replicates the promised GNMA cash flow stream, assuming prepayment is precluded. We regress the option prices on variables found significant in previous prepayment studies, finding that five key regressors explain more than 90% of the prepayment option value in pooled time‐series cross‐sectional analysis. We also show that the time value of the prepayment option calculated by our method displays a pattern similar to that produced by the Black‐Scholes (1973) option pricing model. An additional empirical result is the existence of negative option prices and negative time value of the option prices. We attribute these to the fact that homeowners sometimes exercise their prepayment options when they are out‐of‐the‐money, and to refinancing transaction costs. Our method is independent of assumptions regarding interest rate processes and the homeowner's prepayment behavior, and it provides a benchmark for testing theoretical prepayment models.

The SABR stochastic volatility model with β-volatility β ? (0,1) and an absorbing barrier in zero imposed to the forward prices/rates stochastic process is studied. The presence of (possibly) nonzero correlation between the stochastic differentials that appear on the right hand side of the model equations is considered. A series expansion of the transition probability density function of the model in powers of the correlation coefficient of these stochastic differentials is presented. Explicit formulae for the first three terms of this expansion are derived. These formulae are integrals of known integrands. The zero-th order term of the expansion is a new integral formula containing only elementary functions of the transition probability density function of the SABR model when the correlation coefficient is zero. The expansion is deduced from the final value problem for the backward Kolmogorov equation satisfied by the transition probability density function. Each term of the expansion is defined as the solution of a final value problem for a partial differential equation. The integral formulae that give the solutions of these final value problems are based on the Hankel and on the Kontorovich-Lebedev transforms. From the series expansion of the probability density function we deduce the corresponding expansions of the European call and put option prices. Moreover we deduce closed form formulae for the moments of the forward prices/rates variable. The moment formulae obtained do not involve integrals or series expansions and are expressed using only elementary functions. The option pricing formulae are used to study synthetic and real data. In particular we study a time series (of real data) of futures prices of the EUR/USD currency's exchange rate and of the corresponding option prices. The website: http://www.econ.univpm.it/recchioni/finance/w18 contains material including animations, an interactive application and an app that helps the understanding of the paper. A more general reference to the work of the authors and of their coauthors in mathematical finance is the website:http://www.econ.univpm.it/recchioni/finance.

Pricing European Options Based on the Hesitancy Degree of Investors

The calibration of volatility models from observable option prices is a fundamental problem in quantitative finance. The most common approach among industry practitioners is based on the celebrated Dupire's formula [6], which requires the knowledge of vanilla option prices for a continuum of strikes and maturities that can only be obtained via some form of price interpolation. In this paper, we propose a new local volatility calibration technique using the theory of optimal transport. We formulate a time continuous martingale optimal transport problem, which seeks a martingale diffusion process that matches the known densities of an asset price at two different dates, while minimizing a chosen cost function. Inspired by the seminal work of Benamou and Brenier [1], we formulate the problem as a convex optimization problem, derive its dual formulation, and solve it numerically via an augmented Lagrangian method and the alternative direction method of multipliers (ADMM) algorithm. The solution effectively reconstructs the dynamic of the asset price between the two dates by recovering the optimal local volatility function, without requiring any time interpolation of the option prices.

This paper presents a method to solve the American option pricing problem in the Black Scholes framework that generalizes the Barone-Adesi, Whaley method [1]. An auxiliary parameter is introduced in the American option pricing problem. Power series expansions in this parameter of the option price and of the corresponding free boundary are derived. These series expansions have the Baroni-Adesi, Whaley solution of the American option pricing problem as zero-th order term. The coefficients of the option price series are explicit formulae. The partial sums of the free boundary series are determined solving numerically nonlinear equations that depend from the time variable as a parameter. Numerical experiments suggest that the series expansions derived are convergent. The evaluation of the truncated series expansions on a grid of values of the independent variables is easily parallelizable. The cost of computing the n-th order truncated series expansions is approximately proportional to n as n goes to infinity. The results obtained on a set of test problems with the first and second order approximations deduced from the previous series expansions outperform in accuracy and/or in computational cost the results obtained with several alternative methods to solve the American option pricing problem [1]-[3]. For example when we consider options with maturity time between three and ten years and positive cost of carrying parameter (i.e. when the continuous dividend yield is smaller than the risk free interest rate) the second order approximation of the free boundary obtained truncating the series expansions improves substantially the Barone-Adesi, Whaley free boundary [1]. The website: http://www.econ.univpm.it/recchioni/finance/w20 contains material including animations, an interactive application and an app that helps the understanding of the paper. A general reference to the work of the authors and of their coauthors in mathematical finance is the website: http://www.econ.univpm.it/recchioni/finance.

This paper gives an ordering on option prices under various well‐known martingale measures in an incomplete stochastic volatility model. Our central result is a comparison theorem that proves convex option prices are decreasing in the market price of volatility risk, the parameter governing the choice of pricing measure. The theorem is applied to order option prices under q‐optimal pricing measures. In doing so, we correct orderings demonstrated numerically in Heath, Platen, and Schweizer (Mathematical Finance, 11(4), 2001) in the special case of the Heston model.

This thesis considers two main subjects divided in four problems in the broad field of mathematical finance. The first chapter treats option pricing followed by three chapters on the application of the machine learning algorithm of Random Forests to finance, specifically to risk capital aggregation, portfolio optimization and macro stress testing. In all four chapters new methodologies to treat the respective subjects are developed. All proposed models are benchmarked against commonly applied methods in the respective fields and found to outperform their peers.

- Options are imperative, yet complex instruments in financial markets, primarily deployed to hedge, manage risk and speculation. Pricing these instruments to near accuracy is critical for its use as mispricing can lead to unethical speculation and arbitrage, not to mention financial risk. Since the development of closed form models, numerous analytical, simulation, machine learning and stochastic approaches have been calibrated to improve valuations and address assumptions which may not align to the real world. This paper presents a comprehensive review of pricing strategies; stemming from options and payoff fundamentals, then into exploring various traditional plus algorithmic methods addressing volatility in pricing, and buttressing the theoretical review with computational experiments, with payoff visualizations, and pricing simulations. The objective of the study is to emphasize the advantages and constraints of various option pricing strategies, with an emphasis on their transdisciplinary significance in the fields of computational finance, risk management and applied mathematics. Key Words: option pricing, derivatives, Black-Scholes, Numerical Methods, quantitative finance

Option pricing is essential for market participants as it facilitates risk management and price discovery. This study provides a comprehensive elucidation of diverse option pricing models. The author delineates the development of option pricing models from the fundamental Black-Scholes-Merton model (BSM) to more intricate methodologies. The importance of comprehending option pricing for hedging strategies and risk management is crucial in financial mathematics. The study analyses several models, including the BS model, Jump Diffusion model (JDM), Binomial Tree model (BTM), Monte Carlo simulation (MCS), and stochastic volatility models (SVM), while addressing their applications, limitations, advantages, and disadvantages. This work delineates the history and distinctions between other models and the Black-Scholes-Merton model. The Black-Scholes-Merton model exerts the most substantial influence on financial markets. The alternative models exhibit greater flexibility and enhanced accuracy when confronting complicated and uncertain market trends. The trend of future development is bound to be the continuous updating of models to adapt to more complex situations.

Derivatives have become widely accepted as tools for hedging and risk-management, as well as speculation to some extent. A more recent trend has been gaining ground, namely, arbitrage in derivatives. The critical parameter in derivatives pricing is the volatility of the underlying asset. Exchanges often overestimate volatility in order to cover any sudden changes in market behavior, leading to systematic overpricing of derivatives. Accurate forecasting of volatility would expose systematic overpricing. Unfortunately, volatility is not an easy phenomenon to predict or forecast. One class of models that have proved successful in forecasting volatility in many situations is the Generalized Autoregressive Conditional Heteroscedasticity (GARCH) family of models. The objective of the present study is to analyze systematic bias in the pricing of options derivatives. In order to perform the analysis, data were collected for a sample of stock options traded on the National Stock Exchange (NSE) of India and their underlying stocks. In the study, GARCH models are used to forecast underlying stock volatility, and the forecasted volatility is used in the Black-Scholes model in order to determine whether the corresponding options were fairly priced. Any systematic bias in options pricing would provide evidence for arbitrage opportunities.

Return Dynamics With Lévy Jumps: Evidence from Stock and Option Prices

One of the possible improvements of the classical Black–Scholes option pricing model is to incorporate the stochastic nature of the short rate dynamics in option valuation. In this paper, we present the numerical scheme, based on the discontinuous Galerkin method, for European option pricing when the short rate follows the Merton model. The pricing function satisfies a partial differential equation with two underlying variables — stock price and short rate value. With a localization to a bounded spatial domain, including setting the proper boundary conditions, the governing equation is discretized by the discontinuous Galerkin method over a finite element grid and Crank-Nicolson time integration is applied, consequently. As a result the numerical scheme is represented by a sequence of linear algebraic systems with sparse matrices. Moreover, the numerical simulations reflect the capability of the scheme presented.