Option Pricing Applications Research Articles

Deep Learning of Transition Probability Densities for Stochastic Asset Models with Applications in Option Pricing

Transition probability density functions (TPDFs) are fundamental to computational finance, including option pricing and hedging. Advancing recent work in deep learning, we develop novel neural TPDF generators through solving backward Kolmogorov equations in parametric space for cumulative probability functions. The generators are ultra-fast, very accurate and can be trained for any asset model described by stochastic differential equations. These are “single solve,” so they do not require retraining when parameters of the stochastic model are changed (e.g., recalibration of volatility). Once trained, the neural TDPF generators can be transferred to less powerful computers where they can be used for e.g. option pricing at speeds as fast as if the TPDF were known in a closed form. We illustrate the computational efficiency of the proposed neural approximations of TPDFs by inserting them into numerical option pricing methods. We demonstrate a wide range of applications including the Black-Scholes-Merton model, the standard Heston model, the SABR model, and jump-diffusion models. These numerical experiments confirm the ultra-fast speed and high accuracy of the developed neural TPDF generators. This paper was accepted by Kay Giesecke, finance. Funding: H. Su received research funding support from Nottingham Business School at Nottingham Trent University. M. V. Tretyakov was supported by the Engineering and Physical Sciences Research Council [Grant EP/X022617/1]. D. P. Newton received research funding support from the School of Management at Bath University. Supplemental Material: The online appendices and data files are available at https://doi.org/10.1287/mnsc.2022.01448 .

Evolution of option pricing models: From Black-Scholes to Heston and beyond

This essay explores the evolution of option pricing models, tracing their development from the foundational Black-Scholes model to more advanced frameworks such as the Heston model and beyond. Beginning with an introduction to option pricing theory, the essay discusses the origins of the Black-Scholes model and its assumptions, as well as the challenges and limitations it faces. It then examines the extension of the Black-Scholes model, so-called the Black-Scholes-Merton model. It incorporates dividends, and lays the groundwork for further research into options pricing and financial derivatives. Then various stochastic volatility models emerge, and the essay chooses the Heston model as a typical example for analysis, highlighting its advantages and applications in option pricing. Furthermore, the essay compares the Heston model with other option pricing models, including the SABR model and Bates model. At the end of the essay, recent advances and future directions in option pricing are introduced and discussed. Through this comprehensive exploration, readers can gain a deeper understanding of the evolution of option pricing models and their significance in modern finance.

Numerical Approximation to a Variable-Order Time-Fractional Black–Scholes Model with Applications in Option Pricing

Numerical Approximation to a Variable-Order Time-Fractional Black–Scholes Model with Applications in Option Pricing

Multidimensional Linear and Nonlinear Partial Integro-Differential Equation in Bessel Potential Spaces with Applications in Option Pricing

The purpose of this paper is to analyze solutions of a non-local nonlinear partial integro-differential equation (PIDE) in multidimensional spaces. Such class of PIDE often arises in financial modeling. We employ the theory of abstract semilinear parabolic equations in order to prove existence and uniqueness of solutions in the scale of Bessel potential spaces. We consider a wide class of Lévy measures satisfying suitable growth conditions near the origin and infinity. The novelty of the paper is the generalization of already known results in the one space dimension to the multidimensional case. We consider Black–Scholes models for option pricing on underlying assets following a Lévy stochastic process with jumps. As an application to option pricing in the one-dimensional space, we consider a general shift function arising from a nonlinear option pricing model taking into account a large trader stock-trading strategy. We prove existence and uniqueness of a solution to the nonlinear PIDE in which the shift function may depend on a prescribed large investor stock-trading strategy function.

Saddlepoint approximations to tail expectations under non-Gaussian base distributions: option pricing applications

The saddlepoint approximation formulas provide versatile tools for analytic approximation of the tail expectation of a random variable by approximating the complex Laplace integral of the tail expectation expressed in terms of the cumulant generating function of the random variable. We generalize the saddlepoint approximation formulas for calculating tail expectations from the usual Gaussian base distribution to an arbitrary base distribution. Specific discussion is presented on the criteria of choosing the base distribution that fits better the underlying distribution. Numerical performance and comparison of accuracy are made among different saddlepoint approximation formulas. Improved accuracy of the saddlepoint approximations to tail expectations is revealed when proper base distributions are chosen. We also demonstrate enhanced accuracy of the generalized saddlepoint approximation formulas under non-Gaussian base distributions in pricing European options on continuous integrated variance under the Heston stochastic volatility model.

The Kolmogorov forward fractional partial differential equation for the CGMY-process with applications in option pricing

In this paper we derive the Kolmogorov forward fractional partial differential equation (FPDE) for the CGMY-process. The resulting FPDE is solved numerically with a second order method in space and Backward Differentiation Formula of order two in time. We price options by integrating the resulting probability density function multiplied by the pay-off function of the option. Hence, we only have to solve one FPDE to price several options. This is useful in practical applications where it is common to price many options simultaneously for the same underlying diffusion model.The traditional way to price options requires the solution of one FPDE per option. Since the FPDEs are of similar type and complexity, the advantage of our suggested method is obvious in the case of pricing multiple European type options on the same underlying asset when only the pay-off function differs. The numerical experiments verify that the extra cost of pricing several options compared to only one option is very small with our suggested method.

Error analysis of truncated expansion solutions to high-dimensional parabolic PDEs

We study an expansion method for high-dimensional parabolic PDEs which constructs accurate approximate solutions by decomposition into solutions to lower-dimensional PDEs, and which is particularly effective if there are a low number of dominant principal components. The focus of the present article is the derivation of sharp error bounds for the constant coefficient case and a first and second order approximation. We give a precise characterisation when these bounds hold for (non-smooth) option pricing applications and provide numerical results demonstrating that the practically observed convergence speed is in agreement with the theoretical predictions.

Comonotonic Monte Carlo Simulation and Its Applications in Option Pricing and Quantification of Risk

For many kinds of derivative valuation problems, especially those that try for greater realism using return processes that are more consistent with empirical evidence, Monte Carlo simulation is the only feasible solution technique. Among the well-known strategies to improve its efficiency, the use of a well-chosen control variate is often very effective. But a good selection can make a lot of difference. This article explains how the mathematical concept of comonotonicity can be applied as a new way to create a control variate. A remarkable improvement in performance can be achieved using the comonotonic upper bound as the control variate. The article illustrates the power of Comonotonic Monte Carlo simulation in estimating tail value at risk and pricing arithmetic Asian options.

Exploring the dynamics of financial markets: from stock prices to strategy returns

Exploring the dynamics of financial time-series is an exciting and interesting challenge because of the many truly complex interactions that underly the price formation process. In this contribution we describe some of the anomalous statistical features of such time-series and review models of the price dynamics both across time and across the universe of stocks. In particular we discuss a non-Gaussian statistical feedback process of stock returns which we have developed over the past years with the particular application of option pricing. We then discuss a cooperative model for the correlations of stock dynamics which has its roots in the field of synergetics, where numerical simulations and comparisons with real data are presented. Finally we present summarized results of an empirical analysis probing the dynamics of actual trading strategy return streams.

Instrumental variable regression with variable constraint and its applications in option pricing and portfolio

In nance and economics, the researchers are interested in the regression with time-dependent data that contain variable constraints. Option pricing and portfolio are two important examples of variable constraint. When such a constraint is appended to a classical regression model, new problems arise naturally, including how to build a constraint-dependent model, how to realize its identi ability of the new model, and how to construct an estimation and test. To solve these fundamental problems, in this study we introduce a remodeling method to treat the variable constraint as a quasi instrumental-variable and further to correct the bias and identify the model. A pro le estimation procedure is suggested to estimate the nonparametric function and parameters in the newly de ned model. The consistency and asymptotic normality are obtained. After showing the nite sample property by a simulation study, a real dataset of stock option is analyzed to further illustrate the new methodology.

Importance sampling and statistical Romberg method

The efficiency of Monte Carlo simulations is significantly improved when implemented with variance reduction methods. Among these methods, we focus on the popular importance sampling technique based on producing a parametric transformation through a shift parameter $\theta$. The optimal choice of $\theta$ is approximated using Robbins–Monro procedures, provided that a nonexplosion condition is satisfied. Otherwise, one can use either a constrained Robbins–Monro algorithm (see, e.g., Arouna ( Monte Carlo Methods Appl. 10 (2004) 1–24) and Lelong ( Statist. Probab. Lett. 78 (2008) 2632–2636)) or a more astute procedure based on an unconstrained approach recently introduced by Lemaire and Pages in ( Ann. Appl. Probab. 20 (2010) 1029–1067). In this article, we develop a new algorithm based on a combination of the statistical Romberg method and the importance sampling technique. The statistical Romberg method introduced by Kebaier in ( Ann. Appl. Probab. 15 (2005) 2681–2705) is known for reducing efficiently the complexity compared to the classical Monte Carlo one. In the setting of discritized diffusions, we prove the almost sure convergence of the constrained and unconstrained versions of the Robbins–Monro routine, towards the optimal shift $\theta^{*}$ that minimizes the variance associated to the statistical Romberg method. Then, we prove a central limit theorem for the new algorithm that we called adaptive statistical Romberg method. Finally, we illustrate by numerical simulation the efficiency of our method through applications in option pricing for the Heston model.

Density approximations for multivariate affine jump-diffusion processes

We introduce closed-form transition density expansions for multivariate affine jump-diffusion processes. The expansions rely on a general approximation theory which we develop in weighted Hilbert spaces for random variables which possess all polynomial moments. We establish parametric conditions which guarantee existence and differentiability of transition densities of affine models and show how they naturally fit into the approximation framework. Empirical applications in option pricing, credit risk, and likelihood inference highlight the usefulness of our expansions. The approximations are extremely fast to evaluate, and they perform very accurately and numerically stable.

Option Pricing Applications of Quadratic Volatility Models

Recently there has been a surge of interest in higher order moment properties of time varying volatility models. Various GARCH-type models have been developed and successfully applied in empirical finance. Moment properties are important because the existence of moments permit verification of how well theoretical models match stylized facts such as fat tails in most financial data. In this paper, we consider various types of random coefficient autoregressive (RCA) models with quadratic generalized autoregressive conditional heteroscedasticity (GARCH) errors and study the moments, mean, variance and kurtosis. We also consider the Black-Scholes model with RCA GARCH volatility and show that these moments can be used to evaluate the call price for European options.

RCA model with quadratic GARCH innovation distribution

Rapid development of time series models addressing volatility has recently been reported in the financial literature. Often the standardized residuals from an RCA (Random coefficient autoregressive) model still has fat tails, thus suggesting using a fat-tailed error distribution instead. Kurtosis of GARCH model plays an important role in option pricing applications with real data. This paper considers some volatility models with quadratic GARCH innovations and derive the kurtosis of the process.

Variationally consistent discretization schemes and numerical algorithms for contact problems

We consider variationally consistent discretization schemes for mechanical contact problems. Most of the results can also be applied to other variational inequalities, such as those for phase transition problems in porous media, for plasticity or for option pricing applications from finance. The starting point is to weakly incorporate the constraint into the setting and to reformulate the inequality in the displacement in terms of a saddle-point problem. Here, the Lagrange multiplier represents the surface forces, and the constraints are restricted to the boundary of the simulation domain. Having a uniform inf-sup bound, one can then establish optimal low-ordera prioriconvergence rates for the discretization error in the primal and dual variables. In addition to the abstract framework of linear saddle-point theory, complementarity terms have to be taken into account. The resulting inequality system is solved by rewriting it equivalently by means of the non-linear complementarity function as a system of equations. Although it is not differentiable in the classical sense, semi-smooth Newton methods, yielding super-linear convergence rates, can be applied and easily implemented in terms of a primal–dual active set strategy. Quite often the solution of contact problems has a low regularity, and the efficiency of the approach can be improved by using adaptive refinement techniques. Different standard types, such as residual- and equilibrated-baseda posteriorierror estimators, can be designed based on the interpretation of the dual variable as Neumann boundary condition. For the fully dynamic setting it is of interest to apply energy-preserving time-integration schemes. However, the differential algebraic character of the system can result in high oscillations if standard methods are applied. A possible remedy is to modify the fully discretized system by a local redistribution of the mass. Numerical results in two and three dimensions illustrate the wide range of possible applications and show the performance of the space discretization scheme, non-linear solver, adaptive refinement process and time integration.

Jumps and stochastic volatility in oil prices: Time series evidence

In this paper we examine the empirical performance of affine jump diffusion models with stochastic volatility in a time series study of crude oil prices. We compare four different models and estimate them using the Markov Chain Monte Carlo method. The support for a stochastic volatility model including jumps in both prices and volatility is strong and the model clearly outperforms the others in terms of a superior fit to data. Our estimation method allows us to obtain a detailed study of oil prices during two periods of extreme market stress included in our sample; the Gulf war and the recent financial crisis. We also address the economic significance of model choice in two option pricing applications. The implied volatilities generated by the different estimated models are compared and we price a real option to develop an oil field. Our findings indicate that model choice can have a material effect on the option values.

Density Approximations for Multivariate Affine Jump-Diffusion Processes

We introduce closed-form transition density expansions for multivariate affine jump-diffusion processes. The expansions rely on a general approximation theory which we develop in weighted Hilbert spaces for random variables which possess all polynomial moments. We establish parametric conditions which guarantee existence and differentiability of transition densities of affine models and show how they naturally fit into the approximation framework. Empirical applications in option pricing, credit risk, and likelihood inference highlight the usefulness of our expansions. The approximations are extremely fast to evaluate, and they perform very accurately and numerically stable.

Market dynamics immediately before and after financial shocks: Quantifying the Omori, productivity, and Bath laws

We study the cascading dynamics immediately before and immediately after 219 market shocks. We define the time of a market shock T{c} to be the time for which the market volatility V(T{c}) has a peak that exceeds a predetermined threshold. The cascade of high volatility "aftershocks" triggered by the "main shock" is quantitatively similar to earthquakes and solar flares, which have been described by three empirical laws-the Omori law, the productivity law, and the Bath law. We analyze the most traded 531 stocks in U.S. markets during the 2 yr period of 2001-2002 at the 1 min time resolution. We find quantitative relations between the main shock magnitude M≡log{10} V(T{c}) and the parameters quantifying the decay of volatility aftershocks as well as the volatility preshocks. We also find that stocks with larger trading activity react more strongly and more quickly to market shocks than stocks with smaller trading activity. Our findings characterize the typical volatility response conditional on M , both at the market and the individual stock scale. We argue that there is potential utility in these three statistical quantitative relations with applications in option pricing and volatility trading.

Jumps and Stochastic Volatility in Oil Prices: Time Series Evidence

In this paper we examine the empirical performance of affine jump diffusion models with stochastic volatility in a time series study of crude oil prices. We compare four different models and estimate them using the Markov Chain Monte Carlo method. The support for a stochastic volatility model including jumps in both prices and volatility is strong and the model clearly outperforms the others in terms of a superior fit to data. Using this model and our estimation methodology we obtain detailed insight into two periods of market stress that are included in our sample; the Gulf war and the recent financial crisis. We also address the economic significance of model choice in two option pricing applications. First we compare the implied volatilities generated by the different estimated models. As a final application we price the real option to develop an oil field. Our findings indicate that model choice can have a material effect on the option values.

On a generalization of the Gerber–Shiu function to path-dependent penalties

The Expected Discounted Penalty Function (EDPF) was introduced in a series of now classical papers ( Gerber and Shiu, 1997, 1998a,b). Motivated by applications in option pricing and risk management, and inspired by recent developments in fluctuation theory for Lévy processes, we study an extended definition of the expected discounted penalty function that takes into account a new ruin-related random variable. In addition to the surplus before ruin and deficit at ruin, we extend the EDPF to include the surplus at the last minimum before ruin. We provide an expression for the generalized EDPF in terms of convolutions in a setting involving a subordinator and a spectrally negative Lévy process. Some expressions for the classical EDPF are recovered as special cases of the generalized EDPF.