Artificial neural networks (ANNs) have recently also been applied to solve partial differential equations (PDEs). In this work, the classical problem of pricing European and American financial options, based on the corresponding PDE formulations, is studied. Instead of using numerical techniques based on finite element or difference methods, we address the problem using ANNs in the context of unsupervised learning. As a result, the ANN learns the option values for all possible underlying stock values at future time points, based on the minimization of a suitable loss function. For the European option, we solve the linear Black-Scholes equation, whereas for the American option, we solve the linear complementarity problem formulation. Two-asset exotic option values are also computed, since ANNs enable the accurate valuation of high-dimensional options. The resulting errors of the ANN approach are assessed by comparing to the analytic option values or to numerical reference solutions (for American options, computed by finite elements).
This paper deals with the numerical approximation of American-style option values governed by partial differential complementarity problems. For a variety of one- and two-asset American options we investigate by ample numerical experiments the temporal convergence behaviour of three modern splitting methods: the explicit payoff approach, the Ikonen-Toivanen approach and the Peaceman-Rachford method. In addition, the temporal accuracy of these splitting methods is compared to that of the penalty approach.
Much research in systemic risk is focused on default contagion. While this demands an understanding of valuation, fewer articles specifically deal with the existence, the uniqueness, and the computation of equilibrium prices in structural models of interconnected financial systems. However, beyond contagion research, these topics are also essential for risk-neutral pricing. In this article, we therefore study and compare valuation algorithms in the standard model of debt and equity cross-ownership which has crystallized in the work of several authors over the past one and a half decades. Since…
In this paper we discuss the basket options valuation for a jump-diffusion model. The underlying asset prices follow some correlated local volatility diffusion processes with systematic jumps. We derive a forward partial integral differential equation (PIDE) for general stochastic processes and use the asymptotic expansion method to approximate the conditional expectation of the stochastic variance associated with the basket value process. The numerical tests show that the suggested method is fast and accurate in comparison with the Monte Carlo and other methods in most cases.
We introduce a new probabilistic method for solving a class of impulse control problems based on their representations as Backward Stochastic Differential Equations (BSDEs for short) with constrained jumps. As an example, our method is used for pricing Swing options. We deal with the jump constraint by a penalization procedure and apply a discrete-time backward scheme to the resulting penalized BSDE with jumps. We study the convergence of this numerical method, with respect to the main approximation parameters: the jump intensity $λ$, the penalization parameter $p > 0$ and the time step. In pa…
This paper adresses the valuation of the Paris barrier options proposed by Yor, Jeanblanc-Picque, and Chesnay (Advances in Applied Probability, 29(1997), 165-184) using the Laplace transform approach. Based on suggestions by Pliska the notion of Paris options is extended such that their valuation is possible at any point during their lifespan. The Laplace transforms derived by Yor et al. are modified when necessary, and their basic analytic properties are discussed.
This paper considers the valuation of exotic path-dependent options in Lévy models, in particular options on the supremum and the infimum of the asset price process. Using the Wiener--Hopf factorization, we derive expressions for the analytically extended characteristic function of the supremum and the infimum of a Lévy process. Combined with general results on Fourier methods for option pricing, we provide formulas for the valuation of one-touch options, lookback options and equity default swaps in Lévy models.
We present a numerical method for the frequent pricing of financial derivatives that depends on a large number of variables. The method is based on the construction of a polynomial basis to interpolate the value function of the problem by means of a hierarchical orthogonalization process that allows to reduce the number of degrees of freedom needed to have an accurate representation of the value function. In the paper we consider, as an example, a GARCH model that depends on eight parameters and show that a very large number of contracts for different maturities and asset and parameters values…
This paper provides a methodology for fast and accurate pricing of the long-dated contracts that arise as the building blocks of insurance and pension fund agreements. It applies the recursive marginal quantization (RMQ) and joint recursive marginal quantization (JRMQ) algorithms outside the framework of traditional risk-neutral methods by pricing options under the real-world probability measure, using the benchmark approach. The benchmark approach is reviewed, and the real-world pricing theorem is presented and applied to various long-dated claims to obtain less expensive prices than suggeste…
Sequential Monte Carlo (SMC) methods have successfully been used in many applications in engineering, statistics and physics. However, these are seldom used in financial option pricing literature and practice. This paper presents SMC method for pricing barrier options with continuous and discrete monitoring of the barrier condition. Under the SMC method, simulated asset values rejected due to barrier condition are re-sampled from asset samples that do not breach the barrier condition improving the efficiency of the option price estimator; while under the standard Monte Carlo many simulated ass…
A common approach to valuing exotic options involves choosing a model and then determining its parameters to fit the volatility surface as closely as possible. We refer to this as the model calibration approach (MCA). A disadvantage of MCA is that some information in the volatility surface is lost during the calibration process and the prices of exotic options will not in general be consistent with those of plain vanilla options. We consider an alternative approach where the structure of the user's preferred model is preserved but points on the volatility are features input to a neural network…
We propose a model to quantify the effect of parameter uncertainty on the option price in the Heston model. More precisely, we present a Hamilton-Jacobi-Bellman framework which allows us to evaluate best and worst case scenarios under an uncertain market price of volatility risk. For the numerical approximation the Hamilton--Jacobi--Bellman equation is reformulated to enable the solution with a finite element method. A case study with butterfly options exhibits how the dependence of Delta on the magnitude of the uncertainty is nonlinear and highly varied across the parameter regime. Keywords: …
Using data on 17 listed public banks from Russia over the period 2008 to 2016, we analyze whether international oil prices affect the bank stability in an oil-dependent country. We posit that a decrease in international oil prices has a negative long-run macroeconomic impact for an oil-exporting country, which further deteriorates the bank financial stability. More specifically, a decrease in international oil prices leads for an oil-exporting country as Russia to a currency depreciation and to a deterioration of the fiscal stance. In addition, given the positive correlation of oil and stock p…
We present a parsimonious stochastic model for valuation of options on the fraction of infected individuals during an epidemic. The underlying stochastic dynamical system is a stochastic differential version of the SIR model of mathematical epidemiology.
Market-Informed Valuation of GMMB Riders with Surrender Options under a Heston Stochastic-Local Volatility Model
Laboratoire de Mathématiques et Modélisation d’Évry (LaMME), CNRS UMR 8071, Université Paris-Saclay Èvry, France（巴黎萨克雷大学）；Dipartimento di Scienze Economiche e Statistiche, Università degli Studi di Udine, Udine, Italy（乌迪内大学）；China Institute for Actuarial Science & School of Insurance, Central University of Finance and Economics, China（中央财经大学保险学院）
We study the principal component analysis based approach introduced by Reisinger & Wittum (2007) and the comonotonic approach considered by Hanbali & Linders (2019) for the approximation of American basket option values via multidimensional partial differential complementarity problems (PDCPs). Both approximation approaches require the solution of just a limited number of low-dimensional PDCPs. It is demonstrated by ample numerical experiments that they define approximations that lie close to each other. Next, an efficient discretisation of the pertinent PDCPs is presented that leads to a favo…
Despite being described as a medium of exchange, cryptocurrencies do not have the typical attributes of a medium of exchange. Consequently, cryptocurrencies are more appropriately described as crypto assets. A common investment attribute shared by the more than 2,500 crypto assets is that they are highly volatile. An investor interested in reducing price volatility of a portfolio of crypto assets can do so by constructing an optimal portfolio through standard optimization techniques that minimize tail risk. Because crypto assets are not backed by any real assets, forming a hedge to reduce the …
Recently, a novel adaptive wave model for financial option pricing has been proposed in the form of adaptive nonlinear Schrödinger (NLS) equation [Ivancevic a], as a high-complexity alternative to the linear Black-Scholes-Merton model [Black-Scholes-Merton]. Its quantum-mechanical basis has been elaborated in [Ivancevic b]. Both the solitary and shock-wave solutions of the nonlinear model, as well as its linear (periodic) quantum simplification are shown to successfully fit the Black-Scholes data, and define the financial Greeks. This initial wave model (called the Ivancevic option pricing mod…
In March 2020, the world was thrown into financial distress. This manifested itself in increased uncertainty in the financial markets. Many interest rates collapsed, and funding spreads surged significantly, which increased due to the market turmoil. In light of these events, it is essential to understand and model Wrong-Way Risk (WWR) in a Funding Valuation Adjustment (FVA) context. WWR may currently be absent from FVA calculations in banks' Valuation Adjustment (xVA) engines. However, in this letter, we demonstrate that WWR effects are non-negligible in FVA modelling from a risk-management p…
We present a new approximation scheme for the price and exercise policy of American options. The scheme is based on Hermite polynomial expansions of the transition density of the underlying asset dynamics and the early exercise premium representation of the American option price. The advantages of the proposed approach are threefold. First, our approach does not require the transition density and characteristic functions of the underlying asset dynamics to be attainable in closed form. Second, our approach is fast and accurate, while the prices and exercise policy can be jointly produced. Thir…
The coupled nonlinear volatility and option pricing model presented recently by Ivancevic is investigated, which generates a leverage effect, i.e., stock volatility is (negatively) correlated to stock returns, and can be regarded as a coupled nonlinear wave alternative of the Black-Scholes option pricing model. In this short report, we analytically propose the two-component financial rogue waves of the coupled nonlinear volatility and option pricing model without an embedded w-learning. Moreover, we exhibit their dynamical behaviors for chosen different parameters. The two-component financial …
In this article, we show how the scaling symmetry of the SABR model can be utilized to efficiently price European options. For special kinds of payoffs, the complexity of the problem is reduced by one dimension. For more generic payoffs, instead of solving the 1+2 dimensional SABR PDE, it is sufficient to solve $N_V$ uncoupled 1+1 dimensional PDE's, where $N_V$ is the number of points used to discretize one dimension. Furthermore, the symmetry argument enables us to obtain prices of multiple options, whose payoffs are related to each other by convolutions, by valuing one of them. The results o…
Option contracts are a type of financial derivative that allow investors to hedge risk and speculate on the variation of an asset's future market price. In short, an option has a particular payout that is based on the market price for an asset on a given date in the future. In 1973, Black and Scholes proposed a valuation model for options that essentially estimates the tail risk of the asset price under the assumption that the price will fluctuate according to geometric Brownian motion. More recently, DeMarzo et al., among others, have proposed more robust valuation schemes, where we can even …
This paper investigates the optimal choices of financial derivatives to complete a financial market in the framework of stochastic volatility (SV) models. We introduce an efficient and accurate simulation-based method, applicable to generalized diffusion models, to approximate the optimal derivatives-based portfolio strategy. We build upon the double optimization approach (i.e. expected utility maximization and risk exposure minimization) proposed in Escobar-Anel et al. (2022); demonstrating that strangle options are the best choices for market completion within equity options. Furthermore, we…
Self-Evolving Multi-Agent Symbolic Discovery for Financial Fundamental Analysis
National University of Singapore（新加坡国立大学）；Asian Institute of Digital Finance（亚洲数字金融研究院）；KTH Royal Institute of Technology（KTH皇家理工学院）；University of Oxford（牛津大学）
In this paper, we study the valuation of American type derivatives in the stochastic volatility model of Barndorff-Nielsen and Shephard (2001). We characterize the value of such derivatives as the unique viscosity solution of an integral-partial differential equation when the payoff function satisfies a Lipschitz condition.
When solving the American options with or without dividends, numerical methods often obtain lower convergence rates if further treatment is not implemented even using high-order schemes. In this article, we present a fast and explicit fourth-order compact scheme for solving the free boundary options. In particular, the early exercise features with the asset option and option sensitivity are computed based on a coupled of nonlinear PDEs with fixed boundaries for which a high order analytical approximation is obtained. Furthermore, we implement a new treatment at the left boundary by introducing…
Pricing of financial derivatives, in particular early exercisable options such as Bermudan options, is an important but heavy numerical task in financial institutions, and its speed-up will provide a large business impact. Recently, applications of quantum computing to financial problems have been started to be investigated. In this paper, we first propose a quantum algorithm for Bermudan option pricing. This method performs the approximation of the continuation value, which is a crucial part of Bermudan option pricing, by Chebyshev interpolation, using the values at interpolation nodes estima…
We prove existence, regularity and a Feynman-Kač representation formula of the strong solution to the free boundary problem arising in the financial problem of the pricing of the American Asian option with arithmetic average.
Subdiffusion is a well established phenomenon in physics. In this paper we apply the subdiffusive dynamics to analyze financial markets. We focus on the financial aspect of time fractional diffusion model with moving boundary i.e. American and barrier option pricing in the subdiffusive Black-Scholes (B-S) model. Two computational methods for valuing American options in the considered model are proposed - the weighted finite difference (FD) and the Longstaff-Schwartz method. In the article it is also shown how to valuate numerically wide range of barrier options using the FD approach.
Autonomous AI Agents for Option Hedging: Enhancing Financial Stability through Shortfall Aware Reinforcement Learning
Cornell Ann S. Bowers College of Computing and Information Science, Cornell University（康奈尔大学安·索斯·博尔斯计算与信息科学学院）；Department of Mathematics, University of Texas at Austin（德克萨斯大学奥斯汀分校数学系）；School of Social Sciences, Nanyang Technological University（南洋理工大学社会科学学院）；Krieger School of Arts and Sciences, Johns Hopkins University（约翰霍普金斯大学克里eger艺术与科学学院）
