We discuss in this note applications of the Multidimensional Positive Definite Advection Transport Algorithm (MPDATA) to numerical solutions of partial differential equations arising from stochastic models in quantitative finance. In particular, we develop a framework for solving Black-Scholes-type equations by first transforming them into advection-diffusion problems, and numerically integrating using an iterative explicit finite-difference approach, in which the Fickian term is represented as an additional advective term. We discuss the correspondence between transport phenomena and financial models, uncovering the possibility of expressing the no-arbitrage principle as a conservation law. We depict second-order accuracy in time and space of the embraced numerical scheme. This is done in a convergence analysis comparing MPDATA numerical solutions with classic Black-Scholes analytical formulae for the valuation of European options. We demonstrate in addition a way of applying MPDATA to solve the free boundary problem (leading to a linear complementarity problem) for the valuation of American options. We finally comment on the potential the MPDATA framework has with respect to being applied in tandem with more complex models typically used in quantitive finance.
In this paper we investigate a nonlinear generalization of the Black-Scholes equation for pricing American style call options in which the volatility term may depend on the underlying asset price and the Gamma of the option. We propose a numerical method for pricing American style call options by means of transformation of the free boundary problem for a nonlinear Black-Scholes equation into the so-called Gamma variational inequality with the new variable depending on the Gamma of the option. We apply a modified projective successive over relaxation method in order to construct an effective nu…
his paper presents finite element methods for solving numerically the Risk-Adjusted Pricing Methodology (RAPM) Black-Scholes model for option pricing with transaction costs. Spatial finite element models based on P1 and/or P2 elements are formulated using some group finite elements and numerical quadrature to handle the nonlinear term, in combination with a Crank-Nicolson-type temporal scheme. The temporal scheme is implemented using the Rannacher approach. Spatial-temporal mesh-size ratios are observed for controlling the stability of our method. Our results compare favorably with the finite …
Contrary to the common view that exact pricing is prohibitive owing to the curse of dimensionality, this study proposes an efficient and unified method for pricing options under multivariate Black-Scholes-Merton (BSM) models, such as the basket, spread, and Asian options. The option price is expressed as a quadrature integration of analytic multi-asset BSM prices under a single Brownian motion. Then the state space is rotated in such a way that the quadrature requires much coarser nodes than it would otherwise or low varying dimensions are reduced. The accuracy and efficiency of the method is …
In this paper we present a simple, but new, approximation methodology for pricing a call option in a Black \& Scholes market characterized by stochastic interest rates. The method, based on a straightforward Gaussian moment matching technique applied to a conditional Black \& Scholes formula, is quite general and it applies to various models, whether affine or not. To check its accuracy and computational time, we implement it for the CIR interest rate model correlated with the underlying, using the Monte Carlo simulations as a benchmark. The method's performance turns out to be quite remarkabl…
Valuing Guaranteed Minimum Withdrawal Benefit (GMWB) has attracted significant attention from both the academic field and real world financial markets. As remarked by Yang and Dai, the Black and Scholes framework seems to be inappropriate for such a long maturity products. Also Chen Vetzal and Forsyth in showed that the price of these products is very sensitive to interest rate and volatility parameters. We propose here to use a stochastic volatility model (Heston model) and a Black Scholes model with stochastic interest rate (Hull White model). For this purpose we present four numerical metho…
We analyze and calculate the early exercise boundary for a class of stationary generalized Black-Scholes equations in which the volatility function depends on the second derivative of the option price itself. A motivation for studying the nonlinear Black Scholes equation with a nonlinear volatility arises from option pricing models including, e.g., non-zero transaction costs, investors preferences, feedback and illiquid markets effects and risk from unprotected portfolio. We present a method how to transform the problem of American style of perpetual put options into a solution of an ordinary …
Investors always want to know about the profit and the risk that they will be get before buying some assets. Our main focus is getting the profit and the probability of getting that profit using the differential evolution algorithm for vanilla option pricing on data before and during COVID-19 pandemic. Therefore, we model the pricing of an option using a bi-objective optimization problem using data before and during COVID-19 pandemic for one year expiration date. We change this problem into an optimization problem using adaptive weighted sum method. We use metaheuristics algorithm like Differe…
We present a simple, fast, and accurate method for pricing a variety of discretely monitored options in the Black-Scholes framework, including autocallable structured products, single and double barrier options, and Bermudan options. The method is based on a quadrature technique, and it employs only elementary calculations and a fixed one-dimensional uniform grid. The convergence rate is $O(1/N^4)$ and the complexity is $O(MN\log N)$, where $N$ is the number of grid points and $M$ is the number of observation dates.
In the standard Black-Scholes-Merton framework, dividends are represented as a continuous dividend yield and the pricing of Vanilla options on a stock is achieved through the well-known Black-Scholes formula. In reality however, stocks pay a discrete fixed cash dividend at each dividend ex-date. This leads to the so-called piecewise lognormal model, where the asset jumps from a fixed known amount at each dividend date. There is however no exact closed-form formula for the pricing of Vanilla options under this model. Approximations must be used. While there exists many approximations taylored t…
This paper presents a discrete-time option pricing model that is rooted in Reinforcement Learning (RL), and more specifically in the famous Q-Learning method of RL. We construct a risk-adjusted Markov Decision Process for a discrete-time version of the classical Black-Scholes-Merton (BSM) model, where the option price is an optimal Q-function, while the optimal hedge is a second argument of this optimal Q-function, so that both the price and hedge are parts of the same formula. Pricing is done by learning to dynamically optimize risk-adjusted returns for an option replicating portfolio, as in …
Quantum Speedups for Derivative Pricing Beyond Black-Scholes
Global Technology Applied Research, JPMorganChase（摩根大通全球应用技术研究）；Center for Theoretical Physics – a Leinweber Institute, MIT（麻省理工学院莱因韦伯理论物理中心）；Quantitative Trading & Research, JPMorganChase（摩根大通量化交易与研究）
The author presents alternatives to the Black-Scholes european call option pricing model by incorporating different transaction cost structures in the replicating strategy. In particular, an exponentially decreasing structure is proposed and developed.
Trading option strangles is a highly popular strategy often used by market participants to mitigate volatility risks in their portfolios. In this paper we propose a measure of the relative value of a delta-Symmetric Strangle and compute it under the standard Black-Scholes option pricing model. This new measure accounts for the price of the strangle, relative to the Present Value of the spread between the two strikes, all expressed, after a natural re-parameterization, in terms of delta and a volatility parameter. We show that under the standard BS option pricing model, this measure of relative…
Financial markets based on Lévy processes are typically incomplete and option prices depend on risk attitudes of individual agents. In this context, the notion of utility indifference price has gained popularity in the academic circles. Although theoretically very appealing, this pricing method remains difficult to apply in practice, due to the high computational cost of solving the nonlinear partial integro-differential equation associated to the indifference price. In this work, we develop closed form approximations to exponential utility indifference prices in exponential Lévy models. To th…
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.
We present a novel method for the numerical pricing of American options based on Monte Carlo simulation and the optimization of exercise strategies. Previous solutions to this problem either explicitly or implicitly determine so-called optimal exercise regions, which consist of points in time and space at which a given option is exercised. In contrast, our method determines the exercise rates of randomized exercise strategies. We show that the supremum of the corresponding stochastic optimization problem provides the correct option price. By integrating analytically over the random exercise de…
In this paper we focus on the subdiffusive Black Scholes model. The main part of our work consists of the finite difference method as a numerical approach to the option pricing in the considered model. We derive the governing fractional differential equation and the related weighted numerical scheme being a generalization of the classical Crank-Nicolson scheme. The proposed method has $2-α$ order of accuracy with respect to time where $α\in(0,1)$ is the subdiffusion parameter, and $2$ with respect to space. Further, we provide the stability and convergence analysis. Finally, we present some nu…
This paper presents the Runge-Kutta-Legendre finite difference scheme, allowing for an additional shift in its polynomial representation. A short presentation of the stability region, comparatively to the Runge-Kutta-Chebyshev scheme follows. We then explore the problem of pricing American options with the Runge-Kutta-Legendre scheme under the one factor Black-Scholes and the two factor Heston stochastic volatility models, as well as the pricing of butterfly spread and digital options under the uncertain volatility model, where a Hamilton-Jacobi-Bellman partial differential equation needs to b…
In this Article, a fast numerical numerical algorithm for pricing discrete double barrier option is presented. According to Black-Scholes model, the price of option in each monitoring date can be evaluated by a recursive formula upon the heat equation solution. These recursive solutions are approximated by using Legendre multiwavelets as orthonormal basis functions and expressed in operational matrix form. The most important feature of this method is that its CPU time is nearly invariant when monitoring dates increase. Besides, the rate of convergence of presented algorithm was obtained. The n…
Transformation methods for evaluating approximations to the optimal exercise boundary for linear and nonlinear Black-Scholes equations
Comenius University（布拉迪斯拉发大学）；Department of Applied Mathematics and Statistics, Faculty of Mathematics, Physics & Informatics, Comenius University（布拉迪斯拉发大学数学、物理与信息学院应用数学与统计系）
