Option Pricing Method Research Articles

Estimating the Option Value of managed lanes

The option value of managed lanes (MLs) refers to the value travelers' place on having MLs as a backup option for possible use in the future, even if they are not using the MLs for their current trip. This research used detailed travel data from Katy Freeway MLs and general purpose lanes (GPLs) in Huston to estimate the option value of MLs. Based on these data, this research recommends ML option users to be defined as only those travelers who used MLs at least once a year as opposed to all travelers who could have used the MLs (even those who never used MLs).This research used the Small-Rosen log sum method (1981) and the Black-Scholes option pricing method (1973) to estimate the option value of MLs. The log sum method frequently provided an unrealistically high option value based on times when the option value should logically be close to $0. Thus, the Black-Scholes method was recommended. The option value of the MLs was found to be similar to the value of travel time savings from the MLs for the ML option users. Thus, the option value of MLs is an important component of the total value of MLs.

Efficient BEM-Based Algorithm for Pricing Floating Strike Asian Barrier Options (with MATLAB® Code)

This paper aims to illustrate how SABO (Semi-Analytical method for Barrier Option pricing) is easily applicable for pricing floating strike Asian barrier options with a continuous geometric average. Recently, this method has been applied in the Black–Scholes framework to European vanilla barrier options with constant and time-dependent parameters or barriers and to geometric Asian barrier options with a fixed strike price. The greater efficiency of SABO with respect to classical finite difference methods is clearly evident in numerical simulations. For the first time, a user-friendly MATLAB® code is made available here.

A reduced PDE method for European option pricing under multi-scale, multi-factor stochastic volatility

The number of tailor-made hybrid structured products has risen more prominently to fit each investor’s preferences and requirements as they become more diversified. The structured products entail synthetic derivatives such as combinations of bonds and/or stocks conditional on how they are backed up by underlying securities, stochastic volatility, stochastic interest rates or exchanges rates. The complexity of these multi-asset structures yields lots of difficulties of pricing the products. Because of the complexity, Monte-Carlo simulation is a possible choice to price them but it may not produce stable Greeks leading to a trouble in hedging against risks. In this light, it is desirable to use partial differential equations with relevant analytic and numerical techniques. Even if the partial differential equation method would generate stable security prices and Greeks for single asset options, however, it may result in the curse of dimensionality when pricing multi-asset derivatives. In this study, we make the best use of multi-scale nature of stochastic volatility to lift the curse of dimensionality for up to three asset cases. Also, we present a transformation formula by which the pricing group parameters required for the multi-asset options in illiquid market can be calculated from the underlying market parameters.

Turbocharging Monte Carlo pricing for the rough Bergomi model

The rough Bergomi model, introduced by Bayer et al. [Quant. Finance, 2016, 16(6), 887–904], is one of the recent rough volatility models that are consistent with the stylised fact of implied volatility surfaces being essentially time-invariant, and are able to capture the term structure of skew observed in equity markets. In the absence of analytical European option pricing methods for the model, we focus on reducing the runtime-adjusted variance of Monte Carlo implied volatilities, thereby contributing to the model’s calibration by simulation. We employ a novel composition of variance reduction methods, immediately applicable to any conditionally log-normal stochastic volatility model. Assuming one targets implied volatility estimates with a given degree of confidence, thus calibration RMSE, the results we demonstrate equate to significant runtime reductions—roughly 20 times on average, across different correlation regimes.

Sum of all Black–Scholes–Merton models: An efficient pricing method for spread, basket, and Asian options

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 illustrated through various numerical experiments.

Hybrid intelligent algorithm based option pricing method

This study investigates the relative rate of price discovery in Chinese Shanghai Stock Exchange (SSE) 50 spot index and SSE 50 ETF options, proposing a put–call parity (PCP) approach to recover the spot prices embedded in the options premiums. The PCP approach offers the benefits of reducing model risk and alleviating the burden of volatility estimation. Empirical results reveal that the contribution of SSE 50 ETF options to price discovery exceeds the contributions of SSE 50 index and SSE 50 ETF. However, ETF contributes larger price discovery than ETF options in the high volatility which is opposite from the leverage hypothesis.

Business Model Innovation or Imitation? Strategy Study Based on Real Option Game Theory

The business model is constantly changing in the enterprise life cycle. A large number of studies show that business model innovation enables enterprises to improve their processes and operation modes, reduce costs, create new markets, improve market efficiency, gain competitive advantages, and achieve higher value and business performance. However, since returns on business model innovation are uncertain and business model innovation can easily be imitated by competitors, it is more difficult to evaluate the value of business model innovation. This paper solves the problem of business model innovation or imitation strategy choice under uncertain conditions by constructing a real option game model. The model is based on the assumption of a duopoly market, namely when enterprise A carries out business model innovation, enterprise B can choose to innovate alone or imitate in the middle period. The results of two-stage binary tree analysis show that the value of business model innovation or imitation is only related to enterprise market share α and base period multiple I（a ratio of base period cash flow to innovation investment）. When the base period multiple is very high and the enterprise shares are not much different, business model innovation is the best choice for both the two companies. When the base period multiple is medium and the enterprise shares are not much different, it is possible that there is a combination of equilibrium strategies that the small enterprise participates in business model innovation and the large enterprise business model imitation. When the base period multiple is not low and the enterprise shares are much different, there is a ‘pig game’ that large enterprise participates in business model innovation, and the small enterprise business model imitation; when the base period multiple is low, both small and large enterprises do not have the motivation for innovation. Therefore, business model innovation or imitation needs to take current market shares and initial returns of a project into account. This paper expands the application of real option, which is of great significance to the construction of business model innovation value evaluation model. In the field of business model innovation, theoretical concept construction, measurement scale development and empirical research have been basically completed, which focus on the impact of business model innovation on business performance. Few studies have analyzed the value of business model innovation from the perspectives of mathematical models and game theory, so it is a relatively new perspective to evaluate the value of innovation and imitation by using the real option game theory. Business model innovation is a systematic change, which is more complex than common project investment. This paper highlights consideration of uncertainty and opponent decision-making, which is more in line with the actual situation. On the other hand, the existing research on real option mostly adopts Black-Scholes option pricing method. However, the returns on business model innovation are sustained, belonging to American-style option, so we adopt the binary tree pricing method to take the revenues during the project period into account, providing a directional basis for future research and development. In the existing literature, the NPV method is commonly used in enterprise investment analysis and business model profit forecasting. The real option game has not been widely used, and more researchers are required to enrich the method for more rational use in practice. Firstly, the problem becomes more complex and closer to reality when considering multiple actors and multiple policy choices. When the binary option is exceeded, the binary tree method will not be applicable. Secondly, another complicated task is about the measurement of uncertainty in real option. There are many factors that have risks. It is a future research direction for scholars to unify them into one variable or independent comprehensive analysis of multiple variables.

A second-order discretization with Malliavin weight and Quasi-Monte Carlo method for option pricing

This paper shows a second-order discretization scheme for expectations of stochastic differential equations. We introduce a smart Malliavin weight which is given by a sum of simple polynomials of Brownian motions as an improvement of the scheme of Yamada [J. Comput. Appl. Math., 2017, 321, 427–447]. A new quasi-Monte Carlo simulation is proposed to obtain an efficient option pricing scheme. Numerical examples for the SABR model are shown to illustrate the validity of the scheme.

COS method for option pricing under a regime-switching model with time-changed Lévy processes

We extend the regime-switching model to the rich class of time-changed Lévy processes and use the Fourier cosine expansion (COS) method to price several options under the resulting models. The extension of the COS method to price under the regime-switching model is not straightforward because it requires the evaluation of the characteristic function which is based on a matrix exponentiation which is not an easy task. For a two-state economy, we give an analytical expression for computing this matrix exponential, and for more than two states, we use the Carathéodory–Fejér approximation to find the option prices efficiently. In the new framework developed here, it is possible to allow switches not only in the model parameters as is commonly done in literature, but we can also completely switch among various popular financial models under different regimes without any additional computational cost. Calibration of the different regime-switching models with real market data shows that the best models are the regime-switching time-changed Lévy models. As expected by the error analysis, the COS method converges exponentially and thus outperforms all other numerical methods that have been proposed so far.

Multinomial method for option pricing under Variance Gamma

ABSTRACTThis paper presents a multinomial method for option pricing when the underlying asset follows an exponential Variance Gamma (VG) process. The continuous time VG process is approximated by a continuous time process with the same first four cumulants and then discretized in time and space. This approach is particularly convenient for pricing American and Bermudan options, which can be exercised before the expiration date. Numerical computations of European and American options are presented and compared with results obtained with finite differences method and with the Black–Scholes prices.

The least squares method for option pricing revisited

It is shown that the the popular least squares method of option pricing converges even if the underlying is non-Markovian, the pay-offs are path dependent and with a very flexible setup for approximation of conditional expectations. The main benefit is the increase of freedom in creating specific implementations of the method, but depending on the extent of adopted generality and complexity, the method may become very demanding computationally. It is argued, however, that in many practical applications even modest but computationally viable extensions of standard linear regression may produce satisfactory results from the empirical point of view. This claim is illustrated with several empirical examples.

QLBS: Q-Learner in the Black-Scholes (-Merton) Worlds

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 the Markowitz portfolio theory. Using Q-Learning and related methods, once created in a parametric setting, the model is able to go model-free and learn to price and hedge an option directly from data generated from a dynamic replicating portfolio which is rebalanced at discrete times. If the world is according to BSM, our risk-averse Q-Learner converges, given enough training data, to the true BSM price and hedge ratio of the option in the continuous time limit, even if hedges applied at the stage of data generation are completely random (i.e. it can learn the BSM model itself, too!), because Q-Learning is an off-policy algorithm. If the world is different from a BSM world, the Q-Learner will find it out as well, because Q-Learning is a model-free algorithm. For finite time steps, the Q-Learner is able to efficiently calculate both the optimal hedge and optimal price for the option directly from trading data, and without an explicit model of the world. This suggests that RL may provide efficient data-driven and model-free methods for optimal pricing and hedging of options, once we depart from the academic continuous-time limit, and vice versa, option pricing methods developed in Mathematical Finance may be viewed as special cases of model-based Reinforcement Learning. Further, due to simplicity and tractability of our model which only needs basic linear algebra (plus Monte Carlo simulation, if we work with synthetic data), and its close relation to the original BSM model, we suggest that our model could be used for benchmarking of different RL algorithms for financial trading applications. A supplement to this paper can be found here: http://ssrn.com/abstract=3090608.

Polynomial Jump-Diffusion Models

We develop a comprehensive mathematical framework for polynomial jump-diffusions in a semimartingale context, which nest affine jump-diffusions and have broad applications in finance. We show that the polynomial property is preserved under polynomial transformations and Levy time change. We present a generic method for option pricing based on moment expansions. As an application, we introduce a large class of novel financial asset pricing models with excess log returns that are conditional Levy based on polynomial jump-diffusions.

Analytical and numerical studies on the second-order asymptotic expansion method for European option pricing under two-factor stochastic volatilities

ABSTRACTThe celebrated Black–Scholes model made the assumption of constant volatility but empirical studies on implied volatility and asset dynamics motivated the use of stochastic volatilities. Christoffersen in 2009 showed that multi-factor stochastic volatilities models capture the asset dynamics more realistically. Fouque in 2012 used it to price European options. In 2013, Chiarella and Ziveyi considered Christoffersen’s ideas and introduced an asset dynamics where the two volatilities of the Heston type act separately and independently on the asset price, and using Fourier transform for the asset price process and double Laplace transform for the two volatilities processes, solved a pricing problem for American options. This paper considers the Chiarella and Ziveyi model and parameterizes it so that the volatilities revert to the long-run-mean with reversion rates that mimic fast (for example daily) and slow (for example seasonal) random effects. Applying asymptotic expansion method presented by Fouque in 2012, we make an extensive and detailed derivation of the approximation prices for European options. We also present numerical studies on the behavior and accuracy of our first- and second-order asymptotic expansion formulas.

A Dimension Reduction Shannon-Wavelet Based Method for Option Pricing

We present a robust and highly efficient dimension reduction Shannon-wavelet method for computing European option prices and hedging parameters under a general jump-diffusion model with square-root stochastic variance and multi-factor Gaussian interest rates. Within a dimension reduction framework, the option price can be expressed as a two-dimensional integral that involves only (i) the value of the variance at the terminal time, and (ii) the time-integrated variance process conditional on this value. A Shannon wavelet inverse Fourier technique is developed to approximate the conditional density of the time-integrated variance process. Furthermore, thanks to the excellent approximation properties of Shannon wavelets, the overall pricing procedure is reduced to the evaluation of just a single integral that involves only the density of the terminal variance value. This single integral can be accurately evaluated, since the density of the variance at the terminal time is known in closed-form. We develop sharp approximation error bounds for the option price and hedging parameters. Numerical experiments confirm the robustness and impressive efficiency of the method.

A Shannon Wavelet Method for Pricing Foreign Exchange Options under the Heston Multi-Factor CIR Model

We present a robust and highly efficient Shannon wavelet pricing method for plain-vanilla foreign exchange European options under the jump-extended Heston model with multi-factor CIR interest rate dynamics. Under a Monte Carlo and partial differential equation hybrid computational framework, the option price can be expressed as an expectation, conditional on the variance factor, of a convolution product that involves the densities of the time-integrated domestic and foreign multi-factor CIR interest rate processes. We propose an efficient treatment to this convolution product that effectively results in a significant dimension reduction, from two multi-factor interest rate processes to only a one-factor process. By means of a state-of-the-art Shannon wavelet inverse Fourier technique, the resulting convolution product is approximated analytically and the conditional expectation can be computed very efficiently. We develop sharp approximation error bounds for the option price and hedging parameters. Numerical experiments confirm the robustness and impressive efficiency of the method.

A Simple and Efficient Two-Factor Willow Tree Method for Convertible Bond Pricing with Stochastic Interest Rate and Default Risk

Most numerical methods for option pricing (lattice methods, Monte Carlo simulation) focus on the market’s short-run dynamics, in which instantaneous returns cumulate to a probability distribution at maturity, as the Black–Scholes model does. If the risk-neutral density is available, it is possible to jump immediately to maturity, but this only works for options without path-dependence and for a small number of returns processes. The willow tree method offers a solution between these two cases. The option’s life is split into a small number of discrete chunks, and the price dimension at each of these time steps is represented by a finite and relatively small set of possible values. Each node allows a transition to every price node at the next step; thus, the “tree” resembles a willow, with many long branches connecting the sets of prices separated by discrete time intervals. For the problems it can handle, the willow tree approach has been shown to be accurate and much faster than procedures that require modeling an asymptotically infinite number of instantaneous transitions. In this article, Lu and Xu show how to extend a willow tree to two stochastic factors, an underlying stock price and the interest rate, to apply the model to pricing convertible bonds. The procedure is amenable to many different returns processes. In a numerical illustration, the authors show that the two-factor willow tree is as accurate as a binomial or a Monte Carlo approach but requires much less computing time.

A Hybrid Monte Carlo and Finite Difference Method for Option Pricing

We propose an accurate, efficient, and robust hybrid finite difference method, with a Monte Carlo boundary condition, for solving the Black–Scholes equations. The proposed method uses a far-field boundary value obtained from a Monte Carlo simulation, and can be applied to problems with non-linear payoffs at the boundary location. Numerical tests on power, powered, and two-asset European call option pricing problems are presented. Through these numerical simulations, we show that the proposed boundary treatment yields better accuracy and robustness than the most commonly used linear boundary condition. Furthermore, the proposed hybrid method is general, which means it can be applied to other types of option pricing problems. In particular, the proposed Monte Carlo boundary condition algorithm can be implemented easily in the code of the existing finite difference method, with a small modification.

An Actuarial Mathematics Approach to Option Pricing

We derive and test a new option pricing method based on statistics. We show how such a method allows to a) analytically price options with risk measures - such as Value-at-Risk or Expected Shortfall - on assets with stochastic volatility; and b) build several new structural models for the credit spread. We discuss how the method entails a new put-call parity relation, and how the option price is affected by the issuer’s credit spread. Finally, we discuss several extensions to the formalism developed here, such as to assets with interdependencies, and to any model for the asset’s returns distribution.

Black-Scholes Equation with the Variable Risk-free Interest Rate

As for B-S model with constant coefficients, it can be obtained analytical solution directly by converting into parabolic partial differential equation with constant coefficients, however, the research on the explicit solution of the parabolic partial differential equation under the condition of time-varying is not yet mature. The risk-free interest rate as the discount rate is not always fixed, which is changeable. Consequently, the real option pricing method palys an important role both in theoretical and practical. There are two different methods which dealing with the B-S equation under the condition of the risk-free interest rate is time-varying, one of them is the method of infinite series expansion which is put forward by Kazemi. Another is the method based on decomposition of the path which is put forward by Yun Feng. The two methods both are the summary and improvement of the predecessors' experience. In this paper, we studied the Black-Scholes equation under certain assumptions, we obtain the solution to the problem. And we have proved the compare principle by using advantage function method. Under the compare principle, which is the innovation of this paper, we can find the solution of the Black-Scholes equation with the variable risk-free interest rate by combining with the idea of stage by stage.