Classical Option Pricing Research Articles

Alternatives to classical option pricing

Alternatives to classical option pricing

Pricing cryptocurrency options with machine learning regression for handling market volatility

Pricing cryptocurrency options, crucial for risk management and market stabilization, presents unique challenges due to specific underlying dynamics like the inversion of the leverage effect. Classical option pricing models like Black–Scholes and Heston struggle to address these dynamics due to their set of assumptions. This study introduces machine learning models for options pricing, specifically regression-tree methods. A data-driven machine learning model can incorporate high-frequency volatility estimators into the input set to enhance pricing accuracy. By integrating these estimators, machine learning models can capture the complex dynamics of cryptocurrency markets more effectively than classical pricing approaches. The comparative analysis reveals that equity options are easier to price, clearly indicating inefficiencies in the cryptocurrency option market, which confirms the challenges in achieving accurate pricing. Our results highlight the effectiveness of machine learning models in adapting to the unique characteristics of emerging asset classes, suggesting a shift towards more data-oriented pricing methodologies

A Review of the Option Pricing Model and Further Development

The Black Scholes model and binomial tree model have been the main research objects of scholars in the past fifty years. This article summarizes the optimization process of these two classic option pricing models to understand the different directions of optimizing the models and to provide ideas for future model improvement. Improvements from scholars to the Black-Scholes model mainly focus on the basic model and the relevant variables involved in option pricing, while the optimization of the binomial tree model focuses on the reduction of pricing errors as well as the improvement of model fitting speed. Empirical studies of option pricing models have shown that improved models while enhancing the accuracy of pricing in various financial markets, unavoidably increase computational complexity and reduce efficiency. This article also demonstrates the future integration of financial derivatives pricing models in multiple fields by describing the application of real options in the agriculture, technology, and biology industries.

Research on the Development of Implied Volatility in Option Pricing

Option, as an important financial derivative, has received more and more attention from investors in recent years. This paper includes three parts, the first part introduces the concept of options, including the birth and history of options, the trading of options, and different types of options. Then it discusses the history of option pricing and two typical and classic option pricing models and then introduces implied volatility. The second part of this paper introduces the specific development of the option pricing models, the correction process of option pricing models, and volatility models. Different types of volatility, and their comparisons are included. And this paper also discusses the two functions of implied volatility, one is predicting the future and the other is for risk management and portfolio management. The third part concludes the first two-part and discusses the recent progress of option pricing and its volatility.

Conversion risk on 19th century French consols and embedded options: A simple exercise

This paper investigates the conversion risk exposure of French government consols on the Paris Stock Exchange market during the 19th century. Using the classic option pricing framework, our results indicate that market investors were able to detect the presence of an embedded conversion option and price it quite accurately almost two centuries ago.

Tighter Robust Upper Bounds for Options via No-Regret Learning

Classic option pricing models, such as the Black-Scholes formula, often depend on some rigid assumptions on the dynamics of the underlying asset prices. These assumptions are inevitably violated in practice and thus induce the model risk. To mitigate this, robust option pricing that only requires the no-arbitrage principle has attracted a great deal of attention among researchers. In this paper, we give new robust upper bounds for option prices based on a novel η-momentum trading strategy. Our bounds for European options are tighter for most common moneyness, volatility, and expiration date setups than those presented in the existing literature. Our bounds for average strike Asian options are the first closed-form robust upper bounds for those options. Numerical simulations demonstrate that our bounds significantly outperform the benchmarks for both European and Asian options.

Analysis about the Black-Scholes asset price under the regime-switching framework

Assuming that the macroeconomic environment can be transformed into a two-district system, that is, the path of financial asset prices is uncertain, we track and study the motion of stocks and other asset price process under the conditional Black-Scholes model, and give the economical explanation of the mathematical formula. Further, we derive and analyze an option pricing formula for the Black-Scholes asset model under the condition that the risk-free interest rate is regime-switching too. The method in this article is applied to model the log rate of return of the Tencent stock in a two-district market environment. And the obtained parameter values are used to calculate the option price. In narrowing the gap with actual option prices, our method outperforms the classical option pricing model point by point. Compared with the general and pure mathematical model derived work and the empirical study work, our study does more work on the economic characteristics analysis and interpretation of the mathematical models, and plays a certain role in linking the results of mathematical models with empirical research.

Option pricing generators

We characterize a class of option pricing models by their algebraic structure. Option prices are monoids, that is operators endowed with the commutativity and associativity property and an identity element. If the price of the underlying asset is bounded, the operator corresponds to the concept of t-conorm, while if it is defined on the positive real line the operator is a pseudo-addition. These operators have the same no-arbitrage properties as the classical option pricing models, but are also associative. Each model in this class is characterized by a univariate increasing function that is defined the generator of the model. The generator encodes a synthetic representation of the probability structure of the underlying asset. We provide no arbitrage conditions for the generators and practical guidelines to construct them.

Valuation of bid and ask prices for European options under mixed fractional Brownian motion

<abstract> In this paper, we study the valuation of the bid and ask prices for European options under the mixed fractional Brownian motion with Hurst index $ H &gt; 3/4 $, which is able to capture the long range dependence of the underlying asset returns in real markets. As we know, the classical option pricing theories are usually built on the law of one price, while ignoring the impact of market liquidity on bid-ask spreads. The theory of conic finance replaces the law of one price by the law of two prices, allowing for market participants sell to the market at the bid price and buy from the market at the higher ask price. Within the framework of conic finance, we then derive the explicit formulas for the bid and ask prices of European call and put options by using WANG-transform as a distortion function. Moreover, numerical experiment is performed to illustrate the effects of the Hurst index and market liquidity level on bid and ask prices. </abstract>

Generalised Geometric Brownian Motion: Theory and Applications to Option Pricing.

Classical option pricing schemes assume that the value of a financial asset follows a geometric Brownian motion (GBM). However, a growing body of studies suggest that a simple GBM trajectory is not an adequate representation for asset dynamics, due to irregularities found when comparing its properties with empirical distributions. As a solution, we investigate a generalisation of GBM where the introduction of a memory kernel critically determines the behaviour of the stochastic process. We find the general expressions for the moments, log-moments, and the expectation of the periodic log returns, and then obtain the corresponding probability density functions using the subordination approach. Particularly, we consider subdiffusive GBM (sGBM), tempered sGBM, a mix of GBM and sGBM, and a mix of sGBMs. We utilise the resulting generalised GBM (gGBM) in order to examine the empirical performance of a selected group of kernels in the pricing of European call options. Our results indicate that the performance of a kernel ultimately depends on the maturity of the option and its moneyness.

Classical Option Pricing and Some Steps Further

This paper considers the asset price p as relations C=pV between the value C and the volume V of the executed transactions and studies the consequences of this definition for the option pricing equations. We show that the classical BSM model implicitly assumes that value C and volume V of transactions follow identical Brownian processes. Violation of this identity leads to 2-dimensional BSM-like equation with two constant volatilities. We show that agents expectations those approve execution of transactions can further increase the dimension of the BSM model. We study the case when agents expectations may depend on the option price data and show that such assumption can lead to the nonlinear BSM-like equations. We reconsider the Heston stochastic volatility model for the price determined by the value and the volume and derive 3-dimensional BSM-like model with stochastic value volatility and constant volume volatility. Variety of the BSM-like equations states the problem of reasonable balance between the accuracy and the complexity of the option pricing equations.

An analytical approximation approach for pricing European options in a two-price economy

Classical option pricing theories are usually built on the law of one price, while ignoring the impact of market liquidity on bid-ask spreads. The theory of conic finance replaces the law of one price by the law of two prices, allowing for market participants sell to the market at the bid price and buy from the market at the higher ask price. In this paper, we present a numerical method to calculate the bid and ask prices for the European options. The numerical method is based on the combination of Fourier cosine approximations and numerical integration, which provides an efficient and fast way to compute the bid and ask prices and calibrate the model parameters in real market. Numerical experiments demonstrate the accuracy of our presented numerical method by comparing with Monte Carlo simulation. To better illustrate the practicability of our presented numerical method, we consider the European options written on SSE 50 ETF traded at the Shanghai Stock Exchange.

Quantum option pricing and data analysis

The paper proposes to treat financial models using techniques of quantum mechanics. The methodology relies on the Dirac matrix formalism and the Feynman path integral approach. This leads us to reexamine in this framework the classical option pricing models of Cox-Ross-Rubinstein and Black-Scholes. Moreover, financial data are classified with respect to the spectrum of a certain observable and then analyzed to identify price jumps using supervised machine learning tools.

Pricing discrete barrier options under jump-diffusion model with liquidity risk

Classical option pricing theories are usually built on the paradigm of competitive and frictionless markets, while ignoring the impact of market liquidity on underlying asset prices. In this paper, the importance of liquidity risk on discrete barrier option pricing is analyzed. First, we propose a new model for describing the asset price dynamics in the presence of jumps and liquidity risks, which is able to capture empirically observed patterns. Based on the COS method, we then derive the analytical approximation formulas for the prices of the discrete barrier options. Numerical experiments demonstrate the accuracy of our proposed pricing model by comparing the analytical approximation solutions with Monte Carlo simulation. Finally, empirically studies are carried out to show the superiority of our model based on SSE 50 ETF options. The numerical and empirical results support our idea of introducing liquidity risk and jumps into the underlying asset price dynamics.

Analytical valuation for geometric Asian options in illiquid markets

Classical option pricing theories are usually built on the paradigm of competitive and frictionless markets, while ignoring the impact of market liquidity on asset prices. In this paper, we study the pricing problems of the continuously monitored fixed and floating strike Asian options in imperfectly liquid markets. First, we assume that the dynamics of the underlying asset price follow a liquidity-adjusted Black–Scholes model. And then we derive the partial differential equation for the continuous fixed and floating strike geometric Asian options based on the delta-hedging strategy. Meanwhile, we also present the PDE governing the price of the continuous fixed and floating strike arithmetic Asian options. Second, the analytical pricing formulas for the continuous fixed and floating strike geometric Asian call options are derived by using the PDE method. Besides, we demonstrate the put–call parity relations for the continuous fixed and floating strike geometric Asian options. Finally, numerical experiments are performed to illustrate the accuracy and efficiency of the proposed liquidity-adjusted option pricing model through comparing the analytical solution with Monte Carlo simulation. Furthermore, we investigate the sensitivity of the continuous arithmetic and geometric Asian option prices to the liquidity factors. The numerical results support our idea of introducing market liquidity effect into option pricing framework.

Analysis of a jump-diffusion option pricing model with serially correlated jump sizes

ABSTRACTThis paper extends the classical jump-diffusion option pricing model to incorporate serially correlated jump sizes which have been documented in recent empirical studies. We model the series of jump sizes by an autoregressive process and provide an analysis on the underlying stock return process. Based on this analysis, the European option price and the hedging parameters under the extended model are derived analytically. Through numerical examples, we investigate how the autocorrelation of jump sizes influences stock returns, option prices and hedging parameters, and demonstrate its effects on hedging portfolios and implied volatility smiles. A calibration example based on real market data is provided to show the advantage of incorporating the autocorrelation of jump sizes.

The Hybrid Stochastic-Local Volatility Model with Applications in Pricing FX Options

This thesis presents our study on using the hybrid stochastic-local volatility model for option pricing. Many researchers have demonstrated that stochastic volatility models cannot capture the whole volatility surface accurately, although the model parameters have been calibrated to replicate the market implied volatility data for near at-the-money strikes. On the other hand, the local volatility model can reproduce the implied volatility surface, whereas it does not consider the stochastic behaviour of the volatility. To combine the advantages of stochastic volatility (SV) and local volatility (LV) models, a class of stochastic-local volatility (SLV) models has been developed. The SLV model contains a stochastic volatility component represented by a volatility process and a local volatility component represented by a so-called leverage function. The leverage function can be roughly seen as a ratio between local volatility and conditional expectation of stochastic volatility. The difficulty of implementing the SLV model lies in the calibration of the leverage function. In the thesis, we first review the fundamental theories of stochastic differential equations and the classic option pricing models, and study the behaviour of the volatility in the context of FX market. We then introduce the SLV model and illustrate our implementation of the calibration and pricing procedure. We apply the SLV model to exotic option pricing in the FX market and compare pricing results of the SLV model with pure local volatility and pure stochastic volatility models. Numerical results show that the SLV model can match the implied volatility surface very well as well as improve the pricing performance for barrier options. In addition, we further discuss some extensions of the SLV project, such as parallelization potential for accelerating option pricing and pricing techniques for window barrier options. Although the SLV model we use in the thesis is not entirely new, we contribute to the research in the following aspects: 1) we investigate the hybrid volatility modeling thoroughly from theoretical backgrounds to practical implementations; 2) we resolve some critical issues in implementing the SLV model such as developing a fast and stable numerical method to derive the leverage function; and 3) we build a robust calibration and pricing platform under the SLV model, which can be extended for practical uses.

Managing Employee Stock Option Expense: A Fair Value Approach

Owing to special characteristics, classic option pricing models are not well suited to the valuation of employee stock options (ESOs). This paper attempts to conduct a more general fair value estimation based on attaching performance targets to option vesting. Considering a setting that includes factors such as options that may be exercised early at employee discretion, employee exit rates and firm default risk, this paper presents a sensitivity analysis and empirical tests of option value. The results highlight the importance of considering the characteristics of ESOs in the design of performance‐vested option plans so as to provide the most attractive incentives for employees.

Discrete-Time Option Pricing with Stochastic Liquidity

Classical option pricing theories are usually built on the law of one price, neglecting the impact of market liquidity that may contribute to significant bid-ask spreads. Within the framework of conic finance, we develop a stochastic liquidity model, extending the discrete-time constant liquidity model of Madan (2010). With this extension, we can replicate the term and skew structures of bid-ask spreads typically observed in option markets. We show how to implement such a stochastic liquidity model within our framework using multidimensional binomial trees and we calibrate it to call and put options on the S&P 500.

ON A DIVIDEND-PAYING STOCK OPTIONS PRICING MODEL (SOPM) USING CONSTANT ELASTICITY OF VARIANCE STOCHASTIC DYNAMICS

In this paper, we propose a pricing model for stock option valuation. The model is derived from the classical Black-Scholes option pricing equation via the application of the constant elasticity of variance (CEV) model with dividend yield. This modifies the Black- Scholes equation by incorporating a non-constant volatility power function of the underlying stock price, and a dividend yield parameter.