European Options Research Articles

Quantile Hedging in the complete financial market under the mixed fractional Brownian motion model and the liquidity constraint

This article proposes a pricing framework for European option that utilizes a Quantile hedging strategy in a complete financial market. The methodology involves applying the long memory Geometric Brownian motion model, utilizing the generalized mixed fractional Girsanov theorem, and incorporating relevant findings related to quasi-conditional expectation. The first step in this framework is to derive the option price formula using the equivalent martingale measure. This approach enables investors to use the entire option price without any restrictions to hedge their contingent claim. Then, we add the initial wealth condition to hedge and determine the option price with the Quantile hedging strategy. To do this, we find the maximum probability of success in complete risk hedging for different amounts of model parameters and then use the probability values to determine the option price. Moreover, we apply the minimum variance algorithms and the error correction method to find the hedge ratio. Finally, we illustrate the hedging strategy and provide numerical results that assist investors in determining the required capital for covering the option risk under market data. This calculation takes into account both the level of risk tolerance and the anticipated return on investment.

On some semi-parametric estimates for European option prices

AbstractWe show that an estimate by de la Peña, Ibragimov, and Jordan for ${\mathbb{E}}(X-c)^+$ , with c a constant and X a random variable of which the mean, the variance, and $\mathbb{P}(X \leqslant c)$ are known, implies an estimate by Scarf on the infimum of ${\mathbb{E}}(X \wedge c)$ over the set of positive random variables X with fixed mean and variance. This also shows, as a consequence, that the former estimate implies an estimate by Lo on European option prices.

Review of the Fractional Black-Scholes Equations and Their Solution Techniques

The pioneering work in finance by Black, Scholes and Merton during the 1970s led to the emergence of the Black-Scholes (B-S) equation, which offers a concise and transparent formula for determining the theoretical price of an option. The establishment of the B-S equation, however, relies on a set of rigorous assumptions that give rise to several limitations. The non-local property of the fractional derivative (FD) and the identification of fractal characteristics in financial markets have paved the way for the introduction and rapid development of fractional calculus in finance. In comparison to the classical B-S equation, the fractional B-S equations (FBSEs) offer a more flexible representation of market behavior by incorporating long-range dependence, heavy-tailed and leptokurtic distributions, as well as multifractality. This enables better modeling of extreme events and complex market phenomena, The fractional B-S equations can more accurately depict the price fluctuations in actual financial markets, thereby providing a more reliable basis for derivative pricing and risk management. This paper aims to offer a comprehensive review of various FBSEs for pricing European options, including associated solution techniques. It contributes to a deeper understanding of financial model development and its practical implications, thereby assisting researchers in making informed decisions about the most suitable approach for their needs.

Derivation of discrete analog of Breeden–Litzenberger relation for risk-neutral density

This short paper derives a formula for the risk-neutral density function at discrete points over the strike range using Abel’s summation formula in terms of the arbitrage-free call price function in a European option contingent. The estimate of risk-neutral density is crucial from a risk management perspective and other crucial assessments.

Optimizing Investment Strategies: A Case Study on JPMorgan Chase & Co. Stock Options Using the Black-Scholes Model and What-If Analysis in Excel

This research focuses on applying the Black-Scholes Model to evaluate European options on JPMorgan Chase & Co. stocks. This model has been a critical foundation in evaluating financial instruments, especially options, since its development in 1973 by Fisher Black, Myron Scholes, and Robert Merton. The study utilizes secondary data from some sources to obtain current information regarding stock prices, strike prices, expiration time, volatility, and relevant risk-free interest rates for option valuation as of December 19, 2023. Through this approach, our aim is to gain a better understanding of how the Black-Scholes Model is used as a framework in determining option prices for these stocks. The research methodology involves What-If analysis, exploring variations in key variables such as current stock price, strike price, expiration time, stock price volatility, and risk-free interest rates to assess how these changes affect the prices of both call and put options. Additionally, the study presents graphs representing stock prices, strike prices, interest rates, time, and volatility to visually support the research findings. The analysis results reveal that the prices of both call and put options are highly responsive to changing market conditions. An increase in the current stock price tends to raise the call option price while reducing the put option price. Conversely, an increase in the strike price has the opposite effect. Moreover, variations in the risk-free interest rates influence the option prices, with rising rates increasing the call option price and decreasing the put option price. Furthermore, as the expiration time approaches or stock price volatility increases, both call and put option prices tend to rise. These findings provide a comprehensive understanding of the dynamics of JPMorgan Chase & Co.'s stock option pricing, offering a foundation for investors to make informed and adaptable investment decisions amid constantly evolving financial markets. Sensitivity to changes in key variables is an essential aspect to consider in designing effective investment strategies in the face of ever-changing financial markets.

Simultaneous Calibration of European Option Volatility and Fractional Order under the Time Fractional Vasicek Model

In this paper, we recover the European option volatility function σ(t) of the underlying asset and the fractional order α of the time fractional derivatives under the time fractional Vasicek model. To address the ill-posed nature of the inverse problem, we employ Tikhonov regularization. The Alternating Direction Multiplier Method (ADMM) is utilized for the simultaneous recovery of the parameter α and the volatility function σ(t). In addition, the existence of a solution to the minimization problem has been demonstrated. Finally, the effectiveness of the proposed approach is verified through numerical simulation and empirical analysis.

A Study on the Comparative Advantages and Disadvantages of Different Options

This paper delves into the world of options, a type of derivative financial product that provides the right to buy or sell a specified asset at a predetermined price by a specified date. The historical evolution of options, from informal agreements among farmers to structured contracts in medieval Europe, is explored. The study highlights the versatility of options, including the creation of new options through combinations and constraints. It also critiques existing options pricing models, such as the Black-Scholes model, pointing out their limitations in capturing real market data. The paper categorizes options into call and put options, emphasizing their distinct purposes and risk profiles. It also examines American and European options, focusing on their differences and the factors influencing their suitability. The use of options in hedging strategies is discussed through various real-world cases, showcasing their effectiveness in managing risks, particularly in volatile markets. While options offer diverse opportunities for risk management and profit, they also come with challenges, including the cost of monitoring the options market and the potential pitfalls for retail investors. The paper concludes by highlighting the ongoing evolution of the options market and its potential to become more accessible and comprehensible in the future.

An analytical approximation of European option prices under a hybrid GARCH-Vasicek model with double exponential jump in the bid-ask price economy

<abstract><p>Conic finance theory, which has been developed over the past decade, replaces classical one-price theory with the bid-ask price economy in option pricing since the one-price principle ignores the bid-ask spread created by market liquidity. Within this framework, we investigate the European option pricing problem when stochastic interest rate, stochastic volatility, and double exponential jump are all taken into account. We show that the corresponding bid and ask prices can be formulated into a semi-analytical form with the Fourier-cosine method once the solution to the characteristic function is obtained. Some interesting properties regarding the new results are displayed via numerical implementation.</p></abstract>

EUROPEAN OPTION PRICING IN THE GENERALIZED MIXED WEIGHTED FRACTIONAL BROWNIAN MOTION

In order to describe the self-similarity and long-range dependence of financial asset prices, this paper adopts a new fractional-type process, i.e, the generalized mixed weighted fractional Brownian motion to describe the dynamic change process of risky asset prices. A European option pricing model driven by the generalized mixed weighted fractional Brownian motion is constructed, and explicit solutions to the pricing formulas of European call options and European put options are derived by using the arbitrage-free pricing theory. Finally, through numerical simulation, the influence of the parameter on the option price is analyzed.

CAUCHY PROBLEM FOR ULTRA-PARABOLIC EQUATIONS OF KOLMOGOROV TYPE WITH BLOCK STRUCTURE

The investigation is devoted to ultra-parabolic equations which appear in Asian options problems. Unlike the European option, the payout of Asian derivative depends on the entire trajectory of the price value, not the final value only. Among methods of researching of the Asian options, the one is to include dependent on the price trajectory variables in the state space. The expansion of the state space by including of dependent on the price trajectory variables transforms the path-dependent problem for the Asian option into an equivalent path-independent Markov problem. However, the increasing of the dimension usually leads to partial differential equations which are not uniformly parabolic. The class of these equations under some conditions is a generalization of the well-known degenerate parabolic A.N.Kolmogorov's equation of diffusion with inertia. Mathematical models of the options have been studied in many works. It has been constructed so called L-type fundamental solutions for considered equations previously, some their properties have been established, the Cauchy problem has been researched. In current work, for the given equations we study the classical solutions of the Cauchy problem. For the coefficients of the equations we apply special Hölder conditions with respect to spatial variables. Under these conditions, we prove the wellposedness of the Cauchy problem in special weighed spaces, obtained integral presentation of classic solutions of the Cauchy problem for homogeneous equations. Classes of well-posedness of the Cauchy problem were described. The results obtained in the work are realization of well-known Eidelman-Ivasyshen approach. Ones can be used to advanced studying of the Cauchy problem and boundary value problems for linear and quasi-linear degenerated parabolic equations, as well as in the theory of stochastic processes when studying Markov processes, the transition probability density of which is the fundamental solution of the Cauchy problem for these equations.

Numerical solutions for the time fractional Black-Scholes model governing European option by using double integral transform decomposition method

Numerical solutions for the time fractional Black-Scholes model governing European option by using double integral transform decomposition method

European option prices driven by standard t-process

We consider a standard t-process model of a financial market and obtain the solution to the stock price process under the equivalent martingale measure. Using the equivalent martingale measure we derive the European option pricing based on standard t-process. At the end of this paper, the pricing formula by standard t-process is compared with B-S's formula by numerical simulation.

Markov chain approximation approach to European option pricing in the presence of proportional transaction costs and regime-switching

Markov chain approximation approach to European option pricing in the presence of proportional transaction costs and regime-switching

Cobertura delta precisa de opciones europeas usando cálculo conformable

Objective: we aim to develop a method for delta hedging portfolios of European options based on the theory of conformable calculus which improves accuracy of risk management of listed options in a first-order approximation. Methodology: we allow the time derivative in the classic Black-Scholes-Merton model to have a fractional order0 ≤α≤ 1 and calculate the corresponding delta of a portfolio of listed options as a function of this conformableparameter. Results: applying this method to a portfolio consisting of eight European options on the SPX index, we find that conformable delta hedging offers more accurate average predictions than classical delta hedging. Limitations: this method is applicable for delta hedging in European options only. Originality: this is the first successful application of conformable calculus to delta hedging in European options. Conclusions: application of Conformable Calculus allows for a greater flexibility in the local approximation to price in delta-hedging European options and offers a new and more precise methodology to this objective.

A preconditioned iterative method for coupled fractional partial differential equation in European option pricing

Abstract Recently, regime-switching option pricing based on fractional diffusion models has been used, which explains many significant empirical facts about financial markets better. There are many methods to solve the problem, but to the best of our knowledge, effective preconditioners for the second-order schemes have not been proposed. Thus, in this article, an implicit numerical scheme is developed for a regime-switching European option pricing problem under a multi-state tempered fractional model. The scheme is proven to be unconditionally stable and converges quadratically in space and linearly in time. Besides, the resulting linear system is solved using an iterative method, and a preconditioner is proposed to accelerate the rate of convergence. The preconditioner is constructed through circulant approximations to the Toeplitz blocks due to the coefficient matrix, which is is a block matrix with Toeplitz blocks. The spectral analysis of the preconditioned matrix is given, which demonstrates that the spectrum of the preconditioned matrix is clustered around 1. Numerical examples show the efficiency of the proposed method, and an empirical study is also provided.

A closed-form pricing formula for European options under a multi-factor nonlinear stochastic volatility model with regime-switching

A closed-form pricing formula for European options under a multi-factor nonlinear stochastic volatility model with regime-switching

European Option Pricing under Sub-Fractional Brownian Motion Regime in Discrete Time

In this paper, the approximate stationarity of the second-order moment increments of the sub-fractional Brownian motion is given. Based on this, the pricing model for European options under the sub-fractional Brownian regime in discrete time is established. Pricing formulas for European options are given under the delta and mixed hedging strategies, respectively. Furthermore, European call option pricing under delta hedging is shown to be larger than under mixed hedging. The hedging error ratio of mixed hedging is shown to be smaller than that of delta hedging via numerical experiments.

Put–call parity in a crypto option market — Evidence from Binance

This paper investigates a special and already well-documented topic within market efficiency, the put–call parity relationship. The analysis is carried out to the cryptocurrency options market, where market microstructure effects are treated. A new concept of the forward and backward PCP breaking for European options is introduced. This is examined in detail for Ethereum options listed on Binance using a one-second resolution data set. The results show that significant put–call parity violations remain even after controlling for bid–ask spread, trading costs and funding costs.

Pricing and Optimization Model of European Options

Financial market forecasts are gradually gaining the attention of investors. The options market is an integral and important part. Commonly used analysis methods include traditional BS formula and CRR model. This article obtained the transaction data of SP&500 from February 2013 to February 2018, used the data of the first 4 years to train the model, and finally explored the actual price and predicted value of the European call option. To forecast the stock price, a hidden Markov model (HMM) is first applied. And on this basis, improve the parameter estimation in the traditional method BS-CRR, and compare each model with the actual value of the option. Compared with the traditional model, to some extent, the Hidden Markov Model can predict the state of the stock market., get rid of the limitations of the BS model assumptions, predict the direction of the price trend, and verify the feasibility of its application in the financial field.

Features of realized volatility analysis and return predicting based on LGBM and RNN model

This paper proposes a method of predicting the realized volatility of financial assets using LGBM and RNN models. The study utilizes Convolutional Neural Networks to construct sub-indicators capturing the liquidity and volatility of financial assets. These sub-indicators are used to develop comprehensive measures of liquidity and volatility. Lognormal random walk theory is applied to each asset dimension to price volatility for multiple assets, and the value of European options independent of path is obtained via multiple integration. Monte Carlo method is applied to solve the integral, which becomes inefficient in the case of high dimension and orthogonality. This study also involves leveraging LGB and other models to efficiently exploit data to create high returns and achieve the highest sharp ratio. The current dataset, which comes from a recognized international market maker, includes stock market data that is important for trade execution in the financial markets, particularly snapshots of the order book and executed trades. The study shows that the proposed method can accurately predict the realized volatility of financial assets.