Black-Scholes Formula Research Articles

Option pricing for Barndorff–Nielsen and Shephard model by supervised deep learning

This paper aims to develop a supervised deep learning scheme to compute call option prices for the Barndorff-Nielsen and Shephard model with a non-martingale asset price process having infinite active jumps. In our deep learning scheme, teaching data are generated through the Monte Carlo method developed by Arai and Imai [(2024). Monte Carlo simulation for Barndorff-Nielsen and Shephard model under change of measure, Mathematics and Computers in Simulation, 218, 223–234]. Moreover, the BNS model includes many parameters, which makes the deep learning accuracy worse. Therefore, we will create another input parameter using the Black–Scholes formula. As a result, the accuracy is improved dramatically.

The Maximal and Minimal Distributions of Wealth Processes in Black–Scholes Markets

The Black–Scholes formula is an important formula for pricing a contingent claim in complete financial markets. This formula can be obtained under the assumption that the investor’s strategy is carried out according to a self-financing criterion; hence, there arise a set of self-financing portfolios corresponding to different contingent claims. The natural questions are: If an investor invests according to self-financing portfolios in the financial market, what are the maximal and minimal distributions of the investor’s wealth on some specific interval at the terminal time? Furthermore, if such distributions exist, how can the corresponding optimal portfolios be constructed? The present study applies the theory of backward stochastic differential equations in order to obtain an affirmative answer to the above questions. That is, the explicit formulations for the maximal and minimal distributions of wealth when adopting self-financing strategies would be derived, and the corresponding optimal (self-financing) portfolios would be constructed. Furthermore, this would verify the benefits of diversified portfolios in financial markets: that is, do not put all your eggs in the same basket.

Quantum Uncertainty and the Black-Scholes Formula

The publication of the Black-Scholes formula in 1973 appeared for the first time to put the pricing of financial options onto a rational and objective basis. Its adoption transformed the perception of option pricing from a form of gambling to scientific risk management, and led to a huge increase in options trading. However while the formula revolutionised the world of finance, and remains the industry-standard pricing model today, its proof relies on a number of assumptions about price behaviour which are often contested, such as that log prices follow a random walk with constant volatility, and that one can constantly buy or sell stocks and options without incurring transaction fees. This paper presents an alternative approach to option pricing, based on a quantum oscillator model of stock prices. In the quantum model, the bid/ask spread between buy and sell prices is treated as a fundamental measure of uncertainty, and volatility is not constant but exhibits a smile-like dependence on strike. We show how the Black-Scholes model and its assumptions lead to a systematic mispricing of commonly-traded options, while results can be improved by adopting the quantum model.

Volatility, Uncertainty, and Option Pricing

Amidst global and Chinese uncertainties, this paper delves into stock market volatility and option pricing within the current economic policy context. Focusing on China's stock market, it calculates returns and volatility changes in key indices, analyzing differences among indices and funds, along with temporal variations. Shared data characteristics and reflections on new market trends emerge. Employing methods such as the GARCH model, Black-Scholes formula, and Monte Carlo algorithm, it analyzes volatility and option pricing using extensive annual and monthly data. Visualizations showcase market change patterns. The study defines volatility as a measure of financial asset price fluctuation extent, reflecting asset risk. Higher volatility indicates pronounced price fluctuations and uncertainty, while lower volatility signifies smoother fluctuations and greater certainty. Merging data with volatility's significance, the study probes China's securities market uncertainty, investigating the link between option pricing and volatility. It concludes by identifying the connection between volatility, uncertainty, and option pricing, pointing to future research directions and challenges. Future work will track the latest market trends to enrich understanding.

SOLUTION OF BLACK-SCHOLES EQUATION FOR STANDARD POWER EUROPEAN OPTIONS WITH DISCRETE DIVIDEND PAYMENT

The Nobel Prize celebrated option pricing formulas derived by Fischer Black and Myron Scholes for plain vanilla payoffs known as Black-Scholes formulas. They developed these formulas for pricing European call and put options based on certain assumptions in order to minimize risk factor. The underlying asset pays a constant dividend payment during the life of option was one of the assumption to derive these formulas. S. P. Zhu and X. J. He tried and succeed to improve this assumption by taking discrete dividend payments for underlying asset at fixed dividend date. They derived approximate Black-Scholes formulas of European options for plain vanilla payoffs. Note that standard power payoffs are the generalization of plain vanilla payoffs. This paper contributes to the derivation of approximate Black-Scholes formulas for standard power payoffs. Further, the accuracy of these formulas have been checked numerically.

Using Simpson’s Rule to Evaluate the Normal Cumulative Distribution Function in an Algorithmic Option Pricing Problem

Algorithmic problems are frequently used to help ensure academic honesty during assessments in quantitative courses, asthey allow instructors to easily generate multiple versions of a test, quiz, or assignment with questions that are similar but whoseinput variables and correct answers differ. In general, instructors who create an algorithmic problem must program its solutionby entering an algebraic expression using the input variables. While this is usually straightforward enough, it can be problematicwhen the solution requires a more advanced function that has no algebraic representation. One such challenge occurs withfinancial option pricing problems based on the Black-Scholes model, which relies on the normal cumulative distribution function(CDF). Although students can easily obtain normal CDF values from a calculator or spreadsheet function, no such function isavailable within testing software, presenting an obstacle for instructors attempting to author algorithmic problems on optionpricing. I demonstrate how to overcome this difficulty by using Simpson’s Rule to numerically integrate the underlying normalprobability density function, thereby generating a highly accurate approximation of the CDF that can be programmed into thesoftware and used in the Black-Scholes formula.

Romancing with Black-Scholes Model: An Overview

This paper is an overview of our previous and recent results associated with the modifications of Black-Scholes model and formula. There were and are attempts to extend the classical Black-Scholes model by introducing the randomness into the coefficients r, �, s (interest rate, drift and volatility, respectfully) or to describe them by SDEs, and so on, and then to find explicit or close-form formulas for the hedging strategies and option prices. If coefficients r, �, s depend on some parameter x ?X, where X is some state space, i.e., r(x), �(x), s(x) then we call it (B, S, X)-security markets or (B, S)-security markets in random environment X. We consider different (B, S, X)-securities markets, including Markov, semi-Markov cases, jumps, and analogues/generalizations of Black-Scholes formulas for them. Black-Scholes formula with delay is also presented. Then we consider two types of telegraph process, classical/symmetric and asymmetric, and give two applications of those telegraph processes in finance by presenting European call and put option prices. Moreover, We introduce a new type of Hawkes process, namely, the exponential one-dimensional general compound Hawkes process (E1DGCHP), consider its limit theorems (LLN and FCLT) and present analogue of Black-Scholes formula in this case. Numerical examples are presented for both telegraph processes and E1DGCHP.

Tail Option Pricing Under Power Laws

We build a methodology for which the heuristic produces relative prices for options, with for sole parameter the tail index, under some mild arbitrage constraints. Usual restrictions such as finiteness of variance are not required. The heuristic allows us to scrutinize the volatility surface and test various theories of relative tail option overpricing (usually built on thin tailed models and minor modifications/fudging of the Black-Scholes formula).

A Review Of "The Pricing of Options and Corporate Liabilities"

This year we celebrate the 50th anniversary of the seminal paper, �The Pricing of Options and Corporate Liabilities�, in which the famous Black-Scholes formula was introduced. We will chronicle and review the literature predating and post-dating the publication of this work in order to identify precisely which new concepts they introduced to the field.

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.

On the Stochastic Volatility in the Generalized Black-Scholes-Merton Model

This paper discusses the generalized Black-Scholes-Merton model, where the volatility coefficient, the drift coefficient of stocks, and the interest rate are time-dependent deterministic functions. Together with it, we make the assumption that the volatility, the drift, and the interest rate depend on a gamma or inverse-gamma random variable. This model includes the models of skew Student’s t- and variance-gamma-distributed stock log-returns. The price of the European forward-start call option is derived from the considered models in closed form. The obtained formulas are compared with the Black-Scholes formula through examples.

Exotic option pricing model of the Black–Scholes formula: a proactive investment strategy

The option is an important derivative tool in financial market, and after decades of development, the option has emerged in various forms. This paper studies an exotic option with a proactive investment strategy. Compared with the classic option theory, we assume that the option holders continuously trade the underlying assets according to a predetermined investment strategy. Taking the European call option as an example, we first give some assumptions and define the loss function in terms of a logarithmic investment strategy. Then, the specific pricing expression of the exotic option is derived from the Black-Scholes option pricing formula. Next, numerical simulations are presented to visualize the mechanism of the exotic option in 2D and 3D forms by selecting appropriate parameters. The results indicate that the exotic option has a significant price advantage (up to 43.9% under specific parameter settings) over the classic option. Moreover, the empirical results illustrate a perfect fit between 50 ETF (issued by the Shanghai Stock Exchange) and our exotic option. The new proposed exotic option extends the Black-Scholes option theory from a no-trading condition (do not buy or sell underlying assets during the validity period) to a dynamic investment condition and has important practical significance in real life.

A Numerical Approximation on Black-Scholes Equation of Option Pricing

This paper considered the notion of European option which is geared towards solving analytical and numerical solutions. In particular, we examined the Black-Scholes closed form solution and modified Black-Scholes (MBS) partial differential equation using Crank-Nicolson finite difference method. These partial differential equations were approximated to obtain Call and Put option prices. The explicit price of both options is found accordingly. The numerical solutions were compared to the closed form prices of Black-Scholes formula. More so, comparisons of other parameters were discussed for the purpose of investment plans. The computational results shows: increase in stock volatility increases the value of options, when the initial stock price is equal to its strike price the values of call option is higher than the put option. This informs the investor about the behavior of stock prices for the purpose of decision making. Finally, all simulation results presented graphically using MATLAB.

The principle of two models: the cases of Black-Scholes formula for interest rates and of Gaussian copula for credit

In this article, I try to show that a given market segment needs at least two models: one as a pricing convention and one to run the portfolio of securities. These models should be related in such a way that the model to manage the books is an extension of the pricing convention. Financial modeling succeeded in the case of options, but it failed with the CDOs, resulting in the credit crunch.

Credible Delta Normal Value at Risk for Risk Evaluation of European Call Option

This paper develops a Credible Delta Normal Value at Risk (CredDN) as a method to assess the options risk. The method is constructed by combining Credible Value at Risk (CrVaR) with Delta Normal VaR. The new method is initiated as an alternative instrument to measure the European call option portfolio risk. Using the Black-Scholes Formula, the new method contemplates the nonlinear dependence of the market risk factors specifying the value of a European call option. Then, utilizing simulated financial data that represents assets profit/loss over ten investment periods, we applicate the proposed method for analysis. The novel method is also employed to assess the portfolio’s risk consisting of the active stocks which are involved in option trading. The performance of CredDN in this study is evaluated by administering Kupiec backtesting. The results of Kupiec backtesting suggest that the proposed method effectively quantifies the risk of each option constructing a portfolio at 80%, 90%, and 95% confidence levels.

The Black-Scholes Exotic Barrier Option Pricing Formula

The paper considers a specific type of such financial instrument as an option, namely an exotic barrier call option of the European type. Exotic options are gaining popularity among ordinary investors due to the development of information and telecommunication technologies, thanks to which such specific financial instruments as options have become readily available. We investigate the hedging problem for such options with some restrictions on the payment function and the availability of dividend payment on a risky asset in the classical Black-Scholes model. An analogue of the Black-Scholes formula for the mentioned variant of the exotic barrier is proved. In the future, it is planned to generalize the obtained results for put options and for more general payment functions.

The Research on the Pricing of Double Barrier Option Based on the Black-Scholes Model

The two barrier values in the double barrier option are the focus in the evaluation simulation and they affect the knock-in and knock-out of the option. This paper, simulates a double barrier option and calculates its option value based on the Black-Scholes Model, using Coca-Cola stock data from January 2022 to the end of July, and verifies the feasibility of the Black-Scholes model. This paper also analyzes the sensitivity of the influence of the two barrier values on the option value, outlining the option value changes under different barrier values. The main finding of this paper is that the Black-Scholes formula is used to verify the value of the Coca-Cola option product. This study enriches the practical examples of the validation of double barrier option products in the Black-Scholes formula.

Quantum Uncertainty and the Black-Scholes Formula

Quantum Uncertainty and the Black-Scholes Formula

THE PRINCIPLE OF EXERCISING OPTIONS ON THE ELECTRICITY MARKET

Continual price fluctuations are possible to hedge by using various financial instruments, including options. An option buyer buys the right to the settlement price of an underlying asset such as electricity. Due to the volatility of the asset price, the buyer is not obliged to exercise the option. In such a case, the buyer’s only loss is the purchased right or the option premium, which is equal to the option price. Mathematical models for option pricing have been developed in the last hundred years. These models were very popular during the 1970s, owing to the application of the Black-Scholes formula for the calculation of theoretical option prices. In this article, options are distinguished according to various criteria, specific exercising methods are described and the historical development of option pricing models is reviewed. An example of option exercising with closing out of the margins on the largest Middle European electricity exchange, EEX, is presented.

An Analysis of the Fractional-Order Option Pricing Problem for Two Assets by the Generalized Laplace Variational Iteration Approach

An option is the right to buy or sell a good at a predetermined price in the future. For customers or financial companies, knowing an option’s pricing is crucial. It is well recognized that the Black–Scholes model is an effective tool for estimating the cost of an option. The Black–Scholes equation has an explicit analytical solution known as the Black–Scholes formula. In some cases, such as the fractional-order Black–Scholes equation, there is no closed form expression for the modified Black–Scholes equation. This article shows how to find the approximate analytic solutions for the two-dimensional fractional-order Black–Scholes equation based on the generalized Riemann–Liouville fractional derivative. The generalized Laplace variational iteration method, which incorporates the generalized Laplace transform with the variational iteration method, is the methodology used to discover the approximate analytic solutions to such an equation. The expression of the two-parameter Mittag–Leffler function represents the problem’s approximate analytical solution. Numerical investigations demonstrate that the proposed scheme is accurate and extremely effective for the two-dimensional fractional-order Black–Scholes Equation in the perspective of the generalized Riemann–Liouville fractional derivative. This guarantees that the generalized Laplace variational iteration method is one of the effective approaches for discovering approximate analytic solutions to fractional-order differential equations.