Black-Scholes Research Articles

Inverse Multiquadric Function to Price Financial Options under the Fractional Black–Scholes Model

The inverse multiquadric radial basis function (RBF), which is one of the most important functions in the theory of RBFs, is employed on an adaptive mesh of points for pricing a fractional Black–Scholes partial differential equation (PDE) based on the modified RL derivative. To solve this problem, discretization along space is carried out on a non-uniform grid in order to focus on the hot area, at which the initial condition of the pricing model, i.e., the payoff, has discontinuity. The L1 scheme having the convergence order 2−α is used along the time fractional variable. Then, our proposed numerical method is built by matrices of differentiations to be as efficient as possible. Computational pieces of evidence are brought forward to uphold the theoretical discussions and show how the presented method is efficient in contrast to the exiting solvers.

Numerical solution of the three-asset Black–Scholes option pricing model using an efficient hybrid method

In this paper, a numerical method for solving three-asset Black–Scholes partial differential equation is presented. The model is based on the Crank–Nicolson and the radial basis function methods. Also, the convergency of the proposed method is proved. Implementation of the proposed method is specially studied on cash-or-nothing option. The numerical results show the efficiency and high accuracy of the proposed method.

Regime recovery using implied volatility in Markov modulated market model

AbstractIn the regime switching extension of Black–Scholes–Merton model of asset price dynamics, one assumes that the volatility coefficient evolves as a hidden pure jump process. Under the assumption of Markov regime switching, we have considered the locally risk minimizing theoretical price of European vanilla options. By pretending these prices or their numerical approximations as traded prices, we have first computed the implied volatility (IV) of the underlying asset. Then by performing several numerical experiments on ternary regime models we have investigated the dependence of IV on the time to maturity (TTM) and strike price of the vanilla options. We have observed a clear dependence that is at par with the empirically observed stylized facts. Furthermore, we have experimentally validated that IV time series, obtained from contracts with moneyness and TTM varying in a particular narrow range, can recover the transition instances of the hidden Markov chain having three states. Such regime recovery for any arbitrary state‐space and transition parameters has also been established in a theoretical setting. Moreover, the novel scheme for computing option price is shown to be stable.

Penentuan Harga Opsi Put dan Call Terhadap Saham Nokia dengan Menggunakan Model Black-Scholes

Options are a derivative instrument. High volatility makes options an attractive investment. Volatility can describe risk or gain. Volatility describes the probability of a stock's price over a period of time. Nokia is the code used in the stock of Nokia Corporation. The research method used in this research is descriptive method. The price of the call option premium is determined by using the Monte Carlo simulation method. The data used is Nokia stock data, with the determination of the 3 month put option expiration period from October 2020 to October 2021. In this study, the recommended or non-recommended stocks were found in the buy and sell options with several option prices (last price) on both options. The researcher uses the black scholes model and it is found that Nokia shares are considered attractive to choose and become a recommendation for both buy and sell options for investors. Further findings can be seen in this study.

Very fast algorithms for implied barriers and moving-barrier options pricing

Two closely related O(nlogn)-time tree algorithms under the Black–Scholes model are presented, where n denotes the tree’s number of time steps. The first finds the implied step barrier that matches the barrier-hitting probabilities exactly. In the constant-barrier case, the implied barrier is surprisingly accurate even for small ns; indeed, n=1 gives good results in typical situations. The second prices options with a time-dependent barrier (i.e., moving-barrier options). In practice, both algorithms are one to three orders faster than the standard algorithms even when n is moderate. As a consequence, large portfolios or datasets can finally be studied in a timely manner. Both algorithms can be easily tailored to handle barriers that are continuously monitored, discretely monitored, a mixture of both, or even when the model parameters are all time varying.

On the practical point of view of option pricing

Abstract In this note, we describe a new approach to the option pricing problem by introducing the notion of the safe (and acceptable) price for the writer of an option, in contrast to the fair price used in the Black–Scholes model. Our starting point is that the option pricing problem is closely related with the hedging problem by practical techniques. Recalling that the Black–Scholes model does not give us the price of the option but the initial value of a replicating portfolio, we observe easily that this has a serious disadvantage because it assumes the building of this replicating portfolio continuously in time, and this is a disadvantage of any model that assumes such a construction. Here we study the problem from the practical point of view concerning mainly the over-the-counter market. This approach is not affected by the number of the underlying assets and is particularly useful for incomplete markets. In the usual Black–Scholes or binomial approach or some other approaches, one assumes that one can invest or borrow at the same risk-free rate r > 0 r>0 , which is not true in general. Even if this is the case, one can immediately observe that this risk-free rate is not a universal constant but is different among different people or institutions. So the fair price of an option is not so much fair! Moreover, the two sides are not, in general, equivalent against the risk; therefore, the notion of a fair price has no meaning at all. We also define a variant of the usual binomial model, by estimating safe upward and downward rates u , d u,d , trying to give a cheaper safe or acceptable price for the option.

Numerical Approximation to a Variable-Order Time-Fractional Black–Scholes Model with Applications in Option Pricing

Numerical Approximation to a Variable-Order Time-Fractional Black–Scholes Model with Applications in Option Pricing

An Efficient Method for Solving Fractional Black-Scholes Model with Index and Exponential Decay Kernels

The Black-Scholes equation (BSe) is fascinating in the business world for predicting the performance of financial investment valuation systems. The Caputo fractional derivative (CFD) and Caputo-Fabrizio fractional derivative operators are used in this research to analyze the BSe. The Adomian decomposition method (ADM) and the new iterative transform (NIM) approach are combined alongside the Yang transform. In addition, the convergence and uniqueness results for the aforementioned framework have been calculated. The existence and uniqueness results have been established and frequently accompanied innovative aspects of the prospective system in fixed point terminologies. To provide additional insight into such concepts, a variety of illustrations and tabulations are used. Additionally, the provided techniques regulate and modify the obtained analytical results in a really productive fashion, allowing us to modify and regulate the converging domains of the series solution in a pragmatic manner.

Pricing Asian Lookback Option based on Monte Carlo simulation

With the emergence and development of exotic options, the diversity of securities has largely increased. However, existing types of popular exotic options seem to be either limited or inflexible to satisfy buyers' personal needs. As for the Lookback option and Asian option, the Lookback option’s application is limited in the OTC area with a high price. The Asian option is cheap but used mostly for hedging, leaving relatively little space for speculating. Thus, we created a new exotic option called the Asian Lookback option (ALB) as a combination and derivative of these two options with a flexible implementing way enabling both the requirements for hedging and speculating. In the process of building the model of ALB, these two options’ models are referenced, together with the widely used option pricing method Monte Carlo Simulations based on the assumptions of the Black-Scholes model. To simulate the practical function and price of ALB, we set a control experiment of the prices of four options: two types of ALB, Asian option, and Lookback option in three different markets (S&P 500, Corn, COMEX Gold). It is shown that the price of ALB can be switched from Asian-like to Lookback-like, realizing the function of customized usage from hedging to speculating. These results shed light on a more flexible and personalized development tendency of derivatives.

Valuation of the Extension Option in Time Charter Contracts in the LNG Market

A rapid transition toward a decarbonized economy is underway, following the Paris Agreement and the International Maritime Organization 2030 decarbonization goals. However, due to the high cost of the rapid transition to eco-friendly energy and the geopolitical conflict in eastern Europe, liquefied natural gas (LNG), which emits less carbon than other fossil fuels, is gaining popularity. As the spot market grows due to increased LNG demand, the usage of period extension options in time charter (T/C) contracts is increasing; however, these options are generally provided free of charge in practice, without economic evaluation; this is because some shipowners want to make their time charter contracts more attractive to the more credible charterers. Essentially, the reason for why this option has not been evaluated is because there is no reliable evaluation model currently used in practice. That is, research on the evaluation model for the T/C extension option has been insufficient. Therefore, this study evaluates the economic value of the extended period option in LNG time charter contracts using machine learning models, such as artificial neural networks, support vector machines, and random forest, and then compares them with the Black–Scholes model that is used for option valuations in financial markets. The results indicate superior valuation performance of the random forest model compared with the other models; particularly, its performance was significantly better than the Black–Scholes model. Since T/C extension options involve significant sums in the balance sheets of both shipowners and charterers, the fair value of these options should be assessed. In this regard, this paper has meaning in proposing valid machine models to efficiently reflect the fair value of period extension options that are provided at no charge in the LNG market.

Uncertain Stochastic Optimal Control with Jump and Its Application in a Portfolio Game

This article describes a class of jump-uncertain stochastic control systems, and derives an Itô–Liu formula with jump. We characterize an optimal control law, that satisfies the Hamilton–Jacobi–Bellman equation with jump. Then, this paper deduces the optimal portfolio game under uncertain stochastic financial markets with jump. The information of players is symmetrical. The financial market is constituted of a risk-free asset and a risky asset whose price process is subjected to the jump-uncertain stochastic Black–Scholes model. The game is formulated by two utility maximization problems, each investor tries to maximize his relative utility, which is the weighted average of terminal wealth difference between his terminal wealth and that of his competitor. Finally, the explicit expressions of equilibrium investment strategies and value functions for the constant absolute risk-averse and constant relative risk-averse utility function are derived by using the dynamic programming principle.

Analytical Solution of Black-Scholes Model for Pricing Barrier Option using Method of Partial Taylor Series Expansion

In this work, Black-Scholes differential equation for barrier/traditional option is solved using partial Taylor series expansion method. The developed solutions are in very good agreement with the closed-form solutions of the Black Scholes equation for the powered ML-payoff functions. Also, the analytical solutions of the new method in this present study give the same expressions as the solutions of projected differential equations and homotopy perturbation method as presented in the literature. Moreover, the reliability, speed, accuracy, and ease of application of the proposed method show its potential for wide areas of applications in science, financial mathematics, and engineering.

Price dispersion and vanilla options in a financial market game

We construct a game-theoretic model characterised by perfect information, no transaction costs, in which agents can borrow at the risk-free interest rate and engage in short selling. Traders can freely and instantaneously eliminate any unexploited profit opportunities through pure arbitrage just as the efficient market hypothesis postulates they should. Yet, in this work, we put forth, in a frictionless framework, a counterexample in which the Law of One Price fails in the underlying asset markets at equilibrium. At a theoretical level, this leads both the Binomial Option Pricing Model (BOPM) and in the limit, the Black–Scholes–Merton Model (BSMM), to misprice the vanilla options written on these assets. This compelling result is pregnant with far-reaching ramifications: (i) theoretically, it establishes that no-arbitrage, while necessary, is not sufficient for any of the BOPM and BSMM to yield consistent results; (ii) practically, it crystallises the need for practitioners to rely on additional, more data-adaptive, methods of option pricing.

A spectral collocation method based on fractional Pell functions for solving time–fractional Black–Scholes option pricing model

The fractional Black–Scholes equation has been widely studied by researchers in recent years. In this article, an efficient spectral collocation method based on fractional Pell functions is proposed for solving the time–fractional Black–Scholes equation. We introduce fractional Pell functions using the transformation x→xβ(β>0) on Pell polynomials, and we look for a solution of the model as a linear combination of these functions. Using operational matrices, we approximate the fractional derivative and other terms in a convenient form of the main equation. A system of algebraic equations is obtained by collocating resultant approximate equations. Convergence analysis of the numerical method has been investigated in Sobolev space. Finally, we have demonstrated the capability of the proposed method by considering numerical experiments in the form of tables and figures.

Shadow Price Approximation for the Fractional Black Scholes Model

In this work, we used Tran Hung Thao’s approximation of fractional Brownian motion to approximate the shadow price of the fractional Black Scholes model. In the case to maximize expectation of the utility function in a portfolio optimization problem under transaction cost, the shadow price is approximated by a Markovian process and semimartingale.

Solution of the Black-Scholes Equation by Finite Difference Schemes

Black-Scholes (BS) equation is a popular mathematical model for determining the value of option in financial derivatives. To predict the option value during the contract of the option is a big problem. Several studies have been shown that the option price value can be determined by applying different methods. In this paper, we have discussed three finite difference methods: Explicit, Implicit and Crank-Nicolson for solving Black-Scholes equation for European call option and compared the obtained results with the exact value. It is found that the Crank-Nicolson method is more accurate and cost effective in comparison with explicit and implicit methods.

VARIANCE GAMMA MODEL AND ITS DEVELOPMENT FOR STOCKS CALL OPTION PRICES ESTIMATION

One of the developments in the options market is the formation of various pricing models for an option to help the buyer determine the fairness of the price. Black-Scholes model uses the assumption that the log return price of stocks is normally distributed, while in reality, the real-world price couldn’t fit into that assumption. To be able to obtain an option price calculation that considers skewness and kurtosis in the stock price data, there are many alternative methods, namely Black-Scholes with Gram-Charlier expansion and Variance Gamma models. As a development of the Variance Gamma model, there are also several methods that are able to reduce the simulations variance generated by the Variance Gamma model, namely Antithetic Variate Variance Gamma model and the Importance Sampling Variance Gamma method. For the results, the Importance Sampling Variance Gamma model and the Antithetic Variate Variance Gamma model are really able to reduce the resulting simulations variance so that it can produce more accurate option prices with market prices compared to the Variance Gamma model. In the end, all Variance Gamma models are able to produce a call option price that is more in line with the market price than all Black Scholes models.

Numerical Approximations of a Nonlinear Volatility Model with European Options

Black-Scholes model plays a very significant role in the world of quantitative finance. In this paper, the focus are on both nonlinear and linear Black-Scholes (BS) equations with numerical approximations. We aim to find an effective numerical approximations for Black-Scholes model. Several models from the most relevant class of nonlinear Black-Scholes equations with European option are analyzed in this study. The problem is approached by transforming the problem into a convection-diffusion equation and later it is approximated with the help of finite difference method (Crank-Nicolson). The result of finite difference schemes (Crank-Nicolson) for several volatility models are presented, including the Risk Adjusted Pricing Methodology (RAPM), Leland’s model and the Barles’-Soner’s Model. At the same time, it is attempted to illustrate a comparison of different volatility models. In the case of linear Black-Scholes model, we approximate the model with finite difference method (FDM) and finite element method (FEM) and compare the results. All the numerical schemes are implemented in MATLAB and corresponding graphs are also presented here. GANIT J. Bangladesh Math. Soc. 42.1 (2022) 050- 068

Approximation of single‐barrier options partial differential equations using feed‐forward neural network

AbstractArtificial neural networks are generally employed in the numerical solution of differential equation problems. In this article, we propose an approach that deals with the combination of the feed‐forward neural network method and the optimization technique in solving the partial differential equation arising from the valuation of barrier options. The methodology entails transforming the extended Black–Scholes partial differential equations (PDE), which defines a barrier option, into a constrained optimization problem, and then proposing a trial solution that reduces the differential equation problem to an unconstrained one. This trial function consists of the adjustable and non‐adjustable neural network parameters. We design it to be differentiable, analytic, and satisfy the initial and boundary conditions of the corresponding option pricing PDE. We compare the corresponding option values to the Monte‐Carlo simulated values, Crank–Nicolson finite‐difference values and the exact Black–Scholes prices. Numerical results presented in this research show that neural networks can sufficiently solve PDE‐related problems with sufficient precision and accuracy. Furthermore, they can be applied in the fast and accurate valuation of financial derivatives without closed analytic forms.

Valuation of volatility derivatives with time-varying volatility: An analytical probabilistic approach using a mixture distribution for pricing nonlinear payoff volatility derivatives in discrete observation case

In this paper, we present an analytical probabilistic approach for pricing nonlinear payoff volatility derivatives with discrete sampling by assuming that the underlying asset price evolves according to the Black–Scholes model with time-varying volatility. A major difficulty to solve the pricing problem analytically is the volatility of the underlying asset price is time-varying, resulting the realized variance is distributed according to a mixture distribution, the probability density function of which is unknown. By utilizing the properties of a linear combination of noncentral chi-square random variables, we can calculate the expectation of square root of the realized variance analytically and provide formulas for pricing volatility swaps, volatility options, and variance options, including put–call parity relationships. Furthermore, we demonstrate an interesting application of our formulas by constructing simple closed-form approximate formulas for pricing the volatility derivatives for the constant elasticity of variance model. Finally, Monte Carlo simulations are conducted to illustrate the performance our approach, and effects of price volatility on the fair strike prices of the volatility derivatives are investigated and analyzed through several numerical experiments.