Option Pricing Equations Research Articles

The modified homotopy perturbation method and its application to the dynamics of price evolution in Caputo-fractional order Black Scholes model

BackgroundFollowing a financial loss in trades due to lack of risk management in previous models from market practitioners, Fisher Black and Myron Scholes visited the academic setting and were able to mathematically develop an option pricing equation named the Black–Scholes model. In this study, we address the solution of a Caputo fractional-order Black–Scholes model using an analytic method named the modified initial guess homotopy perturbation method.MethodologyForemost, the classical Black Scholes model relaxed for European option style is generalized to be of Caputo derivative. The introduced method is established by coupling a power series function of arbitrary order with the renown He’s homotopy perturbation method. The convergence of the method is demonstrated using the fixed point theorem, and its methodology is illustrated by solving a generalized theoretical form of the fractional order Black Scholes model. The applicability of the method is proven by solving three different fractional order Black–Scholes equations derived from different market scenarios and its effectiveness is confirmed as feasible series of arbitrary orders that accelerate fast to the exact solution at an integer order were obtained. The computation of these results was carried out using Mathematica 12 software. Subsequently, the obtained outcomes were utilized in Maple 18 software to conduct a series of numerical simulations. These simulations aimed to analyze the influence of the fractional order on the dynamics of payoff functions regarding the share value as the option approached its expiration date under varying market constraints. In all three scenarios, the results showed that option values decrease as the expiration date approaches the integer order. Furthermore, the comparative outcomes reveal that Caputo fractional order derivatives control the flexibility of the classical Black–Scholes model because its payoff curve exhibits more sensitivity to changes associated with market characteristic parameters, such as volatility and interest rates.RecommendationsWe propose that the results of this work should be examined and implemented by mathematicians and economists to better comprehend the influence of Caputo-fractional order derivatives in understanding the dynamics of option price evolution of financial assets.

Calculations of fractional derivative option pricing models based on neural network

The work adopts the neural network algorithm to derive optimized series solutions of fractional option pricing equations. The studied models include Caputo time-fractional equation and space-time fractional differential equations. The solutions of fractional derivative models are made up of the time variable and the kernel functions of RBF neural network. According to the characteristics of the models, the information function, the test solution, the output function and the loss function are established in turn. And then the optimal parameters of the solution structures of fractional derivative models are calculated by the designed neural network in conjunction with Chinese market data. For Caputo time-fractional model, the pricing results under different kernel functions, with or without cluster analysis, are compared through numerical analysis and illustration. The results of cluster analysis are better than those without cluster analysis. For space-time fractional models, comparative studies between the pricing results under Caputo and conformable fractional derivatives and those from conformable fractional derivative are made and analyzed by the market data. The numerical simulations indicate that the space-time fractional derivative models have strong predictive abilities. Application analyses show that it is feasible and operable for fractional calculus tool and neural network algorithm to jointly act on the pricing problems of financial derivatives.

Analytical Solution of Biological Population of Fractional Differential Equations by Reconstruction of Variational Iteration Method

This article presents a brand-new approximation analytical technique we refer to as the reconstruction of variational iteration method. For the goal of solving fractional biological population option pricing equations, this methodology was created. In certain circumstances, you may actually use the well-known Mittag-Leffer function to get an explicit response. The usage of the three examples below demonstrates the precision and effectiveness of the suggested method. The results show that the RVIM is not only quite straightforward but also very successful at resolving non-linear problems.

Comparative Analysis of the Time-Fractional Black–Scholes Option Pricing Equations (BSOPE) by the Laplace Residual Power Series Method (LRPSM)

The residual power series method is effective for obtaining solutions to fractional-order differential equations. However, the procedure needs the n − 1 ϖ derivative of the residual function. We are all aware of the difficulty of computing the fractional derivative of a function. In this article, we considered the simple and efficient method known as the Laplace residual power series method (LRPSM) to find the analytical approximate and exact solutions of the time-fractional Black–Scholes option pricing equations (BSOPE) in the sense of the Caputo derivative. This approach combines the Laplace transform and the residual power series method. The suggested method just needs the idea of an infinite limit, so the computations required to determine the coefficients are minimal. The obtained results are compared in the sense of absolute errors against those of other approaches, such as the homotopy perturbation method, the Aboodh transform decomposition method, and the projected differential transform method. The results obtained using the provided method show strong agreement with different series solution methods, demonstrating that the suggested method is a suitable alternative tool to the methods based on He’s or Adomian polynomials.

Forecasting the behaviour of fractional Black-Scholes option pricing equation by laplace perturbation iteration algorithm

Financial derivatives plays a major role in all financial deals these days. Black–Scholes option pricing model gives a risk free analysis for investing in options. In the current work, a method called the Laplace Perturbation Iteration Algorithm is being applied on Fractional Black–Scholes Equation to obtain its fractional analytical solutions in series form to analyze its results for the European and American option pricing problem, quickly and accurately. Laplace Perturbation Iteration Algorithm incorporates Laplace transform and Perturbation Iteration Algorithm and forms an iterative scheme that derives the analytical solutions without any inconvenience. The Fractional Black–Scholes Equation analytical solution is obtained as a convergent power series of conveniently computed components. Four examples have been solved and their results have been discussed in this paper, in order to check the efficacy of Laplace Perturbation Iteration Algorithm.

A New Homotopy Transformation Method for Solving the Fuzzy Fractional Black–Scholes European Option Pricing Equations under the Concept of Granular Differentiability

The Black–Scholes option pricing model is one of the most significant achievements in modern investment science. However, many factors are constantly fluctuating in the actual financial market option pricing, such as risk-free interest rate, stock price, option underlying price, and security price volatility may be inaccurate in the real world. Therefore, it is of great practical significance to study the fractional fuzzy option pricing model. In this paper, we proposed a reliable approximation method, the Elzaki transform homotopy perturbation method (ETHPM) based on granular differentiability, to solve the fuzzy time-fractional Black–Scholes European option pricing equations. Firstly, the fuzzy function is converted to a real number function based on the horizontal membership function (HMF). Secondly, the specific steps of the ETHPM are given to solve the fuzzy time-fractional Black–Scholes European option pricing equations. Finally, some examples demonstrate that the new approach is simple, efficient, and accurate. In addition, the fuzzy approximation solutions have been visualized at the end of this paper.

Double-barrier option pricing equations under extended geometric Brownian motion with bankruptcy risk

We consider an extended geometric Brownian motion with bankruptcy risk and solve its double barrier option pricing problem. We establish its partial differential equation and provide its numerical solution. Then we discuss the influence of bankruptcy omission by simulation.

Discounting the Discounted Projection Approach

U.K. equity release actuaries are using a flawed approach to value the no-negative equity guarantees in their equity release mortgages. The approach they use, the discounted projection approach, incorrectly uses projected future house prices as the underlying prices in their put option pricing equations. The correct approach uses forward house prices. The discounted projection approach entails significant undervaluations of no-negative equity guarantees and overvaluations of equity release mortgages and can produce valuations that violate rational pricing principles. The discounted projection approach is also inconsistent with both actuarial and accounting standards. Our results have significant ramifications for equity release industry practice and prudential regulation.

English

In this paper we study both analytic and numerical solutions of option pricing equations using systems of orthogonal polynomials. Using a Galerkin-based method, we solve the parabolic partial diferential equation for the Black-Scholes model using Hermite polynomials and for the Heston model using Hermite and Laguerre polynomials. We compare obtained solutions to existing semi-closed pricing formulas. Special attention is paid to the solution of Heston model at the boundary with vanishing volatility.

Application of the Generalized Laplace Homotopy Perturbation Method to the Time-Fractional Black–Scholes Equations Based on the Katugampola Fractional Derivative in Caputo Type

In the finance market, the Black–Scholes equation is used to model the price change of the underlying fractal transmission system. Moreover, the fractional differential equations recently are accepted by researchers that fractional differential equations are a powerful tool in studying fractal geometry and fractal dynamics. Fractional differential equations are used in modeling the various important situations or phenomena in the real world such as fluid flow, acoustics, electromagnetic, electrochemistry and material science. There is an important question in finance: “Can the fractional differential equation be applied in the financial market?”. The answer is “Yes”. Due to the self-similar property of the fractional derivative, it can reply to the long-range dependence better than the integer-order derivative. Thus, these advantages are beneficial to manage the fractal structure in the financial market. In this article, the classical Black–Scholes equation with two assets for the European call option is modified by replacing the order of ordinary derivative with the fractional derivative order in the Caputo type Katugampola fractional derivative sense. The analytic solution of time-fractional Black–Scholes European call option pricing equation with two assets is derived by using the generalized Laplace homotopy perturbation method. The used method is the combination of the homotopy perturbation method and generalized Laplace transform. The analytic solution of the time-fractional Black–Scholes equation is carried out in the form of a Mittag–Leffler function. Finally, the effects of the fractional-order in the Caputo type Katugampola fractional derivative to change of a European call option price are shown.

A fractional reduced differential transform method for solving time fractional Black Scholes American option pricing equation

In this paper, fractional reduced differential transform method (FRDTM) is operated to solve time fractional Black-Scholes American option pricing equation paying no dividends.The Black-Scholes model plays a significant role in the evaluation of European or American call and put options. The advantage of the proposed method to other existing methods is that it finds the solution without discretization or transformation. While using this method, no recommended assumptions are needed and hence the computational work reduces to a greater extent. Numerical experiments prove that the proposed method is efficient and valid for obtaining the solution of time fractional Black-Scholes equation governing American options. This method proves to be powerful for solving general fractional order partial differential equations (PDEs) existing in the field of Science, Engineering and other related fields.

Stock Return Autocorrelation and Individual Equity Option Prices

This study demonstrates empirically the impact of stock return autocorrelation on the prices of individual equity option. The option prices are characterized by the level and slope of implied volatility curves, and the stock return autocorrelation is measured by variance ratio and first-order serial return autocorrelation. Using a large sample of U.S. stocks, we show that there is a clear link between stock return autocorrelation and individual equity option prices: a higher stock return autocorrelation leads to a lower level of implied volatility (compared to realized volatility) and a steeper implied volatility curve. The stock return autocorrelation is more important in explaining the level of implied volatility curve for relatively small stocks. The relation between stock return autocorrelation and option price structure is more pronounced when market is volatile, especially during financial crisis. The stock return autocorrelation is more important in explaining the level of implied volatility curve for relatively small stocks. Thus, stock return autocorrelation can help differentiate the price structure across individual equity options.

The limitations of estimating implied densities from option prices

Bakshi et al. (Stock Return Characteristics, Skew Laws, and the Differential Pricing of Individual Equity Options. Rev. Financ. Stud., 2003, 16(1), 101–143) develop a novel method for estimating a stocks's τ-period risk-neutral return moments from option prices. Their result relies on the integration of all OTM calls and puts to find the prices of a quadratic, cubic, and quartic contract; from which risk-neutral estimates of the 2nd, 3rd, and 4th moments are derived. Dennis and Mayhew (Risk-neutral skewness: Evidence from stock options. J. Financ. Quant. Anal., 2002, 37(3), 471–493) point out that error is introduced into the BKM estimation in practice since options do not span a continuum and the range of moneyness and distance between strikes varies across firms and maturities. We use Monte Carlo simulation to show that the standard approach of Dennis and Mayhew (Risk-neutral skewness: Evidence from stock options. J. Financ. Quant. Anal., 2002, 37(3), 471–493) used within Conrad et al. (Ex ante skewness and expected stock returns. J. Finance., 2013, 68(1), 85–124) and other leading empirical works leads to very biased, and in some cases inconsistent, estimates of BKM implied moments. More so, we prove the particularly concerning result that estimation error using the Dennis and Mayhew (Risk-neutral skewness: Evidence from stock options. J. Financ. Quant. Anal., 2002, 37(3), 471–493) methodology increases dramatically when a stock's actual return distribution is non-normal or the firm's stock price is not exactly at-the-money. Fortunately, the 2-step interpolation method developed by DeMiguel et al. (Improving portfolio selection using option-implied volatility and skewness. J. Financ. Quant. Anal., 2013, 48(6), 1813–1845) is largely robust to the estimation errors we document. Based on our simulation results, we recommend the use of the DeMiguel et al. (Improving portfolio selection using option-implied volatility and skewness. J. Financ. Quant. Anal., 2013, 48(6), 1813–1845) methodology to estimate the 3rd and 4th implied moments, and either the DeMiguel et al. (Improving portfolio selection using option-implied volatility and skewness. J. Financ. Quant. Anal., 2013, 48(6), 1813–1845) methodology or the methodology of Fukasawa et al. (Model-free implied volatility: From Surface to Index. Int. J. Theor. Appl. Finance, 2011, 14(4), 433–463) to estimate the 2nd implied moment (implied volatility). In addition, we argue that the economically large and robust results of our MC simulations and brief empirical study indicate that the findings of prior studies which use the estimation methodology of Dennis and Mayhew (Risk-neutral skewness: Evidence from stock options. J. Financ. Quant. Anal., 2002, 37(3), 471–493), followed by Conrad et al. (Ex ante skewness and expected stock returns. J. Finance., 2013, 68(1), 85–124), should be reevaluated.

Assessing models of individual equity option prices

This article investigates option models in the encompassing class of stochastic volatility, return-jumps, and volatility-jumps. Relying on individual equity options on the 50 most active firms and maximum likelihood estimation method, we obtain several findings. First, while stochastic volatility is as important for individual equity options as it is for index options, return-jumps and volatility-jumps are also essential in pricing individual equity options. Second, the double-jump model improves pricing performance beyond return-jumps absent volatility-jumps, and beyond volatility-jumps absent return-jumps. Third, between return-jumps and volatility-jumps, the former is empirically more relevant than the latter for pricing options; and fourth, the inverse link between volatility-jumps and return-jumps is instrumental for explaining the valuation of deep out-of-money puts and the option dynamics of firms with high kurtosis.

Does Corporate Investment Respond to the Time-Varying Cost of Capital? Empirical Evidence

AbstractI examine whether the time-varying cost of capital is considered in firms’ capital budgeting decisions. For this test, I measure the conditional cost of equity using individual equity option prices. I find that corporate investment responds negatively to fluctuations in the option-implied cost of equity and the weighted average costs of capital. Furthermore, through decomposing marginal $ q $, I reveal that the cost-of-capital elasticity of empirical investment is almost identical to its productivity elasticity, as theory predicts. These findings suggest that firms’ discount rates are updated accurately in practice despite the failure of conventional frameworks, such as factor-based models, in this regard.

Does Corporate Investment Respond to Time-Varying Risk Premia? Empirical Evidence

I examine whether the time-varying cost of capital is considered in firms' capital budgeting decisions. For this test, I measure the conditional cost of equity, using individual equity option prices. I find that corporate investment responds negatively to fluctuations in the option-implied cost of equity and the weighted average costs of capital. Furthermore, through decomposing marginal $q$, I reveal that the cost-of-capital elasticity of empirical investment is almost identical to its productivity elasticity, as theory predicts. These findings suggest that firms' discount rates are updated accurately in practice despite the failure of conventional frameworks, such as factor-based models, in this regard.

Existence and Uniqueness of Martingale Solutions to Option Pricing Equations with Noise

We introduce a new option pricing equation with noise in a frictional financial market, which is fully different from the classical option pricing equation, and arrive at the existence of martingale solutions of this option pricing equation regardless of incompressibility. Furthermore, we also discuss the uniqueness of martingale solutions.

Equity Option Pricing with Systematic and Idiosyncratic Volatility and Jump Risks

Recently, a large number of empirical studies indicated that individual equity options exhibit a strong factor structure. In this paper, the importance of systematic and idiosyncratic volatility and jump risks on individual equity option pricing is analyzed. First, we propose a new factor structure model for pricing the individual equity options with stochastic volatility and jumps, which takes into account four types of risks, i.e., the systematic diffusion, the idiosyncratic diffusion, the systematic jump, and the idiosyncratic jump. Second, we derive the closed-form solutions for the prices of both the market index and individual equity options by utilizing the Fourier inversion. Finally, empirical studies are carried out to show the superiority of our model based on the S&P 500 index and the stock of Apple Inc. on options. The out-of-sample pricing performance of our proposed model outperforms the other three benchmark models especially for short term and deep out-of-the-money options.

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.

Analytic Equity Option Pricing with Stochastic Volatility

We consider the Heston stochastic volatility model for equity options and use operator formalism and perturbation expansion techniques to derive a formal analytic expression for the Green's function (pricing kernel). This is expanded explicitly as a perturbation series in powers of the vol of vol up to second order and used to obtain an asymptotically accurate pricing formula for vanilla equity options, replicating results obtained previously by Ito and Malliavin techniques.