Option Pricing Equations Research Articles

Numerical study for European option pricing equations with non-levy jumps

ABSTRACT We numerically study partial integro-differential equations that arise from the pricing of options under jump-diffusion processes using finite difference methods. Two kinds of jump-diffusion models are considered: one with non-Levy type feedback jumps and the other with an information effect in the form of jumps. Computational efficiency was achieved by employing an implicit–explicit time stepping scheme based on the Crank–Nicolson and Adams–Bashforth methods. We present some numerical results of the option pricing models with various non-Levy jump distributions for European call options and explain their implications in the financial market.

A reliable treatment of residual power series method for time-fractional Black–Scholes European option pricing equations

Approximate analytical solution of a fractional Black–Scholes pricing model is much relevant due to its practical importance to financial markets. In this article, a powerful approximate iterative mathematical scheme based on residual power series (RPS) algorithm is presented to achieve the numerical results of the nonlinear time fractional Black–Scholes (BS) equations based on European options. The Caputo-type fractional derivatives are considered in the present article. The residual power series method (RPSM) developed by Arqub (2013) is the novel technique for finding the analytical Taylor series solutions of systems of linear and nonlinear ODEs and PDEs. The residual power series technique supplies the approximate analytical solutions of the problem in truncated series form using residual error concept along with given initial conditions. The numerical procedures reveal that only a few iterations are sufficient for better approximations of the solutions, which clearly exhibits the reliability and effectiveness of this iterative scheme. This article shows that the adopted scheme is quite systematic as well as computationally attractive regarding solution procedure. In addition, effects of fractional order of time derivatives on the solutions for various particular cases are also depicted through graphs.

Black–Scholes option pricing equations described by the Caputo generalized fractional derivative

Abstract Fractional Black–Scholes equation is a constructive financial equation. The model is used to determine the value of the option without a transaction cost. The analytical solutions of the fractional Black–Scholes equations have been addressed. The Caputo generalized fractional derivative has been used. The homotopy perturbation method has been developed for obtaining the analytical solutions of the fractional Black–Scholes equation (BSE) and the generalized fractionalBSE. The analytical solutions of the fractionalBSE and the generalized fractionalBSE have been represented graphically. The effect of the order ρ of the generalized fractional derivative in the diffusion processes has been analyzed.

On the Solution of Fractional Option Pricing Model by Convolution Theorem

The classical Black-Scholes equation driven by Brownian motion has no memory, therefore it is proper to replace the Brownian motion with fractional Brownian motion (FBM) which has long-memory due to the presence of the Hurst exponent. In this paper, the option pricing equation modeled by fractional Brownian motion is obtained. It is further reduced to a one-dimensional heat equation using Fourier transform and then a solution is obtained by applying the convolution theorem.

Capital structure and financial flexibility: Expectations of future shocks

Abstract We test one of the main predictions of the financial flexibility paradigm, that expectations about future firm-specific investment shocks affect the firm's leverage. We extract the expectations of small and large future shocks from the market prices of equity options. We find that leverage decreases when expectations for any one of the two types of future shocks increase and the relation is statistically significant even when we control for standard determinants of leverage and the firm's probability of default. Expectations for future shocks explain a greater fraction of leverage variation than most standard determinants of leverage do and they affect more the small and financially constrained firms. Our results are not subject to an endogeneity bias and they confirm DeAngelo et al. (2011) model's predictions and the evidence that managers seek for financial flexibility.

Model-free stochastic collocation for an arbitrage-free implied volatility: Part I

This paper explains how to calibrate a stochastic collocation polynomial against market option prices directly. The method is first applied to the interpolation of short-maturity equity option prices in a fully arbitrage-free manner and then to the joint calibration of the constant maturity swap convexity adjustments with the interest rate swaptions smile. To conclude, we explore some limitations of the stochastic collocation technique.

Multigrid method for pricing European options under the CGMY process

We propose a fast multigrid method for solving the discrete partial integro-differential equations (PIDEs) arising from pricing European options when the underlying asset is driven by an infinite activity Levy process. We consider the CGMY model whose kernel singularity gets worse when the parameter Y approaches two. Due to the integral term, the discretization matrix is dense. In order to obtain an efficient multigrid method, we apply a fixed point iteration as a smoother for multigrid. In each smoothing step, we only need to solve a sparse matrix corresponding to the differential operator and compute a matrix-vector product involving the integral operator by a fast Fourier transform (FFT). We prove that the fixed point iteration smoother is effective reducing the high frequency components. Moreover, we also prove a two-grid convergence of the multigrid method by a local mode analysis. We demonstrate the effectiveness of the multigrid method by solving the option pricing equation under the CGMY model with finite and infinite variation processes.

A new iterative method based solution for fractional Black–Scholes option pricing equations (BSOPE)

In this manuscript, a new expansion technique namely residual power series method is used for finding the analytical solution of the Fractional Black–Scholes equation with an initial condition for European option pricing problem. The Black–Scholes formula is important for estimating European call and put option on a non-dividend paying stock in particular when it contains time-fractional derivatives. The fractional derivative is defined in Caputo sense. This technique is based on fractional power series expansion. The convergence analysis of the present method is also deliberated. Example problems are given to examine the efficacy of the proposed method. Obtained solutions are compared with exact solutions solved by other techniques which demonstrate that the present method is robust and easy to implement for other fractional problems arising in science and engineering.

Indian equity options: Smile, risk premiums, and efficiency

AbstractWe study the pricing of equity options in India which is one of the world's largest options markets. Our findings are supportive of market efficiency: A parsimonious smile‐adjusted Black model fits option prices well, and the implied volatility (IV) has incremental predictive power for future volatility. However, the risk premium embedded in IV for Single Stock Options appears to be higher than in other markets. The study suggests that even a very liquid market with substantial participation of global institutional investors can have structural features that lead to systematic departures from the behavior of a fully rational market while being “microefficient.”

Harnack Inequality and No-Arbitrage Analysis

The present paper attains a Harnack inequality for the option pricing (or Kolmogorov) equation with gradient estimate arguments. We then perform a no-arbitrage analysis of a financial market.

Recovering survival probabilities from equity option prices and credit default swap spreads

Recovering survival probabilities from equity option prices and credit default swap spreads

Foreign equity option pricing under bivariate time-varying coefficient jump-diffusion model

Foreign equity option pricing under bivariate time-varying coefficient jump-diffusion model

FINANCIAL MARKETS WITH NO RISKLESS (SAFE) ASSET

We study markets with no riskless (safe) asset. We derive the corresponding Black–Scholes–Merton option pricing equations for markets where there are only risky assets which have the following price dynamics: (i) continuous diffusions; (ii) jump-diffusions; (iii) diffusions with stochastic volatilities, and; (iv) geometric fractional Brownian and Rosenblatt motions. No-arbitrage and market-completeness conditions are derived in all four cases.

Equity Option Implied Probability of Default and Equity Recovery Rate

AbstractThere is a close link between prices of equity options and the default probability of a firm. We show that in the presence of positive expected equity recovery, standard methods that assume zero equity recovery at default misestimate the option‐implied default probability. We introduce a simple method to detect stocks with positive expected equity recovery by examining option prices and propose a method to extract the default probability from option prices that allows for positive equity recovery. We demonstrate possible applications of our methodology with examples that include financial institutions in the United States during the 2007–09 subprime crisis. © 2016 Wiley Periodicals, Inc. Jrl Fut Mark 37:599–613, 2017

Equilibrium approach of asset and option pricing under Lévy process and stochastic volatility

This paper studies the equity premium and option pricing under the general equilibrium framework taking into account stochastic volatility. We establish analytical expressions for the equity premium and pricing kernel of the stock process. Moreover, the equilibrium option pricing formula is derived by the Fourier transformation method. Numerical results show that our model is superior to the previous model with constant volatility in explaining some financial phenomena, such as negative variance risk premium, implied volatilities and negative skewness risk premium. As the price of the underlying asset is modeled as the exponential of the Lévy process with stochastic volatility, our model is more general than the existing equilibrium pricing models.

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.

A high order finite element scheme for pricing options under regime switching jump diffusion processes

This paper considers the numerical pricing of European, American and Butterfly options whose asset price dynamics follow the regime switching jump diffusion process. In an incomplete market structure and using the no-arbitrage pricing principle, the option pricing problem under the jump modulated regime switching process is formulated as a set of coupled partial integro-differential equations describing different states of a Markov chain. We develop efficient numerical algorithms to approximate the spatial terms of the option pricing equations using linear and quadratic basis polynomial approximations and solve the resulting initial value problem using exponential time integration. Various numerical examples are given to demonstrate the superiority of our computational scheme with higher level of accuracy and faster convergence compared to existing methods for pricing options under the regime switching model.

Equity Option Implied Probability of Default and Equity Recovery Rate

There is a close link between prices of equity options and the probability of default of a firm. We show that in the presence of positive expected equity recovery, the standard methods that assume zero equity recovery at default misestimate the probability of default implicit in option prices. We introduce a simple method to detect stocks with positive expected equity recovery by examining option prices, and propose a method to extract the probability of default from option prices in the presence of positive expected equity recovery. Our empirical results based on six large financial institutions in the US during the 2007-2009 crisis show that assuming zero recovery leads to significant mispricing of options and misestimation of implied probability of default.

A Linear Regression Approach for Determining Explicit Expressions for Option Prices for Equity Option Pricing Models with Dependent Volatility and Return Processes

We consider a risk-neutral stock-price model where the volatility and the return processes are assumed to be dependent. The market is complete and arbitrage-free. Using a linear regression approach, explicit functions of risk-neutral density functions of stock return functions are obtained and closed form solutions of the corresponding Black-Scholes-type option pricing results are derived. Implied volatility skewness properties are illustrated.

Numerical solution for option pricing with stochastic volatility model

The option pricing equations derived from stochatic volatility models in finance are often cast in the form of nonlinear partial differential equations. To solve the equations, we used the upwind finite difference scheme for the spatial discretisation and a fully implicit time-stepping scheme. The result of this scheme is a matrix system in the form of an M-Matrix and we proof that the approximate solution converges to the viscosity solution to the equation by showing that the scheme is monotone, consistent and stable. Numerical experiments are implemented to show that the behavior and the order of convergence of upwind finite difference method.