Cubic Hermite Finite Element Method for Nonlinear Black-Scholes Equation Governing European Options

A numerical algorithm for solving a generalized Black-Scholes partial differential equation, which arises in European option pricing considering transaction costs is developed. The Crank-Nicolson method is used to discretize in the temporal direction and the Hermite cubic interpolation method to discretize in the spatial direction. The efficiency and accuracy of the proposed method are tested numerically, and the results confirm the theoretical behaviour of the solutions, which is also found to be in good agreement with the exact solution.

European Option Pricing Formula in Risk-Aversive Markets

In this study, using the method of discounting the terminal expectation value into its initial value, the pricing formulas for European options are obtained under the assumptions that the financial market is risk-aversive, the risk measure is standard deviation, and the price process of underlying asset follows a geometric Brownian motion. In particular, assuming the option writer does not need the risk compensation in a risk-neutral market, then the obtained results are degenerated into the famous Black–Scholes model (1973); furthermore, the obtained results need much weaker conditions than those of the Black–Scholes model. As a by-product, the obtained results show that the value of European option depends on the drift coefficient μ of its underlying asset, which does not display in the Black–Scholes model only because μ = r in a risk-neutral market according to the no-arbitrage opportunity principle. At last, empirical analyses on Shanghai 50 ETF options and S&P 500 options show that the fitting effect of obtained pricing formulas is superior to that of the Black–Scholes model.

European Option Pricing under Wishart Processes

This study deals with a single risky asset pricing model whose volatility is described by Wishart affine processes. This multifactor model with two dependency matrices describing the correlation between the asset dynamic and Wishart processes makes it more flexible enough to fit the market data for short or long maturities. The aim of the study is to derive and solve the call option pricing problem under the double Wishart stochastic volatility model. The Fourier transform techniques combined with perturbation methods are employed in order to price the European call options. The numerical illustrations on pricing predictions show similar behavior of price movements under the double Wishart model with respect to the market price.

Bond and Option Prices under Skew Vasicek Model with Transaction Cost

This paper studies the European option pricing on the zero-coupon bond in which the Skew Vasicek model uses to predict the interest rate amount. To do this, we apply the skew Brownian motion as the random part of the model and show that results of the model predictions are better than other types of the model. Besides, we obtain an analytical formula for pricing the zero-coupon bond and find the European option price by constructing a portfolio that contains the option and a share of the bond. Since the skew Brownian motion is not a martingale, thus we add transaction costs to the portfolio, where the time between trades follows the exponential distribution. Finally, some numerical results are presented to show the efficiency of the proposed model.

A regression-based Monte Carlo method to solve two-dimensional forward backward stochastic differential equations

The purpose of this paper is to investigate the numerical solutions to two-dimensional forward backward stochastic differential equations(FBSDEs). Based on the Fourier cos-cos transform, the approximations of conditional expectations and their errors are studied with conditional characteristic functions. A new numerical scheme is proposed by using the least-squares regression-based Monte Carlo method to solve the initial value of FBSDEs. Finally, a numerical experiment in European option pricing is implemented to test the efficiency and stability of this scheme.

The Approximate Analytic Solution of the Time-Fractional Black-Scholes Equation with a European Option Based on the Katugampola Fractional Derivative

In the finance market, it is well known that the price change of the underlying fractal transmission system can be modeled with the Black-Scholes equation. This article deals with finding the approximate analytic solutions for the time-fractional Black-Scholes equation with the fractional integral boundary condition for a European option pricing problem in the Katugampola fractional derivative sense. It is well known that the Katugampola fractional derivative generalizes both the Riemann–Liouville fractional derivative and the Hadamard fractional derivative. The technique used to find the approximate analytic solutions of the time-fractional Black-Scholes equation is the generalized Laplace homotopy perturbation method, the combination of the generalized Laplace transform and homotopy perturbation method. The approximate analytic solution for the problem is in the form of the generalized Mittag-Leffler function. This shows that the generalized Laplace homotopy perturbation method is one of the most effective methods to construct approximate analytic solutions of the fractional differential equations. Finally, the approximate analytic solutions of the Riemann–Liouville and Hadamard fractional Black-Scholes equation with the European option are also shown.

Option Pricing under Double Heston Jump-Diffusion Model with Approximative Fractional Stochastic Volatility

Based on the present studies about the application of approximative fractional Brownian motion in the European option pricing models, our goal in the article is that we adopt the creative model by adding approximative fractional stochastic volatility to double Heston model with jumps since approximative fractional Brownian motion is more proper for application than Brownian motion in building option pricing models based on financial market data. We are the first to adopt the creative model. We derive the pricing formula for the options and the formula for the characteristic function. We also estimate the parameters with the loss function for the model and two nested models and compare the performance among those models based on the market data. The outcome illustrates that the model offers the best performance among the three models. It demonstrates that approximative fractional Brownian motion is more proper for application than Brownian motion.