Performance Enhancement of Trinomial Option Pricing Model

This paper is mainly about the research object of early exercise of American options by two-period binomial models. In this work, we use the two-period binomial model to analyze the price of American options and European options, decide whether to exercise early or not, and compare the differences between American options and European options. What is more, we use Python to help us with our calculations. Options are a kind of financial derivatives that help one to arbitrage, speculate, and hedge. To make more profits, people can use the binomial model to calculate the price of options. Through this analysis, we got to know each price of options in different situations, and we know that the price of American options is always equal to or larger than the price of European options. This work can help us to calculate the price of the option and decide if it is exercised early, and make sure to get more profits and apply it to daily life.

This paper reviews the past research on option pricing, including the Bachelier option pricing formula, Black-Scholes model, and the Cox-Ross-Rubinstein (CRR) binomial option pricing model. Then some practical calculations are performed on the Black-Scholes model and CRR binomial model and the differences between the results of these two models are analyzed for investors to choose the appropriate model for option prices. From the practical calculation results, it is verified that the Black-Scholes model can give out the analytical solutions of option prices and decrease the calculation of solution, while it can only be used for European options with stock prices regarded as a continuous process. For the CRR binomial model, although it can only give numerical solutions and requires lots of calculation because of too many branches, but it can solve the American option prices with early exercises and the European option prices with stock prices regarded as a discrete process.

The paper develops a new class of financial market models. These models are based on generalised telegraph processes: Markov random flows with alternating velocities and jumps occurring when the velocities are switching. While such markets may admit an arbitrage opportunity, the model under consideration is arbitrage-free and complete if directions of jumps in stock prices are in a certain correspondence with their velocity and interest rate behaviour. An analog of the Black–Scholes fundamental differential equation is derived, but, in contrast with the Black–Scholes model, this equation is hyperbolic. Explicit formulas for prices of European options are obtained using perfect and quantile hedging.

Numerical solving of the generalized Black-Scholes differential equation using Laguerre neural network

In this paper, some problems about Black-Scholes model with fuzzy number coefficients is studied. Firstly, by using the Ito formula, the asset prices at $T$ or $t$ times are obtained from the Black-Scholes model with fuzzy number coefficients. Then, by operation rules of fuzzy numbers, the expression of the price of European call option in form of level set of fuzzy number is obtained. At last, an example is given to show the feasibility of the method.

ABSTRACTDistributional Divergence and Statistical Experiments are used herein for a positive stochastic process. This framework provides, under mild assumptions, Risk Neutral Probability (-ies) for a stock price process which does not have necessarily either to satisfy a Stochastic Differential Equation or to follow a model, both non-realistic assumptions. The results contribute in understanding the relation between statistical contiguity and market's informational efficiency. -price of European option is obtained, confirming the universal quote of the Black–Scholes–Merton price for the class of calm stock prices that includes log-normal price. Other consequences are presented.

The impact of market frictional costs on pricing insurance and financial products in a regime-switching environment has not been well-explored. This paper introduces a general pricing model for European options which incorporates market frictional costs, regime switches and model uncertainty. Regime switches are due to changes in an economic environment. Model uncertainty is attributed to misspecification of transition intensities for economic regimes. The selling and buying prices of a European option are determined through stochastic optimal control and nonlinear partial differential equations. A fair value is determined by a closed-form solution to a minimization problem based on a relative entropy. The fair value is consistent with the one obtained using the Esscher transform, which is an important tool in actuarial science. Numerical methods and results for implementing the pricing model are presented. The results indicate that after controlling for the model uncertainty, market frictional costs are more significant than regime switches in accounting for the fair, selling and buying prices.

Pricing European Options Based on the Hesitancy Degree of Investors

Bias Reduction in European Option Pricing

We propose a Legendre–Laguerre spectral approximation to price the European and double barrier options in the time-fractional framework. By choosing an appropriate basis function, the spectral discretization is used for the approximation of the spatial derivatives of the time-fractional Black–Scholes equation. For the time discretization, we consider the popular \(L1\) finite difference approximation, which converges with order \(\mathcal{O}((\Delta \tau)^{2-\alpha})\) for functions which are twice continuously differentiable. However, when using the \(L1\) scheme for problems with nonsmooth initial data, only the first-order accuracy in time is achieved. This low-order accuracy is also observed when solving the time-fractional Black–Scholes European and barrier option pricing problems for which the payoffs are all nonsmooth. To increase the temporal convergence rate, we therefore consider a Richardson extrapolation method, which when combined with the spectral approximation in space, exhibits higher order convergence such that high accuracies over the whole discretization grid are obtained. Compared with the traditional finite difference scheme, numerical examples clearly indicate that the spectral approximation converges exponentially over a small number of grid points. Also, as demonstrated, such high accuracies can be achieved in much fewer time steps using the extrapolation approach. doi:10.1017/S1446181121000286

The establishment of option prices is one of the crucial aspects in derivative trade. Black-Scholes model (BSM) is one of the most popular models of option price establishment. The option exchanges such as CBOE, HKEX, and NSE use this model to determine the price options. Black-Scholes Model is modelled with stock price movement as a stochastic process. Another popular model is binomial model (BSM), originated from stock exchange movement model which divides interval time [0, T] into n equal length step. It holds several models to determine the value of up-move, down-move, and probability. Binomial model is categorised as Cox-Ross-Rubinstein, Jarrow-Rudd, and Leisen-Reimer. There are numerous literatures which discuss the relation between BM and BSM, including the convergence of binomial model and BSM. The former’s model which is often put side-by-side with BSM is Cox-Ross-Rubinstein. Even though this model is quite simple, it requires a lot, even thousands of steps to render Cox-Ross-Rubinstein to converge with BSM. It certainly takes a lot of time to calculate. Therefore, in this study, with limited steps, Jarrow-Rudd and Leisen-Reimer models are compared to BSM with the Cox-Ross-Rubinstein model. It aims to check on which binomial model is more convergent to BSM with limited steps in the same period. The data collected were secondary data from finance.yahoo.com. Judgemental sampling was used for technique sampling and several shares, with large market capitalization in Hongkong, India, and Indonesia are chosen. By calculating the MAFE error value from option price of BSM and BM, it is discovered that Leisen-Reimer with 101 steps is more convergent to BSM.

The Indonesian stock market, especially PT Bank Central Asia (BCA) stock experiences high volatility. Therefore, this research is very important to provide a deeper understanding of the pricing of put and call options as a risk management instrument. This study aims to analyze the pricing of put and call options on PT Bank Central Asia (BCA) shares using the Black-Scholes model and identify factors that affect stock price fluctuations on option prices. This study aims to understand the changes in put and call option prices on PT Bank Central Asia (BCA) shares caused by fluctuations in stock prices. This study uses historical data of PT Bank Central Asia (BCA) shares, volatility, and current interest rates as the basis for analysis. The Black-Scholes method is used as a framework to analyze the pricing of put and call options by considering volatility and interest rates. The results of this study analyze the price changes of put and call options in the face of stock price fluctuations and the factors that affect option prices so as to provide insight into risk management and investment decision making. This research is expected to provide practical guidance for investors and risk managers in making decisions related to put and call options on PT Bank Central Asia (BCA) shares.

Option markets, empirical price data, and theoretical arguments all indicate that asset prices in actively traded markets are driven by Lévy flights and/or tempered Lévy flights, not by Brownian motion. So here we model asset prices in the real world by (1) where dZL is a Lévy or tempered Lévy flight. Derivatives based on such assets cannot be made risk free by dynamic hedging, so these derivatives cannot be priced using the standard Black–Scholes–Merton (BSM) arbitrage-free pricing criterion. Therefore, we develop a pricing theory based on a more general pricing criterion: that the expected return of all diversifiable portfolios is at the risk-free rate. We show that even though derivatives based on Lévy or tempered Lévy-driven assets cannot be hedged to make risk-free portfolios, they can be hedged to make diversifiable portfolios. This allows us to conclude that these option prices are given by (2) where the expected value uses the “real-world” probability measure, but with the asset price X(t) replaced by the alternative “pricing process” (3) which grows at the risk-free rate. We analyze these models to obtain explicit asymptotic formulas for European option prices. This analysis shows that as the time to expiry decreases, eventually all these Lévy and tempered Lévy-based models reduce to the same canonical model, and that European option prices and implied volatilities are given by similarity solutions under this canonical model. These similarity solutions are then examined to assess the mishedging that arises from Brownian-based models in a Lévy world.

<abstract><p>In this paper, the pricing of European options under a new two-factor non-affine stochastic volatility model is studied. In order to reduce the computational complexity, we use the Taylor expansion and Fourier-cosine method to derive an analytical approximation formula for European option prices. Numerical experiments prove that the proposed formula is fast and efficient for pricing European options compared with Monte Carlo simulations. The sensitivity of the parameters is analyzed to explain the rationality of the model. Finally, we present some preliminary empirical analysis revealing that the pricing performance of our proposed model is superior to that of the single-factor model.</p></abstract>

The main purpose of this paper is to study the stock price jump process is more common than Poisson process -a kind of special renewal process-the incident time interval is independent random variable sequence and identical subordinate to Gamma distribution. The object of study is European bi-direction option, It also known as the double two-way options. Also, European bi-direction option is an exotic option. Then, build up stochastic differential equation under the circumstance of the market no arbitrage based on stochastic analysis and martingale theory. Based on the above discussion, we obtain the European bi-direction option pricing formulas under jump diffusion model by simple mathematical induce by means of martingale method.