Price Of European Call Option Research Articles

Research on the option pricing of Amazon and Google based on B-S model and Monte Carlo simulation

The emergence of financial derivatives has helped people to effectively hedge the risks present in the financial markets while giving them the ability to make leveraged investments in the financial markets. Although derivatives did not originate in the financial market, it still has a huge role in the financial market so far. In recent years, science and technology innovation is known as a growing concern for governments due to factors such as environmental issues. In this paper, we choose to study the financial derivative, European call option pricing problem of technology-based companies AMZN and GOOGL. In this paper, the European call options of these two companies are priced by using the BSM model, and the results predicted by the BSM model are improved by using Monte Carlo Simulation. Ultimately, the paper finds that Monte Carlo Simulation prices European call options on technology-based firms nearly three times higher than those priced by the BSM model. The results suggest that using a single model to price European call options on technology-based firms is inaccurate. The purpose of this paper is to help investors in pricing European call options on technology-based firms and to warn investors about the inaccuracy of a single option pricing model.

Option Pricing under a Generalized Black–Scholes Model with Stochastic Interest Rates, Stochastic Strings, and Lévy Jumps

We introduce a novel option pricing model that features stochastic interest rates along with an underlying price process driven by stochastic string shocks combined with pure jump Lévy processes. Substituting the Brownian motion in the Black–Scholes model with a stochastic string leads to a class of option pricing models with expiration-dependent volatility. Further extending this Generalized Black–Scholes (GBS) model by adding Lévy jumps to the returns generating processes results in a new framework generalizing all exponential Lévy models. We derive four distinct versions of the model, with each case featuring a different jump process: the finite activity lognormal and double–exponential jump diffusions, as well as the infinite activity CGMY process and generalized hyperbolic Lévy motion. In each case, we obtain closed or semi-closed form expressions for European call option prices which generalize the results obtained for the original models. Empirically, we evaluate the performance of our model against the skews of S&P 500 call options, considering three distinct volatility regimes. Our findings indicate that: (a) model performance is enhanced with the inclusion of jumps; (b) the GBS plus jumps model outperform the alternative models with the same jumps; (c) the GBS-CGMY jump model offers the best fit across volatility regimes.

European Option Pricing under Sub-Fractional Brownian Motion Regime in Discrete Time

In this paper, the approximate stationarity of the second-order moment increments of the sub-fractional Brownian motion is given. Based on this, the pricing model for European options under the sub-fractional Brownian regime in discrete time is established. Pricing formulas for European options are given under the delta and mixed hedging strategies, respectively. Furthermore, European call option pricing under delta hedging is shown to be larger than under mixed hedging. The hedging error ratio of mixed hedging is shown to be smaller than that of delta hedging via numerical experiments.

An Optimization Approach for pricing of Discrete European Call options Based on the Preference of Investors

An Optimization Approach for pricing of Discrete European Call options Based on the Preference of Investors

On regime-switching European option pricing

The concern of this article is to derive a regime switching model that can be utilized to price European call options for a financial market that exhibits structural changes with time. The model is formulated based on the fact that the underlying asset process is described by a geometric Brownian motion that is modulated by a continuous-time Markov chain with two regimes. Moreover, by an application of the change of measure technique, an option price is derived under the risk neutral valuation and the model parameter estimates is performed by use of the maximum likelihood estimation. The model implementation is carried out by utilizing the Russell 2000 and Facebook in dices data sets. The model results are compared with that of the Black-Scholes model in order to establish the model with better results in terms of predicting the European call option prices. In general, the data sets have common characteristics of financial time series across the regimes and the volatility process spends longer time in regime 2 than it stays in regime 1. The predicted call option prices from both models are more or less similar across the market indices; however, the results of the Black-Scholes model are a bit closer to the market prices than that of the regime-switching model across the two markets. Therefore, the Black-Scholes model slightly gives better results for the Russell 2000 and Facebook indices data sets as compared with the RS model.

Noncausal affine processes with applications to derivative pricing

AbstractLinear factor models, where the factors are affine processes, play a key role in Finance, since they allow for quasi‐closed form expressions of the term structure of risks. We introduce the class of noncausal affine linear factor models by considering factors that are affine in reverse time. These models are especially relevant for pricing sequences of speculative bubbles. We show that they feature nonaffine dynamics in calendar time, while still providing (quasi) closed form term structures and derivative pricing formulas. The framework is illustrated with term structure of interest rates and European call option pricing examples.

Application Of Natural Decomposition Method For Solution Of Fractional Black-Scholes Equation

The Black-Scholes equation is a partial differential equation that can model the European call option price problem. This equation can be of the order of natural numbers or fractional. The aim of this paper is to find a solution to the fractional order Black-Scholes partial differential equation. The method used to find solutions to these equations is the Natural decomposition method. Two numerical examples are presented in this paper. The results show that the Natural decomposition method is effective and easy to use to solve the fractional Black-Scholes equation.

Pricing of European call option under fuzzy interest rate

<p style='text-indent:20px;'>Option pricing under fuzzy environment is a hot research topic nowadays. Traditionally, option pricing were made in the case of fixed interest rate, while the fluctuate of interest rate may result in profit loss or bring unexpected risk. Thus, based on credibility theory, a new option pricing model under fuzzy interest rate are constructed in this paper. In fact, almost all fuzzy option pricing uses expected value method. In this paper, a new pricing method, fuzzy term structure and fuzzy affine term structure method, is adopted, and two European call option pricing formulas are obtained, one is that the fuzzy interest rate coefficients are constants, the other is that the fuzzy interest rate drift coefficient is a fuzzy process.</p>

Option Prices Expansions and Applications

Daniel Bloch presents a general pricing approximation technique for European call option prices in a jump-diffusion model with stochastic interest rates. The author considers the dynamics of the logarithm of the forward price, under the forward measure, in the class of multifactor Affine and Quadratic models without jumps, and devise the dynamics of its associated variance swap. Bloch expresses the expected future average volatility as a function of that variance swap, and uses classical Ito's calculus to expand prices around the Black formula. The author then adds jumps to the dynamics of the log-forward price and condition the expectation of the call price with respect to the number of jumps. Applying a change of measure, option prices decompose into a weighted sum of approximated multifactor Affine and Quadratic models. Bloch uses these prices decompositions to define an analytical formula that approximate the implied volatility surface in the class of Affine and Quadratic models with jumps. At last, the author compares a few prices approximations against Monte Carlo simulations for a range of maturities covering one year, and provide accuracy measures. Bloch then uses these prices as benchmark to measure the accuracy of the implied volatility surface expansion. The author obtains very low MAE and RMSE, both on the prices and IVS, demonstrating that the series expansions are very precise for short and long maturities. Applications include: fast prices and Greeks estimation for European options, initial values for numerical computation of IVS, and control variate for complex pricing models in Monte Carlo and Machine Learning.

A numerical comparative analysis between crank-Nicolson finite difference method and binomial model for European call option price

There are several kinds of numerical techniques for solving option valuation problems. In this article Binomial model (BM) and Crank-Nicolson finite difference (CNFD) approach are applied and compared with the Black-Scholes analytic solution (BSAS) to determine the best numerical method. It has been noticed that, for the valuation of European call (EC) options at various factors, the Binomial model (BM) is found to be more accurate than the Crank-Nicolson finite difference (CNFD) technique. In addition, for the impact of high volatility, the technique of Crank-Nicolson finite difference (CNFD) approaches quicker to Black-Scholes analytic solution (BSAS) than the Binomial model (BM). Furthermore, in comparison to the Binomial model (BM), CNFD method consumes more time to determine the option price in each time step.

Option pricing under the fractional stochastic volatility model

We investigate the European call option pricing problem under the fractional stochastic volatility model. The stochastic volatility model is driven by both fractional Brownian motion and standard Brownian motion. We obtain an analytical solution of the European option price via the Itô’s formula for fractional Brownian motion, Malliavin calculus, derivative replication and the fundamental solution method. Some numerical simulations are given to illustrate the impact of parameters on option prices, and the results of comparison with other models are presented. doi:10.1017/S1446181121000225

Option pricing with polynomial chaos expansion stochastic bridge interpolators and signed path dependence

Recent technological advances have made possible the obtention of vast amounts of market data and strong computing power for advanced models which would not have been practicable for use in real market settings before. In this manuscript we devise a model-free empirical risk-neutral distribution based on Polynomial Chaos Expansions coupled with stochastic bridge interpolators that includes information from the entire set of observable European call option prices under all available strikes and maturities for a given underlying asset in a way that is guaranteed by construction to produce a valid state price distribution function at all times. We also obtain a non parametric model for the risk premium behaviour via an optimisation problem that joins the risk-neutral Polynomial Chaos Expansion result with any general model for the real-world distribution. Finally, we show an empirical application on SP500 Options on Futures using a real-world distribution that assumes the presence of signed path dependence in the returns of the underlying asset.

Residue Sum Formula for Pricing Options under the Variance Gamma Model

We present and prove a triple sum series formula for the European call option price in a market model where the underlying asset price is driven by a Variance Gamma process. In order to obtain this formula, we present some concepts and properties of multidimensional complex analysis, with particular emphasis on the multidimensional Jordan Lemma and the application of residue calculus to a Mellin–Barnes integral representation in C3, for the call option price. Moreover, we derive triple sum series formulas for some of the Greeks associated to the call option and we discuss the numerical accuracy and convergence of the main pricing formula.

Option pricing under a Markov-modulated Merton jump-diffusion dividend

In this paper, the valuation of European option is investigated when the discrete dividends are described by Markov-modulated Merton jump-diffusion process. According to the dividend discount theory, we regard the stock price as the net present value of all future dividends. The regime switching Esscher transform is applied to determine a risk-neutral measure. The closed form solution of European option is obtained under the condition that the dividend payments are announced in advance. Numerical simulations for the European call option prices are provided.

OPTION PRICING UNDER THE FRACTIONAL STOCHASTIC VOLATILITY MODEL

AbstractWe investigate the European call option pricing problem under the fractional stochastic volatility model. The stochastic volatility model is driven by both fractional Brownian motion and standard Brownian motion. We obtain an analytical solution of the European option price via the Itô’s formula for fractional Brownian motion, Malliavin calculus, derivative replication and the fundamental solution method. Some numerical simulations are given to illustrate the impact of parameters on option prices, and the results of comparison with other models are presented.

An efficient Monte Carlo simulation for new uncertain Heston–CIR hybrid model

In this paper, we consider two new stock models in which their differential equations are modeled by Liu process in uncertain environment. Firstly, we study the uncertain Schobel–Zhu–Hull–White hybrid model and obtain its closed European call option pricing using Liu calculus. Also, we solve this model by Monte Carlo simulation to ensure the performance of Monte Carlo method. Our main purpose is to present a new model, uncertain Heston–CIR hybrid model, in which its uncertain differential equations cannot be solved and so we can calculate the option value via Monte Carlo simulation. Finally, some examples are stated for illustrating these models to obtain successful results and show the efficiency of Monte Carlo method.

Option Pricing with Markov Switching

In this article, we consider a model of time-varying volatility which generalizes the classical Black-Scholes model to include regime-switching properties. Specically, the unobservable state variables for stock uctu- ations are modeled by a Markov process, and the drift and volatility pa- rameters take dierent values depending on the state of this hidden Markov process. We provide a closed-form formula for the arbitrage-free price of the European call option, when the hidden Markov process has nite number of states. Two simulation methods, the discrete diusion method and the Markovian tree method, for computing the European call option price are presented for comparison.

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.

Option pricing under time interval driven model

A new model which we called time driven model for option pricing is proposed. In this model, the price of underlying asset is driven by different random driving source in different time interval. Explicit option pricing formulas for the European call option are obtained by using the Mellin transform. Moreover, some numerical illustrations are given by computing European call option prices.

CVaR-hedging and its applications to equity-linked life insurance contracts with transaction costs

<p style='text-indent:20px;'>This paper analyzes Conditional Value-at-Risk (CVaR) based partial hedging and its applications on equity-linked life insurance contracts in a Jump-Diffusion market model with transaction costs. A nonlinear partial differential equation (PDE) that an option value process inclusive of transaction costs should satisfy is provided. In particular, the closed-form expression of a European call option price is given. Meanwhile, the CVaR-based partial hedging strategy for a call option is derived explicitly. Both the CVaR hedging price and the weights of the hedging portfolio are based on an adjusted volatility. We obtain estimated values of expected total hedging errors and total transaction costs by a simulation method. Furthermore,our results are implemented to derive target clients’ survival probabilities and age of equity-linked life insurance contracts.</p>