European Call Option Research Articles

Convergence of the Two Point Flux Approximation method and the fitted Two Point Flux Approximation method for options pricing with local volatility function

In this paper, we deal with numerical approximations for solving the Black–Scholes Partial Differential Equation (PDE) for European and American options pricing with local volatility. This PDE is well-known to be degenerated. Local volatility model is a model where the volatility depends locally of both stock price and time. In contrast to constant volatility or time-dependent volatility models for which analytical representations of the exact solution is known for European Call options, there is no analytical solution for local volatility. The space discretization is performed using the classical finite volume method with Two-Point Flux Approximation (TPFA) and a novel scheme called Fitted Two-Point Flux Approximation (FTPFA). The Fitted Two-Point Flux Approximation (FTPFA) combines the fitted finite volume method and the standard TPFA method. More precisely the fitted finite volume method is used when the stock price approaches zero with the goal to handle the degeneracy of the PDE while the TPFA method is used on the rest of space domain. This combination yields our fitted TPFA scheme. The Euler method is used for the time discretization. We provide the rigorous convergence proofs of the two fully discretized schemes. Numerical experiments to support theoretical results are provided.

European Call Option under Stochastic Interest Rate in a Fractional Brownian Motion with Transaction Cost

This paper deals on the valuation of European call option price in a stochastic environment by employing three factors which are the stochastic model of the asset value, the stochastic interest rate and the transaction cost. We specify that our underlying asset and the stochastic interest rate, particularly Hull-White model, follows a fractional Brownian Motion governed by Hurst parameter H. We used the hedging and replicating technique to established the zero-coupon bond on the European option. Finally, we give a closed-form formula of the European call option price.

MODEL INVESTASI POLIS ASURANSI JIWA BERBONUS TIPE DWIGUNA DENGAN AMERICAN PUT OPTION

Insurance is a form of protection against the concept of risk transfer that may occur from an uncertain event. One type of insurance is life insurance. With the changing times, life insurance launched innovative products, one of which is endowment life insurance. Endowment life insurance is a type of life insurance that provides compensation benefits if the policy holder dies during the coverage period, as well as providing cash value and bonuses if the policy holder is still alive until the end of the period. This study aims to determine the fair price of insurance policies and the price of bonus life insurance policies. Determination of the value of the American put option with the binomial method using a numerical approach, the formulation of the portfolio model includes determining the model of the unit price with Geometric Brownian Motion, solving using Black-Scholes, determining the structure of the bonus life insurance policy with the formulation of a single premium payment, bonus rate, and benefits. then analyze the movement of the price of a reasonable bonus life insurance policy with a surrender option based on the age of the insured, technical rate, participation level, and volatility. The surrender value obtained is the difference between the value of the American Put option and the European Call Option. Based on the simulation, the conclusion of this analysis is that the price of the endowment type life insurance policy can be estimated using the binomial method at around 0.78 for a fair policy price and around 0.27 for a policy price with a surrender option. This gives an idea of the relative value of the policy price to the expected benefits under certain conditions, such as the death of the insured in the first year of the contract.

Pricing European Call Options with Visualization Based on the Binomial Model, Monte-Carlo Simulation, and Classical Black-Scholes Model

Pricing European Call Options with Visualization Based on the Binomial Model, Monte-Carlo Simulation, and Classical Black-Scholes Model

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

Credible Delta Normal Value at Risk for Risk Evaluation of European Call Option

This paper develops a Credible Delta Normal Value at Risk (CredDN) as a method to assess the options risk. The method is constructed by combining Credible Value at Risk (CrVaR) with Delta Normal VaR. The new method is initiated as an alternative instrument to measure the European call option portfolio risk. Using the Black-Scholes Formula, the new method contemplates the nonlinear dependence of the market risk factors specifying the value of a European call option. Then, utilizing simulated financial data that represents assets profit/loss over ten investment periods, we applicate the proposed method for analysis. The novel method is also employed to assess the portfolio’s risk consisting of the active stocks which are involved in option trading. The performance of CredDN in this study is evaluated by administering Kupiec backtesting. The results of Kupiec backtesting suggest that the proposed method effectively quantifies the risk of each option constructing a portfolio at 80%, 90%, and 95% confidence levels.

Vulnerable European Call Option Pricing Based on Uncertain Fractional Differential Equation

Vulnerable European Call Option Pricing Based on Uncertain Fractional Differential Equation

European Call Option and the Hamilton-Jacobi Type in Classical and Fractional Forms

European Call Option and the Hamilton-Jacobi Type in Classical and Fractional Forms

Pricing of A European Call Option in Stochastic Volatility Models

Volatility occupies a strategic place in the financial markets. In this context of crisis, and with the great movements of the markets, traders have been forced to turn to volatility trading for the potential gain it provides. The Black-Scholes formula for the value of a European option to purchase the underlying depends on a few parameters which are more or less easy to calculate, except for the realized volatility at maturity which makes a problem, because there is no single value, nor an established way to calculate it. In this article, we exploit the Martingale pricing method to find the expected present value of a given asset relative to a riskneutral probability measure. We consider a bond-stock market that evolves according to the dynamics of the Black-Scholes model, with a risk-free interest rate varying with time. Our methodology has effectively directed us towards interesting formulas that we have derived from the exact calculation, giving the present value of the volatility realized over a period of maturity for a European option in a stochastic volatility model.

Perturbation analysis of sub/super hedging problems

AbstractWe investigate the links between various no‐arbitrage conditions and the existence of pricing functionals in general markets, and prove the Fundamental Theorem of Asset Pricing therein. No‐arbitrage conditions, either in this abstract setting or in the case of a market consisting of European Call options, give rise to duality properties of infinite‐dimensional sub‐ and super‐hedging problems. With a view towards applications, we show how duality is preserved when reducing these problems over finite‐dimensional bases. We also introduce a rigorous perturbation analysis of these linear programing problems, and highlight numerically the influence of smile extrapolation on the bounds of exotic options.

A Finite Difference-Spectral Method for Solving the European Call Option Black–Scholes Equation

In this paper, we present a novel technique based on backward-difference method and Galerkin spectral method for solving Black–Scholes equation. The main propose of this method is to reduce the solution of this problem to the solution of a system of algebraic equations. The convergence order of the proposed method is investigated. Also, we provide numerical experiment to show the validity of proposed method.

Formulation Of A Rational Option Pricing Model using Artificial Neural Networks

This paper inquires on the options pricing modeling using Artificial Neural Networks to price Apple(AAPL) European Call Options. Our model is based on the premise that Artificial Neural Networks can be used as functional approximators and can be used as an alternative to the numerical methods to some extent, for a faster and an efficient solution. This paper provides a neural network solution for two financial models, the BlackScholes-Merton model, and the calibrated-Heston Stochastic Volatility Model, we evaluate our predictions using the existing numerical solutions for the same, the analytic solution for the Black-Scholes equation, COS-Model for Heston’s Stochastic Volatility Model and Standard Heston-Quasi analytic formula. The aim of this study is to find a viable time-efficient alternative to existing quantitative models for option pricing.

Homotopy analysis method and its applications in the valuation of European call options with time-fractional Black-Scholes equation

• HAM and its applications in the valuation of ECO with TFBSE in a Caputo sense are presented. • By means of the marked point process, the classical Black-Scholes equation turns to a time-fractional Black-Scholes equation. • The valuation formula for the price of a European call option with fractional order is obtained via HAM. • Illustrative examples were presented to measure the performance of HAM in terms of accuracy, effectiveness and suitability. • The physical behaviour of the option prices has been shown in terms of plots for different values of fractional order. This paper presents the applications of Homotopy Analysis Method (HAM) in the valuation of a European Call Option (ECO) with Time-Fractional Black-Scholes Equation (TFBSE). The fractional derivative is considered in the sense of Caputo. Also, it is assumed that the stock price pays no dividend and follows the geometric Brownian motion. Based on HAM, a series solution for TFBSE has been obtained successfully. The valuation formula for the price of ECO with fractional order is also obtained. The accuracy, effectiveness and suitability of HAM were tested on two illustrative examples in the context of the Crank Nicolson Method (CRN), Binomial Model (BM) and the Black-Scholes Model (BSM). The comparative study of the results obtained via HAM, CRN, BM and BSM is presented. Furthermore, the physical behaviour of the option prices obtained via HAM has been shown in terms of plots for diverse fractional order. Moreover, HAM is found to be accurate, effective and suitable for the solution of TFBSE. Hence, it can be concluded that HAM converges faster to the analytical solution and is a good alternative tool to determine the price of ECO with fractional order.

Option pricing under multifractional Brownian motion in a risk neutral framework

In this paper, we introduce a new method to compute the European Call Option price (ct) under multi-fractional Brownian motion (mBm) with deterministic Hurst function. We build a mathematical framework using a Lebovits et al. study to approximate mBm to fractional Brownian motion (fBm). As a result we obtain ct , through the simulation of the logarithmic price under mBm, using a Vasicek model for the discount factor. Finally, we compare the results with those computed with the Black Scholes model and Call market price (SPX).

Option Pricing Under Multifractional Process and Long-Range Dependence

We introduced a new method to compute the European Call (and Put) Option price under the assumption of multifractional Brownian motion (mBm). The reason why we need a procedure for estimating the Option price is due to the absence of a closed formula for this process. To compute the Option price, we first simulated the logarithmic price under mBm and, by using a discount factor, we computed the option’s pay-off. Then, we fitted the best probability distribution associated to the discounted pay-off, computing the European Call Option price as its average.

Black-Scholes Model of European Call Option Pricing in Constant Market Condition

Investment is a saving activity with the aim of overcoming price increases or often called inflation. Investments can be in the form of gold, property, silver or stock investments. Stock investment is considered more profitable than just saving at a bank. Currency values are declining due to inflation. This results in a tendency to invest in shares. Stock investment carries a great risk. Therefore, in 2004 stock options began to trade. Stock options are contracts that give the holder the right to buy / sell shares at the agreed time, at a certain price. Stock option prices tend to be cheaper than stock prices. Therefore, determining the right price of stock options is needed. In this study, we will focus on the European type of buying options, the right to buy shares at an agreed price at maturity. The purpose of this study is the completion of the Black-Scholes model of European type option prices at a constant market, assuming stock movements meet the stochastic differential equation, fixed risk-free interest rates, companies distributing dividends, no taxes, no transaction costs, and free market arbitration. The results of this research are in the form of differential equations and the settlement of the Black-Scholes model of European type call option prices, and a case study used by stock option contracts with a maturity of January 4, 2010, PT Aqua Golden Mississippi Tbk.

Explicit expressions to counterparty credit exposures for Forward and European Option

With the fast development of over the counter (OTC) derivatives market, counterparty credit risk (CCR) has become one of the main risks that can even affect the survival of banks. As a result, it is essential to measure and manage CCR exposures. Giant financial institutions apply numerical methods such as the Monte Carlo to calculate exposures. However, the numerical methods usually cost a significant amount of time, and some advanced algorithms like distributed and parallel processing are usually used to accelerate the calculation. Nevertheless, for small banks, they cannot afford the calculation cost. In order to make small banks manage CCR more efficiently, this paper puts forward analytic models to measure CCR exposures and derives the explicit expressions to exposures for Forward contract and European Option, which are represented in the OTC market. The explicit expressions to credit exposures for Forward contract are derived under the assumption that the underlying market risk factors follow Geometric Brown Motion. For European Option case, the problem turns to be difficult since the analytic formula involves double-definite integral of Gaussian function that cannot be simplified into elementary functions. An approximating normal distribution function with integrability is proposed, then the analytic approximations of European Call Option’s credit exposures and maximum errors are presented.

Extended Cubic B-spline Collocation Method for Pricing European Call Option

A numerical method is proposed based on the extended cubic B-spline functions for solving the model of European call option. The Black-Scholes equation is fully-discretized by using the extended cubic B-spline collocation for spatial discretization and the finite difference method for the time discretization. The stability of difference scheme is analysed and the stability condition is given. The accuracy of this method can be improved by properly choosing the values of the parameter λ. A numerical experiment is performed to show the accuracy and stability of the proposed method.

The fractional and mixed-fractional CEV model

The continuous observation of the financial markets has identified some ‘stylized facts’ which challenge the conventional assumptions, promoting the born of new approaches. On the one hand, the long-range dependence has been faced replacing the traditional Gauss–Wiener process (Brownian motion), characterized by stationary independent increments, by a fractional version. On the other hand, the CEV model addresses the Leverage effect and smile-skew phenomena, efficiently. In this paper, these two insights are merging and both the fractional and mixed-fractional extensions for the CEV model, are developed. Using the fractional versions of both the Itô’s calculus and the Fokker–Planck equation, the transition probability density function of the asset price is obtained as the solution of a non-stationary Feller process with time-varying coefficients, getting an analytical valuation formula for a European Call option. Besides, the Greeks are computed and compared with the standard case.

Default Risk and Cross Section of Returns

Prior research uses the basic one-period European call-option pricing model to compute default measures for individual firms and concludes that both the size and book-to-market effects are related to default risk. For example, small firms earn higher return than big firms only if they have higher default risk and value stocks earn higher returns than growth stocks if their default risk is high. In this paper we use a more advanced compound option pricing model for the computation of default risk and provide a more exhaustive test of stock returns using univariate and double-sorted portfolios. The results show that long/short hedge portfolios based on Geske measures of default risk produce significantly larger return differentials than Merton’s measure of default risk. The paper provides new evidence that mediates between the rational and behavioral explanations of value premium.