Price Of Call Option Research Articles

Geometric Average Asian Option Pricing with Paying Dividend Yield under Non-Extensive Statistical Mechanics for Time-Varying Model.

This paper is dedicated to the study of the geometric average Asian call option pricing under non-extensive statistical mechanics for a time-varying coefficient diffusion model. We employed the non-extensive Tsallis entropy distribution, which can describe the leptokurtosis and fat-tail characteristics of returns, to model the motion of the underlying asset price. Considering that economic variables change over time, we allowed the drift and diffusion terms in our model to be time-varying functions. We used the formula, Feynman–Kac formula, and ansatz to obtain a closed-form solution of geometric average Asian option pricing with a paying dividend yield for a time-varying model. Moreover, the simulation study shows that the results obtained by our method fit the simulation data better than that of Zhao et al. From the analysis of real data, we identify the best value for q which can fit the real stock data, and the result shows that investors underestimate the risk using the Black–Scholes model compared to our model.

Proactive Hedging European Call Option Pricing with Linear Position Strategy

Proactive hedging option is an exotic European stock option designed for hedgers. Such option requires option holders to buy in (or sell out) the underlying asset (stock) and allows them to adjust the holdings of the underlying asset per its price changes within an option period. The proactive hedging option is an attractive choice for hedgers because its price is lower than that of classical options and because it completely hedges the risk of exposure for option holders. In this study, the underlying asset price movement is assumed to follow geometric fractional Brownian motion. The pricing formula for proactive hedging call options is derived with a linear position strategy by applying the risk-neutral evaluation principle. We use simulations to confirm that the price of this exotic option is always no more than that of the classical European option under the same parameters.

Barycentric spectral domain decomposition methods for valuing a class of infinite activity Lévy models

A new barycentric spectral domain decomposition methods algorithm for solving partial integro-differential models is described. The method is applied to European and butterfly call option pricing problems under a class of infinite activity Levy models. It is based on the barycentric spectral domain decomposition methods which allows the implementation of the boundary conditions in an efficient way. After the approximation of the spatial derivatives, we obtained the semi-discrete equations. The computation of these equations is performed by using the barycentric spectral domain decomposition method. This is achieved with the implementation of an exponential time integration scheme. Several numerical tests for the pricing of European and butterfly options are given to illustrate the efficiency and accuracy of this new algorithm. We also show that Greek options, such as Delta and Gamma sensitivity, are computed with no spurious oscillation.

Analytical Approximations of American Call Options with Discrete Dividends

In this study, we assume that stock prices follow piecewise geometric Brownian motion, a variant of geometric Brownian motion except the ex-dividend date, and find pricing formulas of American call options. While piecewise geometric Brownian motion can effectively incorporate discrete dividends into stock prices without losing consistency, the process results in the lack of closed-form solutions for option prices. We aim to resolve this by providing analytical approximation formulas for American call option prices under this process. Our work differs from other studies using the same assumption in at least three respects. First, we investigate the analytical approximations of American call options and examine European call options as a special case, while most analytical approximations in the literature cover only European options. Second, we provide both the upper and the lower bounds of option prices. Third, our solutions are equal to the exact price when the size of the dividend is proportional to the stock price, while binomial tree results never match the exact option price in any circumstance. The numerical analysis therefore demonstrates the efficiency of our method. Especially, the lower bound formula is accurate, and it can be further improved by considering second order approximations although it requires more computing time.

PRICING BERMUDAN-TYPE CALL OPTION THROUGH BINOMIAL TREE METHOD

<p class="Default"><em>Bermudan option is a type of option that has characteristics between American option and European option whose its value never exceeds the value of the American option and is never less than the European option. The objective of this research is to calculate Bermudan call option of John Keels Stock through the binomial tree method using statistics software of Matlab R2010a. </em><em>Assessment of Bermudan type option relates to discrete issues, in which the Bermudan type option has a certain number of times of early exercise specified in the option contract, where such times can only be made at some time prior to the option due date. Precise pricing for Bermudan type option can be obtained by discrete models such as the binomial tree method, a numerical method that is one of the most popular approaches for calculating option prices. This research uses time series data obtained from BNI Financial Update Corner, FEB UGM. The Bermudan call option price calculation will be compared with the calculation of European option pricing and American option price with underlying asset without dividend. The results show that the price of John Keels's Bermudan type call option using the binomial tree method yields the same value as American type call option and European type call option.</em><em></em></p><strong><em>Keywords:</em></strong><em> Bermudan Type Option, Binomial Tree Method, Matlab R2010a, Spss 20</em>

Right Tail Hedging: Managing Risk When Markets Melt Up

Popular academic and practitioner lore claims that buying options, whether puts or calls, is a negative expected return investment and hence should not be undertaken by rational, risk-neutral investors. However, real-world considerations such as the possibility of large jumps and imitative behavior of traders in both bull and bear markets can reverse this conclusion. In particular, the potential for positive economic shocks such as those observed since the 2016 U.S. election may make call options–based strategies superior to buy and hold strategies. In this article, the author extends his work published in The Journal of Portfolio Management in 2007 on left tail or downside tail risk hedging to address upside hedging, which has become increasingly relevant in an environment of low yields, elevated asset prices, low credit spreads, indexation, and multidecadal lows in call option prices.

PENENTUAN HARGA OPSI DENGAN MODEL BLACK-SCHOLES MENGGUNALKAN METODE BEDA HINGGA FORWARD TIME CENTRAL SPACE

Option can be defined as a contract between two sides/parties said party one and party two. Party one has the right to buy or sell of stock to party two. Party two can invest by observe the put option price or call option price on a time period in the option contract. Black-Scholes option solution using finite difference method based on forward time central space (FTCS) can be used as the reference for party two in the investment determining. Option price determining by using Black-Scholes was applied on Samsung stock (SSNLF) by using finite difference method FTCS. Daily data of Samsung stock in one year was processed to obtain the volatility of the stock. Then, the call option and put option are calculated by using FTCS method after discretization on the Black-Scholes model. The value of call option was obtained as $1.457695030014260 and the put option value was obtained as $1.476925604670225.

Big moves of mutual funds

Previous research suggests that short-sellers correctly anticipate that big moves by one or more mutual funds cause temporary price distortions. By moving in the opposite direction to those trades, short-sellers outsmart mutual funds. This appears to be healthy for the overall market, but mutual funds’ vulnerability substantially reduces their profits. For informed traders, the equity lending market and the options market are substitutes for one another. In this paper, I empirically investigate how big moves of mutual funds affect the cross-section of option prices. Overall, I find strong empirical evidence for directional trading in index options. The flow-induced bearish trading activity of short-sellers is price destabilizing and greatly influences put and call option prices. Firstly, in line with the short-selling channel, inflows into mutual funds mainly impact the pricing of put options. The demand for out-of-the money put options leads to a significant drop in risk-neutral skewness. Secondly, inflows (outflows) affect the risk-neutral volatility negatively (positively); the impact on prices is consistent across put and call options. Thirdly, while the effect on risk-neutral skewness is associated with the unexpected component (based on same-day flows) only, the effect on risk-neutral volatility is associated with the unexpected and the expected component (based on prior days’ trading). Finally, while the negative relationship of inflows on risk-neutral volatility and skewness is consistent over time, the effect of outflows on risk-neutral volatility is not observable in bearish markets.

Small-time expansions for state-dependent local jump–diffusion models with infinite jump activity

In this article, we consider a Markov process {Xt}t⩾0, which solves a stochastic differential equation driven by a Brownian motion and an independent pure jump component exhibiting both state-dependent jump intensity and infinite jump activity. A second order expansion is derived for the tail probability P[Xt⩾x+y] in small time t, where x is the initial value of the process and y>0. As an application of this expansion and a suitable change of the underlying probability measure, a second order expansion, near expiration, for out-of-the-money European call option prices is obtained when the underlying stock price is modeled as the exponential of the jump–diffusion process {Xt}t⩾0 under the risk-neutral probability measure.

: CAPM (Security Prices in Mean-Variance Equilibrium: A Further Study on CAPM)

Chinese Abstract: CAPM 所在的资本市场是非完全的,并且是 一 个 Hilbert 空间,我们找出了该市场中 SDF 模仿支付的显式表达式。纯风险证券的局部均衡中,CAPM 公式成立,CAPM 等价于市场组合恰好为切点组合,并且证券价格的通解在一维空间中。给定投资者的禀赋和均值方差偏好,结合 CAPM 的价格通解,资本市场的均衡条件转化为单变量的方程,从而求得均值方差均衡价格的解析解。证券价格是内生的,全体风险证券是作为 一 个整体被定价的,貌似真实的风险决定收益是局部均衡的产物,其错误在于把市场收益率当成外生的。我们还通过数值例子,分析均值方差决策与无套利的关系,化解不可能前沿的危机,展示贝塔定价导致欧式看涨期权负价格,以及不可以使用贝塔估计权益资本成本。CAPM 实证方面,我们区分了均衡收益率与时序收益率,讨论被忽视的 DGP 和平稳性等问题,并结合均衡解析解给出针对预测型因子模型的建议。 English Abstract: The capital market for CAPM is incomplete and is a Hilbert space, we find out the analytic expression for the SDF mimicking payoff in this market. CAPM is equivalent to the condition that the market portfolio is the tangent portfolio. CAPM holds under the partial equilibrium of pure risky assets, and the general solution to asset prices has only one dimension. Given investor’s endowment and mean-variance preference, applying the general solution from CAPM, the condition of market equilibrium turns out to be an equation of only one variable, from which we work out the closed form solutions. By the mistake that the return of market portfolio is assumed to be exogenous, the risk-return characteristics is a concept from the partial equilibrium of pure risky assets, thus it is illusory, for the assets are priced as a whole and the prices are endogenous. By way of numerical examples, we explore the relationship between mean-variance equilibrium and free of arbitrage, defuse the crisis of impossible frontier, present the negative price of European call option pricing by beta, and point out the incorrect practice of the estimation of cost of equity by beta. On the empirical study of CAPM, we emphasize the difference between rate of return in mean- variance equilibrium and the one computed by price time series. We discuss the unnoticed issues such as DGP, stationarity and others. For the predictive factor models, we provide suggestions based on the closed form solution of mean-variance equilibrium.

The Call Option Pricing Based on Investment Strategy with Stochastic Interest Rate

In this paper, the call option price is evaluated based on linear investment strategy in order to hedge the risk actively in stock market with stochastic interest rate. The Vasicek model is used to describe the structure of interest rates. The mathematical characterization is discussed for the unique no-arbitrage price associated with any attainable contingent claim. The appropriate numeraire (zero-coupon bond) and measures (T-forward measure) are chosen to simplify the calculations. Based on the designed linear investment strategy with stochastic interest rate, a novel option price approach is obtained under the T-forward measure.

Risk-Neutral Pricing of European Call Options: A Specious Concept

Risk-neutral pricing of European call options is investigated from a mathematical point-of-view and is found to be a specious concept1. Risk-neutral pricing of European call options is an approximation in which all terms of order are ignored, where is the risk premium and σ is the volatility.

Mixed Fractional Merton Model to Evaluate European Options with Transaction Costs

This paper deals with the problem of discrete-time option pricing by the mixed fractional version of Merton model with transaction costs. By a mean-self-financing delta hedging argument in a discrete-time setting, a European call option pricing formula is obtained. We also investigate the effect of the time-step δt and the Hurst parameter H on our pricing option model, which reveals that these parameters have high impact on option pricing. The properties of this model are also explained.

Pricing Options on Ghanaian Stocks Using Black-Scholes Model

We present a succinct new approach to derive the Black-Scholes partial differential equation and subsequently the Black-Scholes formula. We proceed to use the formula to price options using stocks listed on Ghana stock exchange as underlying assets. From one year historical stock prices we obtain volatilities of the listed stocks which are subsequently used to compute prices of three month European call option. The results indicate that it is possible to use the Black Scholes formula to price options on the stocks listed on exchange. However, it was realised that most call option prices tend to zero either due to very low volatilities or very low stock prices. On the other hand put options were found to give positive prices even for stocks with very low volatilities or low stock prices.

A Linear Regression Approach for Determining Option Pricing for Currency-Rate Diffusion Model with Dependent Stochastic Volatility, Stochastic Interest Rate, and Return Processes

A three-factor exchange-rate diffusion model that includes three stochastically-dependent Brownian motion processes, namely, the domestic interest rate process, volatility process and return process is considered. A linear regression approach that derives explicit expressions for the distribution function of log return of foreign exchange rate is derived. Subsequently, a closed form workable formula for the call option price that has an algebraic expression similar to a Black-Scholes model, which facilitates easier study, is discussed.

Corrections in Heston model derivations for bond options

A closed form solution for European call option price has been first proposed by Black and Scholes. Later, Heston modified the model by replacing the constant volatility with a stochastic volatility and applied it on bond options. However, for bond options, we found that the mathematical derivations of the partial differential equations involving probabilities for the call option expiring in the money obtained by Heston contain some errors. This paper rectifies the errors and a new PDE has been derived. The paper uses important theories of stochastic calculus and portfolio theory such as Ito's lemma, portfolio hedging while formulating the PDE for the option price probabilities.

Penentuan Harga Opsi Asia dengan Metode Monte Carlo

An option is a contract between a holder and a writer in which the writer grants the rights (not obligations) to the holder to buy or sell the assets of the writer at a certain price (strike price) at maturity time. Asian options are included in the dependent path option. This means that Asia's payoff option depends not only on the stock price at maturity time, but it is the average stock price during its maturity and symbolized A (average). Monte Carlo is basically used as a numerical procedure to estimate the expected value of pricing product derivatives. The techniques used are the standard Monte Carlo and variance reduction. The result obtained the Asia call option price and put for both techniques with 95% confidence interval. The variance reduction technique looks faster reducing 95% confidence interval than standard method.

A Hybrid Monte Carlo and Finite Difference Method for Option Pricing

We propose an accurate, efficient, and robust hybrid finite difference method, with a Monte Carlo boundary condition, for solving the Black–Scholes equations. The proposed method uses a far-field boundary value obtained from a Monte Carlo simulation, and can be applied to problems with non-linear payoffs at the boundary location. Numerical tests on power, powered, and two-asset European call option pricing problems are presented. Through these numerical simulations, we show that the proposed boundary treatment yields better accuracy and robustness than the most commonly used linear boundary condition. Furthermore, the proposed hybrid method is general, which means it can be applied to other types of option pricing problems. In particular, the proposed Monte Carlo boundary condition algorithm can be implemented easily in the code of the existing finite difference method, with a small modification.

Option Price Decomposition in Spot-Dependent Volatility Models and Some Applications

We obtain a Hull and White type option price decomposition for a general local volatility model. We apply the obtained formula to CEV model. As an application we give an approximated closed formula for the call option price under a CEV model and an approximated short term implied volatility surface. These approximated formulas are used to estimate model parameters. Numerical comparison is performed for our new method with exact and approximated formulas existing in the literature.

EMPIRICAL TEST OF PUT-CALL PARITY ON SPX OVER THE SHORT BAN 2008

Put call parity is a theoretical no-arbitrage condition linking a call option price to a put option price written on the same stock or index. This study finds that Put call parity violations are quite symmetric over the whole sample. However during the ban period 2008 in the U.S., puts are significantly and economically overpriced relative to calls. Some possible explanations are the short selling restriction, momentum trading behaviour and the changes in supply and demand of puts over the short ban. One interesting finding that the relationship between time to expiry, put call parity deviations and returns on the index is highly non-linear.
 Key word: Put-call parity, SPX, short ban 2008
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