Analytical Approximations of American Call Options with Discrete Dividends

2018 ◽  
Vol 26 (3) ◽  
pp. 283-310
Author(s):  
Kwangil Bae
Keyword(s):  
Brownian Motion ◽  
Stock Prices ◽  
Stock Price ◽  
Computing Time ◽  
Option Price ◽  
Geometric Brownian Motion ◽  
Option Prices ◽  
Call Options ◽  
Analytical Approximations ◽  
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.

Research on the American Call Options on the Stocks Paying Multiple Dividends

2019 ◽  
Vol 27 (3) ◽  
pp. 253-274
Author(s):  
Kwangil Bae
Keyword(s):  
Brownian Motion ◽  
Lower Bound ◽  
Stock Prices ◽  
Stock Price ◽  
Numerical Study ◽  
Option Price ◽  
Upper And Lower Bounds ◽  
Multivariate Normal ◽  
Call Options ◽  
Pricing Formula

Cassimon et al. (2007) propose a pricing formula of American call options under the multiple dividends by extending Roll (1977). However, because these studies investigate the option pricing formula under the escrow model, there is inconsistency for the assumption of the stock prices. This paper proposes pricing formulas of American call options under the multiple dividends and piecewise geometric Brownian motion. For the formulas, I approximate the log prices of ex-dividend dates to follow a multivariate normal distribution, and decompose the option price as a function of payoffs and exercise boundaries. Then, I obtain an upper bound of the American call options by substituting approximated log prices into the both of the payoffs and the exercise boundaries. Besides, I obtain a lower bound of the price by substituting approximated price only into the exercise boundaries. These upper and lower bounds are exact prices when the amounts of dividends are linear to the stock prices. According to the numerical study, the lower bound produces relatively small errors. Especially, it produces small errors when the dividends are more sensitive to the stock price changes.

scholarly journals OPTIONS WRITTEN ON STOCKS WITH KNOWN DIVIDENDS

2004 ◽  
Vol 07 (07) ◽  
pp. 901-907
Author(s):  
ERIK EKSTRÖM ◽  
JOHAN TYSK
Keyword(s):  
Stock Price ◽  
Option Price ◽  
Option Prices ◽  
Strike Price ◽  
Call Options ◽  
Alternative Approach ◽  
Black Scholes ◽  
Market Practice ◽  
The Difference ◽  
Risk Free Rate

There are two common methods for pricing European call options on a stock with known dividends. The market practice is to use the Black–Scholes formula with the stock price reduced by the present value of the dividends. An alternative approach is to increase the strike price with the dividends compounded to expiry at the risk-free rate. These methods correspond to different stock price models and thus in general give different option prices. In the present paper we generalize these methods to time- and level-dependent volatilities and to arbitrary contract functions. We show, for convex contract functions and under very general conditions on the volatility, that the method which is market practice gives the lower option price. For call options and some other common contracts we find bounds for the difference between the two prices in the case of constant volatility.

scholarly journals Geometric Brownian Motion and Value at Risk For Analysis Stock Price Of Bumi Serpong Damai Ltd

2021 ◽  
Vol 1 (1) ◽  
pp. 11-21
Author(s):  
Trimono Trimono ◽  
Di Asih I Maruddani ◽  
Prisma Hardi Aji Riyantoko ◽  
I Gede Susrama Mas Diyasa
Keyword(s):  
Brownian Motion ◽  
Capital Market ◽  
Confidence Level ◽  
Stock Prices ◽  
Value At Risk ◽  
Stock Price ◽  
Risk Measure ◽  
Geometric Brownian Motion ◽  
Financial Assets ◽  
The People

Investment is one of the activities that last actually attractive to the people of Indonesia. One of the most widely traded financial assets in the capital market is stocks. Stock prices frequently experience challenges to predict changes, so they can increase or decrease at any time. One method that can be applied to predict stock prices is GBM. Then, the risk can be measured using the VaR risk measure. The GBM model is determined to be accurate in predicting the stock price of BSDE.JK, with a MAPE value of 5.17%. By using VaR-HS and VaR CFE, the prediction of risk of loss at the 95% confidence level for the period 06/07/21 is -0.0597 and -0.0623

scholarly journals A LOGISTIC NONLINEAR BLACK-SCHOLES-MERTON PARTIAL DIFFERENTIAL EQUATION: EUROPEAN OPTION

2018 ◽  
Vol 6 (6) ◽  
pp. 480-487
Author(s):  
Joseph Otula Nyakinda
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Brownian Motion ◽  
Stock Price ◽  
Option Price ◽  
Logistic Models ◽  
Geometric Brownian Motion ◽  
Partial Differential ◽  
Illiquid Markets ◽  
Black Scholes

Nonlinear Black-Scholes equations provide more accurate values by taking into account more realistic assumptions, such as transaction costs, illiquid markets, risks from an unprotected portfolio or large investor's preferences, which may have an impact on the stock price, the volatility, the drift and the option price itself. Most modern models are represented by nonlinear variations of the well-known Black-Scholes Equation. On the other hand, asset security prices may naturally not shoot up indefinitely (exponentially) leading to the use of Verhulst’s Logistic equation. The objective of this study was to derive a Logistic Nonlinear Black Scholes Merton Partial Differential equation by incorporating the Logistic geometric Brownian motion. The methodology involves, analysis of the geometric Brownian motion, review of logistic models, process and lemma, stochastic volatility models and the derivation of the linear and nonlinear Black-Scholes-Merton partial differential equation. Illiquid markets have also been analyzed alongside stochastic differential equations. The result of this study may enhance reliable decision making based on a rational prediction of the future asset prices given that in reality the stock market may depict a nonlinear pattern.

scholarly journals Limit theorems for prices of options written on semi-Markov processes

2021 ◽  
Vol 105 (0) ◽  
pp. 3-33
Author(s):  
E. Scalas ◽  
B. Toaldo
Keyword(s):  
Brownian Motion ◽  
Limit Theorems ◽  
Option Price ◽  
Type Equation ◽  
Geometric Brownian Motion ◽  
Bernstein Function ◽  
Option Prices ◽  
Motion Time ◽  
Semi Markov Processes ◽  
Inverse Stable Subordinator

We consider plain vanilla European options written on an underlying asset that follows a continuous time semi-Markov multiplicative process. We derive a formula and a renewal type equation for the martingale option price. In the case in which intertrade times follow the Mittag-Leffler distribution, under appropriate scaling, we prove that these option prices converge to the price of an option written on geometric Brownian motion time-changed with the inverse stable subordinator. For geometric Brownian motion time changed with an inverse subordinator, in the more general case when the subordinator’s Laplace exponent is a special Bernstein function, we derive a time-fractional generalization of the equation of Black and Scholes.

scholarly journals HOLDER-EXTENDIBLE EUROPEAN OPTION: CORRECTIONS AND EXTENSIONS

2015 ◽  
Vol 56 (4) ◽  
pp. 359-372 ◽  
Author(s):  
PAVEL V. SHEVCHENKO
Keyword(s):  
Brownian Motion ◽  
Interest Rate ◽  
Real Life ◽  
Option Price ◽  
Geometric Brownian Motion ◽  
Fixed Amount ◽  
Underlying Asset ◽  
Closed Form Solutions ◽  
Pricing Options ◽  
Interest Rate Volatility

Financial contracts with options that allow the holder to extend the contract maturity by paying an additional fixed amount have found many applications in finance. Closed-form solutions for the price of these options have appeared in the literature for the case when the contract for the underlying asset follows a geometric Brownian motion with constant interest rate, volatility and nonnegative dividend yield. In this paper, option price is derived for the case of the underlying asset that follows a geometric Brownian motion with time-dependent drift and volatility, which is more important for real life applications. The option price formulae are derived for the case of a drift that includes nonnegative or negative dividend. The latter yields a solution type that is new to the literature. A negative dividend corresponds to a negative foreign interest rate for foreign exchange options, or storage costs for commodity options. It may also appear in pricing options with transaction costs or real options, where the drift is larger than the interest rate.

Option pricing under the fractional stochastic volatility model

2021 ◽  
Vol 63 ◽  
pp. 123-142
Author(s):  
Yuecai Han ◽  
Zheng Li ◽  
Chunyang Liu
Keyword(s):  
Brownian Motion ◽  
Option Pricing ◽  
Fractional Brownian Motion ◽  
Stochastic Volatility ◽  
Option Price ◽  
Stochastic Volatility Model ◽  
Option Prices ◽  
European Call Option ◽  
Volatility Model ◽  
The Impact

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

scholarly journals The Geometric Brownian Motion of Indosat Telecommunications Daily Stock Price During the Covid-19 Pandemic in Indonesia

2021 ◽  
Vol 2084 (1) ◽  
pp. 012012
Author(s):  
Tiara Shofi Edriani ◽  
Udjianna Sekteria Pasaribu ◽  
Yuli Sri Afrianti ◽  
Ni Nyoman Wahyu Astute
Keyword(s):  
Brownian Motion ◽  
Stock Price ◽  
Service Providers ◽  
Main Idea ◽  
Geometric Brownian Motion ◽  
Government Policies ◽  
Percentage Error ◽  
Price Movement ◽  
Data Movement ◽  
Total Data

Abstract One of the major telecommunication and network service providers in Indonesia is PT Indosat Tbk. During the coronavirus (COVID-19) pandemic, the daily stock price of that company was influenced by government policies. This study addresses stock data movement from February 5, 2020 to February 5, 2021, resulted in 243 data, using the Geometric Brownian motion (GBM). The stochastic process realization of this stock price fluctuates and increases exponentially, especially in the 40 latest data. Because of this situation, the realization is transformed into log 10 and calculated its return. As a result, weak stationary in variance is obtained. Furthermore, only data from December 7, 2020 to February 5, 2021 fulfill the GBM assumption of stock price return, as R t 1 * , t 1 * = 1 , 2 , 3 , … , 40 . The main idea of this study is adding datum one by one as much as 10% – 15% of the total data R t 1 * , starting from December 4, 2020 backwards. Following this procedure, and based on the 3% < p-value < 10%, the study shows that its datum can be included in R t 1 * , so t 1 * = − 4. − 3 , − 2 , … , 40 and form five other data groups, R t 2 * , … , R t 6 * . Considering Mean Absolute Percentage Error (MAPE) and amount of data from each group, R t 6 * is selected for modelling. Thus, GBM succeeded in representing the stock price movement of the second most popular Indonesian telecommunication company during COVID-19 pandemic.

scholarly journals PENENTUAN HARGA OPSI BELI ATAS SAHAM PT. ANTAM (PERSERO) MENGGUNAKAN MODEL BINOMIAL FUZZY

2018 ◽  
Vol 19 (1) ◽  
pp. 8-24
Author(s):  
Agung Prabowo ◽  
Zulfatul Mukarromah ◽  
Lisnawati Lisnawati ◽  
Pramono Sidi
Keyword(s):  
Stock Price ◽  
Market Price ◽  
Option Price ◽  
Risk Level ◽  
Binomial Model ◽  
Purchase Price ◽  
Option Prices ◽  
Movement Data ◽  
Curve Representation ◽  
Price Option

Option is a financial instrument where price depends on the underlying stock price. The pricing of options, both selling options and purchase options, may use the CRR (Cox-Ross-Rubinstein) binomial model. Only two possible parameters were used that is u if the stock price rises and d when the stock price down. One of the elements that determine option prices is volatility. In the binomial model CRR volatility is constant. In fact, the financial market price of stocks fluctuates so that volatility also fluctuates. This article discusses volatility of fluctuating stock price movements by modeling it using binomial fuzzy with triangular curve representation. The analysis is carried out in relation to the existence of three interpretations of the triangular curve representation resulting in different degrees of membership. In addition to volatility, this study added the size or risk level ρ. As an illustration, this study used stock price movement data from PT. Antam (Persero) from August 2015 until July 2016. The results of one period obtained from the purchase price option for August 2016 with the largest volatility, medium and smallest respectively were Rp.143,43, Rp.95,49, and Rp.79,00. There was calculated at the risk level of  ρ = 90%. The degree of membership for each option price varies depending on the interpretation of the triangle curve representation.   Opsi merupakan instrumen keuangan yang harganya tergantung pada harga saham yang mendasarinya. Penentuan harga opsi, baik opsi jual maupun opsi beli dapat menggunakan model binomial CRR (Cox-Ross-Rubinstein). Dalam model ini hanya dimungkinkan adanya dua parameter yaitu u apabila harga saham naik dan d pada saat harga saham turun. Salah satu unsur yang menentukan harga opsi adalah volatilitas. Dalam model binomial CRR digunakan volatilitas yang bersifat konstan. Padahal, pada pasar keuangan pergerakan harga saham mengalami fluktuasi sehingga volatilitas juga menjadi fluktuatif. Artikel ini membahas volatilitas pergerakan harga saham yang fluktuatif dengan memodelkannya menggunakan binomial fuzzy dengan representasi kurva segitiga. Analisis dilakukan terkait dengan adanya tiga interpretasi terhadap representasi kurva segitiga tersebut yang menghasilkan derajat keanggotaan yang berbeda. Selain volatilitas, dalam penelitian ini ditambahkan ukuran atau tingkat risiko ρ. Sebagai ilustrasi, digunakan data pergerakan harga saham PT. Antam (Persero) dari Agustus 2015 hingga Juli 2016. Hasil penelitian dengan perhitungan satu periode diperoleh hasil harga opsi beli untuk bulan Agustus 2016 dengan volatilitas terbesar, menengah, dan terkecil masing-masing adalah Rp.143,43, Rp.95,49, dan Rp.79,00 yang dihitung pada tingkat risiko ρ = 90%. Derajat keanggotaan untuk masing-masing harga opsi berbeda-beda tergantung pada interpretasi dari representasi kurva segitiga.

scholarly journals ANALYSIS OF MARKET VOLATILITY VIA A DYNAMICALLY PURIFIED OPTION PRICE PROCESS

2014 ◽  
Vol 09 (03) ◽  
pp. 1450006 ◽  
Author(s):  
CHUONG LUONG ◽  
NIKOLAI DOKUCHAEV
Keyword(s):  
Stock Price ◽  
Implied Volatility ◽  
Financial Time Series ◽  
Option Price ◽  
Option Prices ◽  
Price Process ◽  
Volatility Index ◽  
Quadratic Interpolation ◽  
Dynamic Estimation ◽  
The Impact

The paper studies methods of dynamic estimation of volatility for financial time series. We suggest to estimate the volatility as the implied volatility inferred from some artificial "dynamically purified" price process that in theory allows to eliminate the impact of the stock price movements. The complete elimination would be possible if the option prices were available for continuous sets of strike prices and expiration times. In practice, we have to use only finite sets of available prices. We discuss the construction of this process from the available option prices using different methods. In order to overcome the incompleteness of the available option prices, we suggests several interpolation approaches, including the first order Taylor series extrapolation and quadratic interpolation. We examine the potential of the implied volatility derived from this proposed process for forecasting of the future volatility, in comparison with the traditional implied volatility process such as the volatility index VIX.

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