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.

scholarly journals Fast Fourier Transform Based Power Option Pricing with Stochastic Interest Rate, Volatility, and Jump Intensity

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Jiexiang Huang ◽  
Wenli Zhu ◽  
Xinfeng Ruan
Keyword(s):  
Fourier Transform ◽  
Interest Rate ◽  
Fast Fourier Transform ◽  
Asset Price ◽  
Option Price ◽  
Moment Generating Function ◽  
Interest Rate Volatility ◽  
Jump Model ◽  
Jump Intensity ◽  
The Moment

Firstly, we present a more general and realistic double-exponential jump model with stochastic volatility, interest rate, and jump intensity. Using Feynman-Kac formula, we obtain a partial integrodifferential equation (PIDE), with respect to the moment generating function of log underlying asset price, which exists an affine solution. Then, we employ the fast Fourier Transform (FFT) method to obtain the approximate numerical solution of a power option which is conveniently designed with different risks or prices. Finally, we find the FFT method to compute that our option price has better stability, higher accuracy, and faster speed, compared to Monte Carlo approach.

scholarly journals NUMERICAL STABILITY OF A HYBRID METHOD FOR PRICING OPTIONS

2019 ◽  
Vol 22 (07) ◽  
pp. 1950036
Author(s):  
MAYA BRIANI ◽  
LUCIA CARAMELLINO ◽  
GIULIA TERENZI ◽  
ANTONINO ZANETTE
Keyword(s):  
Interest Rate ◽  
Asset Price ◽  
Option Prices ◽  
Underlying Asset ◽  
Binomial Tree ◽  
Price Process ◽  
Pricing Options ◽  
Model Following ◽  
Jump Model ◽  
The Interest Rate

We develop and study stability properties of a hybrid approximation of functionals of the Bates jump model with stochastic interest rate that uses a tree method in the direction of the volatility and the interest rate and a finite-difference approach in order to handle the underlying asset price process. We also propose hybrid simulations for the model, following a binomial tree in the direction of both the volatility and the interest rate, and a space-continuous approximation for the underlying asset price process coming from a Euler–Maruyama type scheme. We test our numerical schemes by computing European and American option prices.

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.

PRICING EXOTIC OPTIONS

2000 ◽  
Vol 14 (3) ◽  
pp. 317-326 ◽  
Author(s):  
Sheldon M. Ross ◽  
J. George Shanthikumar
Keyword(s):  
Brownian Motion ◽  
Convex Function ◽  
Option Price ◽  
Geometric Brownian Motion ◽  
Exotic Options ◽  
European Option ◽  
No Arbitrage ◽  
Efficient Simulation ◽  
Lookback Options ◽  
Fixed Set

We show that if the payoff of a European option is a convex function of the prices of the security at a fixed set of times, then the geometric Brownian motion risk neutral option price is increasing in the volatility of the security. We also give efficient simulation procedures for determining the no-arbitrage prices of European barrier, Asian, and lookback options.

scholarly journals Pricing Catastrophe Equity Put Options in a Mixed Fractional Brownian Motion Environment

2020 ◽  
Vol 2020 ◽  
pp. 1-15
Author(s):  
Guohe Deng
Keyword(s):  
Brownian Motion ◽  
Interest Rate ◽  
Fractional Brownian Motion ◽  
Long Range ◽  
Stock Price ◽  
Option Price ◽  
Put Options ◽  
Put Option ◽  
Mixed Fractional Brownian Motion ◽  
Short Interest

This paper considers the pricing of the CatEPut option (catastrophe equity put option) in a mixed fractional model in which the stock price is governed by a mixed fractional Brownian motion (mfBM model), which manifests long-range correlation and fluctuations from the financial market. Using the conditional expectation and the change of measure technique, we obtain an analytical pricing formula for the CatEPut option when the short interest rate is a deterministic and time-dependent function. Furthermore, we also derive analytical pricing formulas for the catastrophe put option and the influence of the Hurst index when the short interest rate follows an extended Vasicek model governed by another mixed fractional Brownian motion so that the environment captures the long-range dependence of the short interest rate. Based on the numerical experiments, we analyze quantitatively the impacts of different parameters from the mfBM model on the option price and hedging parameters. Numerical results show that the mfBM model is more close to the realistic market environment, and the CatEPut option price is evaluated accurately.

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.

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