scholarly journals A Comparative Analysis of the Black-Scholes- Merton Model and the Heston Stochastic Volatility Model

2019 ◽  
Vol 39 ◽  
pp. 127-140
Author(s):  
Tahmid Tamrin Suki ◽  
ABM Shahadat Hossain
Keyword(s):  
Comparative Analysis ◽  
Stochastic Volatility ◽  
Stock Prices ◽  
Computer Algebra System ◽  
Stochastic Volatility Model ◽  
Affecting Factors ◽  
Pricing Models ◽  
Merton Model ◽  
Volatility Model ◽  
Black Scholes

This paper compares the performance of two different option pricing models, namely, the Black-Scholes-Merton (B-S-M) model and the Heston Stochastic Volatility (H-S-V) model. It is known that the most popular B-S-M Model makes the assumption that volatility of an asset is constant while the H-S-V model considers it to be random. We examine the behavior of both B-S-M and H-S-V formulae with the change of different affecting factors by graphical representations and hence assimilate them. We also compare the behavior of some of the Greeks computed by both of these models with changing stock prices and hence constitute 3D plots of these Greeks. All the numerical computations and graphical illustrations are generated by a powerful Computer Algebra System (CAS), MATLAB. GANIT J. Bangladesh Math. Soc.Vol. 39 (2019) 127-140

scholarly journals SWITCHING TO NONAFFINE STOCHASTIC VOLATILITY: A CLOSED-FORM EXPANSION FOR THE INVERSE GAMMA MODEL

2016 ◽  
Vol 19 (05) ◽  
pp. 1650031 ◽  
Author(s):  
NICOLAS LANGRENÉ ◽  
GEOFFREY LEE ◽  
ZILI ZHU
Keyword(s):  
Closed Form ◽  
Stochastic Volatility ◽  
Stochastic Volatility Model ◽  
Heston Model ◽  
Time Dependent ◽  
Market Data ◽  
Affine Models ◽  
Inverse Gamma ◽  
Volatility Model ◽  
Black Scholes

We examine the inverse gamma (IGa) stochastic volatility model with time-dependent parameters. This nonaffine model compares favorably in terms of volatility distribution and volatility paths to classical affine models such as the Heston model, while being as parsimonious (only four stochastic parameters). In practice, this means more robust calibration and better hedging, explaining its popularity among practitioners. Closed-form volatility-of-volatility expansions are obtained for the price of vanilla options, which allow for very fast pricing and calibration to market data. Specifically, the price of a European put option with IGa volatility is approximated by a Black–Scholes price plus a weighted combination of Black–Scholes Greeks, with weights depending only on the four time-dependent parameters of the model. The accuracy of the expansion is illustrated on several calibration tests on foreign exchange market data. This paper shows that the IGa model is as simple, more realistic, easier to implement and faster to calibrate than classical transform-based affine models. We therefore hope that the present work will foster further research on nonaffine models favored by practitioners such as the IGa model.

scholarly journals ESTIMASI VALUE AT RISK MENGGUNAKAN VOLATILITAS DISPLACED DIFFUSION

2019 ◽  
Vol 8 (4) ◽  
pp. 298
Author(s):  
MIRANDA NOVI MARA DEWI ◽  
KOMANG DHARMAWAN ◽  
KARTIKA SARI
Keyword(s):  
At Risk ◽  
Stochastic Process ◽  
Stochastic Volatility ◽  
Diffusion Model ◽  
Stock Prices ◽  
Value At Risk ◽  
Implied Volatility ◽  
Stochastic Volatility Model ◽  
Financial Asset ◽  
Volatility Model

Value at Risk (VaR) is a measure of risk that is able to calculate the worst possible loss that can occurs to stock prices with a certain level of confidence and within a certain period of time. The purpose of this study was to determine the VaR estimate from PT. Indonesian Telecommunications by using Displaced Diffusion volatility. The Displaced Diffusion Model is a stochastic volatility model that describes changes in a financial asset assuming volatility is not constant, but follows a stochastic process. Displaced Diffusion model are capable of modelling skewed implied volatility structures and frequently applied by interest rate quants. Based on the estimation of Displaced Diffusion volatility, it is found that volatility for PT. Indonesian Telecommunications is 0.010168 and VaR estimation using Displaced Diffusion volatility with a confidence level of  95 percent of 1.63%.

scholarly journals Pricing Various Types of Power Options under Stochastic Volatility

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1911
Author(s):  
Youngrok Lee ◽  
Yehun Kim ◽  
Jaesung Lee
Keyword(s):  
Financial Markets ◽  
Stochastic Volatility ◽  
Asset Price ◽  
Stochastic Volatility Model ◽  
Exotic Options ◽  
Option Prices ◽  
Black Scholes Model ◽  
Volatility Model ◽  
Black Scholes ◽  
Pricing Power

The exotic options with curved nonlinear payoffs have been traded in financial markets, which offer great flexibility to participants in the market. Among them, power options with the payoff depending on a certain power of the underlying asset price are widely used in markets in order to provide high leverage strategy. In pricing power options, the classical Black–Scholes model which assumes a constant volatility is simple and easy to handle, but it has a limit in reflecting movements of real financial markets. As the alternatives of constant volatility, we focus on the stochastic volatility, finding more exact prices for power options. In this paper, we use the stochastic volatility model introduced by Schöbel and Zhu to drive the closed-form expressions for the prices of various power options including soft strike options. We also show the sensitivity of power option prices under changes in the values of each parameter by calculating the resulting values obtained from the formulas.

FROM THE IMPLIED VOLATILITY SKEW TO A ROBUST CORRECTION TO BLACK-SCHOLES AMERICAN OPTION PRICES

2001 ◽  
Vol 04 (04) ◽  
pp. 651-675 ◽  
Author(s):  
JEAN-PIERRE FOUQUE ◽  
GEORGE PAPANICOLAOU ◽  
K. RONNIE SIRCAR
Keyword(s):  
Free Boundary ◽  
Boundary Problem ◽  
Stochastic Volatility ◽  
Free Boundary Problem ◽  
Implied Volatility ◽  
Stochastic Volatility Model ◽  
Option Prices ◽  
One Dimensional ◽  
Volatility Model ◽  
Black Scholes

We describe a robust correction to Black-Scholes American derivatives prices that accounts for uncertain and changing market volatility. It exploits the tendency of volatility to cluster, or fast mean-reversion, and is simply calibrated from the observed implied volatility skew. The two-dimensional free-boundary problem for the derivative pricing function under a stochastic volatility model is reduced to a one-dimensional free-boundary problem (the Black-Scholes price) plus the solution of a fixed boundary-value problem. The formal asymptotic calculation that achieves this is presented here. We discuss numerical implementation and analyze the effect of the volatility skew.

SIMPLIFIED HEDGE FOR PATH-DEPENDENT DERIVATIVES

2016 ◽  
Vol 19 (07) ◽  
pp. 1650045 ◽  
Author(s):  
CAROLE BERNARD ◽  
JUNSEN TANG
Keyword(s):  
Stochastic Volatility ◽  
Stochastic Volatility Model ◽  
Asian Option ◽  
Distributional Properties ◽  
Lookback Option ◽  
Volatility Model ◽  
Black Scholes ◽  
Delta Hedging ◽  
Path Dependent

Path-dependent derivatives are typically difficult to hedge. Traditional dynamic delta hedging does not perform well because of the difficulty to evaluate the Greeks and the high cost of constantly rebalancing. We propose to price and hedge path-dependent derivatives by constructing simplified alternatives that preserve certain distributional properties of their terminal payoffs, and that can be hedged by semi-static replication. The method is illustrated by a geometric Asian option and by a lookback option in the Black–Scholes setting, for which explicit forms of the simplified alternatives exist. Extensions to a Lévy market and to a Heston stochastic volatility model are discussed as well.

RISK MANAGEMENT UNDER A FACTOR STOCHASTIC VOLATILITY MODEL

2011 ◽  
Vol 28 (01) ◽  
pp. 65-80 ◽  
Author(s):  
MARCOS ESCOBAR ◽  
PABLO OLIVARES
Keyword(s):  
Risk Management ◽  
Stochastic Volatility ◽  
Value At Risk ◽  
Factor Model ◽  
Risk Measures ◽  
Stochastic Volatility Model ◽  
Dependence Structure ◽  
Black Scholes Model ◽  
Volatility Model ◽  
Black Scholes

In this paper, we study risk measures and portfolio problems based on a Stochastic Volatility Factor Model (SVFM). We analyze the sensitivity of Value at Risk (VaR) and Expected Shortfall (ES) to the changes in the parameters of the model. We compare the positions of a linear portfolio under assets following a SVFM, a Black–Scholes Model and a model with constant dependence structure. We consider an application to a portfolio of three selected Asian funds.

scholarly journals THE 3/2 MODEL AS A STOCHASTIC VOLATILITY APPROXIMATION FOR A LARGE-BASKET PRICE-WEIGHTED INDEX

2015 ◽  
Vol 18 (06) ◽  
pp. 1550041 ◽  
Author(s):  
BEN HAMBLY ◽  
JUOZAS VAICENAVICIUS
Keyword(s):  
Stochastic Volatility ◽  
Common Factor ◽  
Stochastic Volatility Model ◽  
Model Approximation ◽  
Model Driven ◽  
Weighted Index ◽  
Volatility Model ◽  
Black Scholes ◽  
Volatility Derivatives ◽  
Jump Diffusion Model

We derive large-basket approximations of a price-weighted index whose component prices follow a single sector jump-diffusion model. As the basket size approaches infinity, a suitable average converges to a Black–Scholes model driven by the common factor process. We extend this by considering the behavior of the residual idiosyncratic noise and show that a version of the 3/2 model emerges as a natural stochastic volatility model approximation. This provides a theoretical justification for its use as a model for jointly pricing index and volatility derivatives.

PRICING PERPETUAL TIMER OPTION UNDER THE STOCHASTIC VOLATILITY MODEL OF HULL–WHITE

2017 ◽  
Vol 58 (3-4) ◽  
pp. 406-416
Author(s):  
JICHAO ZHANG ◽  
XIAOPING LU ◽  
YUECAI HAN
Keyword(s):  
Analytical Solution ◽  
Stochastic Volatility ◽  
Joint Distribution ◽  
Bessel Process ◽  
Time Change ◽  
Stochastic Volatility Model ◽  
Explicit Analytical Solution ◽  
Volatility Model ◽  
Black Scholes ◽  
Instantaneous Volatility

The valuation of perpetual timer options under the Hull–White stochastic volatility model is discussed here. By exploring the connection between the Hull–White model and the Bessel process and using time-change techniques, the triple joint distribution for the instantaneous volatility, the cumulative reciprocal volatility and the cumulative realized variance is obtained. An explicit analytical solution for the price of perpetual timer call options is derived as a Black–Scholes–Merton-type formula.

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