LOGNORMAL-MIXTURE DYNAMICS AND CALIBRATION TO MARKET VOLATILITY SMILES

2002 ◽  
Vol 05 (04) ◽  
pp. 427-446 ◽  
Author(s):  
DAMIANO BRIGO ◽  
FABIO MERCURIO
Keyword(s):  
General Class ◽  
Implied Volatility ◽  
Asset Price ◽  
Market Volatility ◽  
Option Prices ◽  
Price Model ◽  
Analytical Approximations ◽  
Mixture Dynamics ◽  
Real Market ◽  
Volatility Function

We introduce a general class of analytically tractable models for the dynamics of an asset price based on the assumption that the asset-price density is given by the mixture of known basic densities. We consider the lognormal-mixture model as a fundamental example, deriving explicit dynamics, closed form formulas for option prices and analytical approximations for the implied volatility function. We then introduce the asset-price model that is obtained by shifting the previous lognormal-mixture dynamics and investigate its analytical tractability. We finally consider a specific example of calibration to real market option data.

scholarly journals Markov Chain Monte Carlo Method for Estimating Implied Volatility in Option Pricing

2018 ◽  
Vol 10 (6) ◽  
pp. 108
Author(s):  
Yao Elikem Ayekple ◽  
Charles Kofi Tetteh ◽  
Prince Kwaku Fefemwole
Keyword(s):  
Option Pricing ◽  
Implied Volatility ◽  
Call Option ◽  
Option Prices ◽  
Options Market ◽  
European Call Option ◽  
Black Scholes Model ◽  
Black Scholes ◽  
Real Market ◽  
Independent Path

Using market covered European call option prices, the Independence Metropolis-Hastings Sampler algorithm for estimating Implied volatility in option pricing was proposed. This algorithm has an acceptance criteria which facilitate accurate approximation of this volatility from an independent path in the Black Scholes Model, from a set of finite data observation from the stock market. Assuming the underlying asset indeed follow the geometric brownian motion, inverted version of the Black Scholes model was used to approximate this Implied Volatility which was not directly seen in the real market: for which the BS model assumes the volatility to be a constant. Moreover, it is demonstrated that, the Implied Volatility from the options market tends to overstate or understate the actual expectation of the market. In addition, a 3-month market Covered European call option data, from 30 different stock companies was acquired from Optionistic.Com, which was used to estimate the Implied volatility. This accurately approximate the actual expectation of the market with low standard errors ranging between 0.0035 to 0.0275.

CONVEX REGULARIZATION OF LOCAL VOLATILITY ESTIMATION

2017 ◽  
Vol 20 (01) ◽  
pp. 1750006 ◽  
Author(s):  
VINICIUS ALBANI ◽  
ADRIANO DE CEZARO ◽  
JORGE P. ZUBELLI
Keyword(s):  
Implied Volatility ◽  
Convergence Rates ◽  
Surface Model ◽  
Option Prices ◽  
Local Volatility ◽  
Market Data ◽  
Regularization Techniques ◽  
Convex Regularization ◽  
Real Market ◽  
Volatility Surface

We apply convex regularization techniques to the problem of calibrating Dupire’s local volatility surface model taking into account the practical requirement of discrete grids and noisy data. Such requirements are the consequence of bid and ask spreads, quantization of the quoted prices and lack of liquidity of option prices for strikes far away from the at-the-money level. We obtain convergence rates and results comparable to those obtained in the idealized continuous setting. Our results allow us to take into account separately the uncertainties due to the price noise and those due to discretization errors, thus, allowing estimating better discretization levels both in the domain and in the image of the parameter to solution operator by a Morozov’s discrepancy principle. We illustrate the results with simulated as well as real market data. We also validate the results by comparing the implied volatility prices of market data with the computed prices of the calibrated model.

A NEW REPRESENTATION OF THE LOCAL VOLATILITY SURFACE

2008 ◽  
Vol 11 (07) ◽  
pp. 691-703
Author(s):  
MARIANITO R. RODRIGO ◽  
ROGEMAR S. MAMON
Keyword(s):  
Explicit Solution ◽  
Implied Volatility ◽  
Option Prices ◽  
Strike Price ◽  
Local Volatility ◽  
Forward Equation ◽  
Volatility Surface ◽  
Volatility Function ◽  
Ansatz Approach ◽  
Local Volatility Surface

In this paper, we address the problem of recovering the local volatility surface from option prices consistent with observed market data. We revisit the implied volatility problem and derive an explicit formula for the implied volatility together with bounds for the call price and its derivative with respect to the strike price. The analysis of the implied volatility problem leads to the development of an ansatz approach, which is employed to obtain a semi-explicit solution of Dupire's forward equation. This solution, in turn, gives rise to a new expression for the volatility surface in terms of the price of a European call or put. We provide numerical simulations to demonstrate the robustness of our technique and its capability of accurately reproducing the volatility function.

scholarly journals Estimation of Tail Risk and Moments Using Option Prices with a Novel Pricing Model under a Distorted Lognormal Distribution

2020 ◽  
Vol 2020 ◽  
pp. 1-25
Author(s):  
Yan Chen ◽  
Ya Cai ◽  
Chengli Zheng
Keyword(s):  
Lognormal Distribution ◽  
Risk Measures ◽  
Empirical Test ◽  
Asset Price ◽  
Test Results ◽  
Option Prices ◽  
Tail Risk ◽  
Option Markets ◽  
Novel Method ◽  
Real Market

Risk measures based on the trading option prices in the market are forward-looking, such as VIX. We propose a new method combining distorted lognormal distribution with interpolation to price options accurately and then estimate tail risk. Our method can price the option of any strikes between the maximum and the minimum value of strikes in the real market, which reduces the instability and inaccuracy of using the limited option to measure the risk. In addition, our novel method treats the underlying asset price as a stochastic indicator rather than a fixed indicator as described in previous research studies for risk measurement. Moreover, even if the available sample size is very small, we can measure the risk stably and precisely after interpolation. Finally, the empirical test results of SP500 market show that this method has good performance, especially for the option markets with sparse strikes.

scholarly journals A modification term for Black-Scholes model based on discrepancy calibrated with real market data

2021 ◽  
Vol 1 (4) ◽  
pp. 313-326
Author(s):  
Xiaozheng Lin ◽  
◽  
Meiqing Wang ◽  
Choi-Hong Lai ◽  
Keyword(s):  
Option Pricing ◽  
Implied Volatility ◽  
Option Prices ◽  
Market Prices ◽  
Market Data ◽  
Black Scholes Model ◽  
Black Scholes ◽  
Option Pricing Model ◽  
Real Market ◽  
S Model

<abstract><p>The Black-Scholes option pricing model (B-S model) generally requires the assumption that the volatility of the underlying asset be a piecewise constant. However, empirical analysis shows that there are discrepancies between the option prices obtained from the B-S model and the market prices. Most current modifications to the B-S model rely on modelling the implied volatility or interest rate. In contrast to the existing modifications to the Black-Scholes model, this paper proposes the concept of including a modification term to the B-S model itself. Using the actual discrepancies of the results of the Black-Scholes model and the market prices, the modification term related to the implied volatility is derived. Experimental results show that the modified model produces a better option pricing results when compare to market data.</p></abstract>

THE PRICING OF EUROPEAN OPTIONS UNDER THE CONSTANT ELASTICITY OF VARIANCE WITH STOCHASTIC VOLATILITY

2013 ◽  
Vol 12 (01) ◽  
pp. 1350004 ◽  
Author(s):  
BOUNGHUN BOCK ◽  
SUN-YONG CHOI ◽  
JEONG-HOON KIM
Keyword(s):  
Numerical Experiment ◽  
Stochastic Volatility ◽  
Hybrid Model ◽  
Multiscale Analysis ◽  
Asset Price ◽  
European Options ◽  
Barrier Option ◽  
Price Model ◽  
Constant Elasticity ◽  
Constant Elasticity Of Variance

This paper considers a hybrid risky asset price model given by a constant elasticity of variance multiplied by a stochastic volatility factor. A multiscale analysis leads to an asymptotic pricing formula for both European vanilla option and a Barrier option near the zero elasticity of variance. The accuracy of the approximation is provided in a rigorous manner. A numerical experiment for implied volatilities shows that the hybrid model improves some of the well-known models in view of fitting the data for different maturities.

THE HEAT-KERNEL MOST-LIKELY-PATH APPROXIMATION

2012 ◽  
Vol 15 (01) ◽  
pp. 1250001 ◽  
Author(s):  
JIM GATHERAL ◽  
TAI-HO WANG
Keyword(s):  
Heat Kernel ◽  
Implied Volatility ◽  
Order Term ◽  
Time Integration ◽  
Natural Extension ◽  
Approximation Formula ◽  
Local Volatility ◽  
Improved Performance ◽  
Volatility Function ◽  
Definition Of

In this article, we derive a new most-likely-path (MLP) approximation for implied volatility in terms of local volatility, based on time-integration of the lowest order term in the heat-kernel expansion. This new approximation formula turns out to be a natural extension of the well-known formula of Berestycki, Busca and Florent. Various other MLP approximations have been suggested in the literature involving different choices of most-likely-path; our work fixes a natural definition of the most-likely-path. We confirm the improved performance of our new approximation relative to existing approximations in an explicit computation using a realistic S&P500 local volatility function.

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.

CALIBRATION OF STOCHASTIC VOLATILITY MODELS VIA SECOND-ORDER APPROXIMATION: THE HESTON CASE

2015 ◽  
Vol 18 (06) ◽  
pp. 1550036 ◽  
Author(s):  
ELISA ALÒS ◽  
RAFAEL DE SANTIAGO ◽  
JOSEP VIVES
Keyword(s):  
Term Structure ◽  
Implied Volatility ◽  
Order Approximation ◽  
Calibration Procedure ◽  
Computational Effort ◽  
Heston Model ◽  
Option Prices ◽  
Volatility Models ◽  
Short And Long Term ◽  
Long Term Behavior

In this paper, we present a new, simple and efficient calibration procedure that uses both the short and long-term behavior of the Heston model in a coherent fashion. Using a suitable Hull and White-type formula, we develop a methodology to obtain an approximation to the implied volatility. Using this approximation, we calibrate the full set of parameters of the Heston model. One of the reasons that makes our calibration for short times to maturity so accurate is that we take into account the term structure for large times to maturity: We may thus say that calibration is not "memoryless," in the sense that the option's behavior far away from maturity does influence calibration when the option gets close to expiration. Our results provide a way to perform a quick calibration of a closed-form approximation to vanilla option prices, which may then be used to price exotic derivatives. The methodology is simple, accurate, fast and it requires a minimal computational effort.

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