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 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>

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.

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.

Implied Volatility Surface Estimation Using Transductive Gaussian Fields Regression

2011 ◽  
Vol 467-469 ◽  
pp. 1781-1786
Author(s):  
Hye Jin Park ◽  
Dae Won Lee ◽  
Jae Wook Lee
Keyword(s):  
Implied Volatility ◽  
Global Financial Crisis ◽  
Gaussian Field ◽  
Implied Volatility Surface ◽  
Before And After ◽  
Unlabelled Data ◽  
Regression Techniques ◽  
Real Market ◽  
Volatility Surface ◽  
The Global Financial Crisis

Implied volatility estimation is one of the fundamental tasks for asset pricing and risk management. In this paper, we investigate the applicability of semi-supervised regression techniques to estimate an implied volatility surface from the real market option data. Specifically, we employ a transductive Gaussian field regression method since it is able to predict a distribution of the implied volatilities for unlabelled data using only partially labeled data. We've conducted simulation on S&P 500 index data before and after the global financial crisis with discussions of the observed empirical properties of the method.

scholarly journals A Bayesian nonparametric approach to option pricing

2020 ◽  
Vol 18 (4) ◽  
pp. 115-137
Author(s):  
Zhang Qin ◽  
Caio Almeida
Keyword(s):  
Option Pricing ◽  
Implied Volatility ◽  
Kernel Functions ◽  
General Covariance ◽  
Bayesian Nonparametric ◽  
Market Data ◽  
Nonparametric Approach ◽  
Covariance Functions ◽  
Implied Volatility Surface ◽  
Volatility Surface

Accurately modeling the implied volatility surface is of great importance to option pricing, trading and hedging. In this paper, we investigate the use of a Bayesian nonparametric approach to fit and forecast the implied volatility surface with observed market data. More specifically, we explore Gaussian Processes with different kernel functions characterizing general covariance functions. We also obtain posterior distributions of the implied volatility and build confidence intervals for the predictions to assess potential model uncertainty. We apply our approach to market data on the S&P 500 index option market in 2018, analyzing 322,983 options. Our results suggest that the Bayesian approach is a powerful alternative to existing parametric pricing models

IMPLIED AND LOCAL VOLATILITIES UNDER STOCHASTIC VOLATILITY

2001 ◽  
Vol 04 (01) ◽  
pp. 45-89 ◽  
Author(s):  
ROGER W. LEE
Keyword(s):  
Asymptotic Expansion ◽  
Stochastic Volatility ◽  
Implied Volatility ◽  
Asymptotic Methods ◽  
Stochastic Volatility Model ◽  
Slow Variation ◽  
Option Prices ◽  
Local Volatility ◽  
Volatility Model ◽  
Zero Correlation

For asset prices that follow stochastic-volatility diffusions, we use asymptotic methods to investigate the behavior of the local volatilities and Black–Scholes volatilities implied by option prices, and to relate this behavior to the parameters of the stochastic volatility process. We also give applications, including risk-premium-based explanations of the biases in some naïve pricing and hedging schemes. We begin by reviewing option pricing under stochastic volatility and representing option prices and local volatilities in terms of expectations. In the case that fluctuations in price and volatility have zero correlation, the expectations formula shows that local volatility (like implied volatility) as a function of log-moneyness has the shape of a symmetric smile. In the case of non-zero correlation, we extend Sircar and Papanicolaou's asymptotic expansion of implied volatilities under slowly-varying stochastic volatility. An asymptotic expansion of local volatilities then verifies the rule of thumb that local volatility has the shape of a skew with roughly twice the slope of the implied volatility skew. Also we compare the slow-variation asymptotics against what we call small-variation asymptotics, and against Fouque, Papanicolaou, and Sircar's rapid-variation asymptotics. We apply the slow-variation asymptotics to approximate the biases of two naïve pricing strategies. These approximations shed some light on the signs and the relative magnitudes of the biases empirically observed in out-of-sample pricing tests of implied-volatility and local-volatility schemes. Similarly, we examine the biases of three different strategies for hedging under stochastic volatility, and we propose ways to implement these strategies without having to specify or estimate any particular stochastic volatility model. Our approximations suggest that a number of the empirical pricing and hedging biases may be explained by a positive premium for the portion of volatility risk that is uncorrelated with asset risk.

scholarly journals Online local volatility calibration by convex regularization

2014 ◽  
Vol 8 (2) ◽  
pp. 243-268 ◽  
Author(s):  
Vinicius Albani ◽  
Jorge Zubelli
Keyword(s):  
Inverse Problem ◽  
Stock Price ◽  
Convergence Rates ◽  
Regularization Parameter ◽  
Option Price ◽  
Local Volatility ◽  
Regularity Properties ◽  
Volatility Model ◽  
Volatility Surface ◽  
Online Methods

We address the inverse problem of local volatility surface calibration from market given option prices. We integrate the ever-increasing flow of option price information into the well-accepted local volatility model of Dupire. This leads to considering both the local volatility surfaces and their corresponding prices as indexed by the observed underlying stock price as time goes by in appropriate function spaces. The resulting parameter to data map is defined in appropriate Bochner-Sobolev spaces. Under this framework, we prove key regularity properties. This enable us to build a calibration technique that combines online methods with convex Tikhonov regularization tools. Such procedure is used to solve the inverse problem of local volatility identification. As a result, we prove convergence rates with respect to noise and a corresponding discrepancy-based choice for the regularization parameter. We conclude by illustrating the theoretical results by means of numerical tests.

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