scholarly journals Asymptotic formulae for implied volatility in the Heston model

2010 ◽  
Vol 466 (2124) ◽  
pp. 3593-3620 ◽  
Author(s):  
Martin Forde ◽  
Antoine Jacquier ◽  
Aleksandar Mijatović
Keyword(s):  
Approximate Formula ◽  
Implied Volatility ◽  
Holomorphic Functions ◽  
Heston Model ◽  
Elementary Functions ◽  
First Order ◽  
Asymptotic Formulae ◽  
Volatility Function

In this paper, we prove an approximate formula expressed in terms of elementary functions for the implied volatility in the Heston model. The formula consists of the constant and first-order terms in the large maturity expansion of the implied volatility function. The proof is based on saddlepoint methods and classical properties of holomorphic functions.

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.

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.

scholarly journals MULTISCALE STOCHASTIC VOLATILITY MODEL FOR DERIVATIVES ON FUTURES

2014 ◽  
Vol 17 (07) ◽  
pp. 1450043 ◽  
Author(s):  
JEAN-PIERRE FOUQUE ◽  
YURI F. SAPORITO ◽  
JORGE P. ZUBELLI
Keyword(s):  
Stochastic Volatility ◽  
Implied Volatility ◽  
Order Approximation ◽  
Perturbation Technique ◽  
Stochastic Volatility Model ◽  
Spot Price ◽  
Additional Hypothesis ◽  
First Order ◽  
Volatility Model ◽  
First Order Approximation

In this paper, we present a new method for computing the first-order approximation of the price of derivatives on futures in the context of multiscale stochastic volatility studied in Fouque et al. (2011). It provides an alternative method to the singular perturbation technique presented in Hikspoors & Jaimungal (2008). The main features of our method are twofold: firstly, it does not rely on any additional hypothesis on the regularity of the payoff function, and secondly, it allows an effective and straightforward calibration procedure of the group market parameters to implied volatilities. These features were not achieved in previous works. Moreover, the central argument of our method could be applied to interest rate derivatives and compound derivatives. The only pre-requisite of our approach is the first-order approximation of the underlying derivative. Furthermore, the model proposed here is well-suited for commodities since it incorporates mean reversion of the spot price and multiscale stochastic volatility. Indeed, the model was validated by calibrating the group market parameters to options on crude-oil futures, and it displays a very good fit of the implied volatility.

scholarly journals VOLATILITY DERIVATIVES AND MODEL-FREE IMPLIED LEVERAGE

2014 ◽  
Vol 17 (01) ◽  
pp. 1450002 ◽  
Author(s):  
MASAAKI FUKASAWA
Keyword(s):  
Implied Volatility ◽  
Asset Price ◽  
Simple Relation ◽  
Heston Model ◽  
Model Free ◽  
Implied Volatility Surface ◽  
Black Scholes ◽  
Volatility Derivatives ◽  
Quadratic Covariation ◽  
Volatility Surface

We revisit robust replication theory of volatility derivatives and introduce a broader class which may be considered as the second generation of volatility derivatives. One of them is a swap contract on the quadratic covariation between an asset price and the model-free implied variance (MFIV) of the asset. It can be replicated in a model-free manner and its fair strike may be interpreted as a model-free measure for the covariance of the asset price and the realized variance. The fair strike is given in a remarkably simple form, which enable to compute it from the Black–Scholes implied volatility surface. We call it the model-free implied leverage (MFIL) and give several characterizations. In particular, we show its simple relation to the Black–Scholes implied volatility skew by an asymptotic method. Further to get an intuition, we demonstrate some explicit calculations under the Heston model. We report some empirical evidence from the time series of the MFIV and MFIL of the Nikkei stock average.

scholarly journals The Adjoint Method for the Inverse Problem of Option Pricing

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Shou-Lei Wang ◽  
Yu-Fei Yang ◽  
Yu-Hua Zeng
Keyword(s):  
Inverse Problem ◽  
Option Pricing ◽  
Implied Volatility ◽  
Adjoint Method ◽  
Numerical Examples ◽  
Limited Memory ◽  
Minimization Procedure ◽  
Volatility Function ◽  
The Cost ◽  
Quasi Newton

The estimation of implied volatility is a typical PDE inverse problem. In this paper, we propose theTV-L1model for identifying the implied volatility. The optimal volatility function is found by minimizing the cost functional measuring the discrepancy. The gradient is computed via the adjoint method which provides us with an exact value of the gradient needed for the minimization procedure. We use the limited memory quasi-Newton algorithm (L-BFGS) to find the optimal and numerical examples shows the effectiveness of the presented method.

scholarly journals A HOLOMORPHIC REPRESENTATION OF THE JACOBI ALGEBRA

2006 ◽  
Vol 18 (02) ◽  
pp. 163-199 ◽  
Author(s):  
STEFAN BERCEANU
Keyword(s):  
Hilbert Space ◽  
Differential Operators ◽  
Holomorphic Functions ◽  
First Order ◽  
Polynomial Coefficients ◽  
Holomorphic Representation

A representation of the Jacobi algebra 𝔥1 ⋊ 𝔰𝔲(1, 1) by first-order differential operators with polynomial coefficients on the manifold [Formula: see text] is presented. The Hilbert space of holomorphic functions on which the holomorphic first-order differential operators with polynomials coefficients act is constructed.

scholarly journals APPROXIMATE SOLUTION OF ONE PROBLEM ON ELECTRICAL OSCILLATIONS IN WIRES WITH THE USE OF POLYLOGARITHMS

2017 ◽  
Vol 60 (4) ◽  
pp. 334-340 ◽  
Author(s):  
P. G. Lasy ◽  
I. N. Meleshko
Keyword(s):  
Approximate Solution ◽  
Initial Conditions ◽  
Fourier Method ◽  
Special Kind ◽  
Elementary Functions ◽  
The Real ◽  
First Order ◽  
Homogeneous Wave ◽  
Transcendental Functions ◽  
Current Flows

The article considers a mixed problem with homogeneous boundary conditions for onedimensional homogeneous wave equation. Such a problem can arise, for example, when studying oscillations of current and voltage in the conductor through which electric current flows, while the line is free from distortion. The solution can be found with the use of the Fourier method in the form of trigonometric series. This representation is of purely theoretical interest, because the real calculation should be, first, to find a large number of coefficients of the integrals, which in itself is not a trivial task and, second, it is almost impossible to assess the error of the calculations. An alternative way of solving this problem based on the use of transcendental functions i. e. polylogarithms that represent complex power series of a special kind. The exact solution of the problem is expressed through the imaginary part of a polylogarithm of the first order on the single circle and the approximate one – via the real part of the dilogarithm. In addition, if the initial conditions in the problem are elementary functions, then the solution is also computed using elementary functions. A simple and effective error estimate of the approximate solution has been found. It does not depend on time and it has the first-order of accuracy regarding the step of a partitioning segment of the numerical axis on which the problem is considered. This valuation is uniform with respect to the variables of the problem – both spatial and temporal. 

scholarly journals Approximation Formula for Option Prices under Rough Heston Model and Short-Time Implied Volatility Behavior

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1878
Author(s):  
Siow Woon Jeng ◽  
Adem Kilicman
Keyword(s):  
Term Structure ◽  
Taylor Expansion ◽  
Implied Volatility ◽  
Order Approximation ◽  
Heston Model ◽  
Approximation Formula ◽  
Option Prices ◽  
Second Order Approximation ◽  
Time Term ◽  
Short Time

Rough Heston model possesses some stylized facts that can be used to describe the stock market, i.e., markets are highly endogenous, no statistical arbitrage mechanism, liquidity asymmetry for buy and sell order, and the presence of metaorders. This paper presents an efficient alternative to compute option prices under the rough Heston model. Through the decomposition formula of the option price under the rough Heston model, we manage to obtain an approximation formula for option prices that is simpler to compute and requires less computational effort than the Fourier inversion method. In addition, we establish finite error bounds of approximation formula of option prices under the rough Heston model for 0.1≤H<0.5 under a simple assumption. Then, the second part of the work focuses on the short-time implied volatility behavior where we use a second-order approximation on the implied volatility to match the terms of Taylor expansion of call option prices. One of the key results that we manage to obtain is that the second-order approximation for implied volatility (derived by matching coefficients of the Taylor expansion) possesses explosive behavior for the short-time term structure of at-the-money implied volatility skew, which is also present in the short-time option prices under rough Heston dynamics. Numerical experiments were conducted to verify the effectiveness of the approximation formula of option prices and the formulas for the short-time term structure of at-the-money implied volatility skew.

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.

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