scholarly journals Model-Free Stochastic Collocation for an Arbitrage-Free Implied Volatility, Part II

Risks ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 30
Author(s):  
Fabien Le Floc’h ◽  
Cornelis Oosterlee
Keyword(s):  
Implied Volatility ◽  
Initial Guess ◽  
Stochastic Collocation ◽  
Option Prices ◽  
Spline Collocation ◽  
Model Free ◽  
Free Interpolation ◽  
Collocation Technique ◽  
Interpolation Technique ◽  
The Stability

This paper explores the stochastic collocation technique, applied on a monotonic spline, as an arbitrage-free and model-free interpolation of implied volatilities. We explore various spline formulations, including B-spline representations. We explain how to calibrate the different representations against market option prices, detail how to smooth out the market quotes, and choose a proper initial guess. The technique is then applied to concrete market options and the stability of the different approaches is analyzed. Finally, we consider a challenging example where convex spline interpolations lead to oscillations in the implied volatility and compare the spline collocation results with those obtained through arbitrage-free interpolation technique of Andreasen and Huge.

An Empirical Research on the Informational Efficiency of Model Free Implied Volatility

2008 ◽  
Vol 16 (2) ◽  
pp. 67-94
Author(s):  
Byung Kun Rhee ◽  
Sang Won Hwang
Keyword(s):  
Implied Volatility ◽  
Realized Volatility ◽  
Informational Efficiency ◽  
Option Prices ◽  
Index Options ◽  
Model Free ◽  
Black Scholes ◽  
Option Pricing Model ◽  
Korean Market ◽  
Better Than

Black-Scholes Imolied volatility (8SIV) has a few drawbacks. One is that the model Is not much successful in fitting the option prices. and It Is n야 guaranteed the model is correct one. Second. the usual tradition in using the BSIV is that only at-the-money Options are used. It is well-known that IV's of In-the-money or Qut-of-the-money ootions are much different from those estimated from near-the-money options. In this regard, a new model is confronted with Korean market data. Brittenxmes and Neuberger (2000) derive a formula for volatility which is a function of option prices‘ Since the formula is derived without using any option pricing model. volatility estimated from the formula is called model-tree implied volatillty (MFIV). MFIV overcomes the two drawbacks of BSIV. Jiang and Tian (2005) show that. with the S&P index Options (SPX), MFIV is suoerlor to historical volatility (HV) or BSIV in forecasting the future volatllity. In KOSPI 200 index options, when the forecasting performances are compared, MFIV is better than any other estimated volatilities. The hypothesis that MFIV contains all informations for realized volatility and the other volatilities are redundant is oot rejected in any cases.

FORWARD AND FUTURE IMPLIED VOLATILITY

2011 ◽  
Vol 14 (03) ◽  
pp. 407-432 ◽  
Author(s):  
PAUL GLASSERMAN ◽  
QI WU
Keyword(s):  
Implied Volatility ◽  
Transition Density ◽  
Option Prices ◽  
Piecewise Constant ◽  
Model Based ◽  
Model Free ◽  
Out Of Sample ◽  
Sabr Model ◽  
Using Data ◽  
Better Than

We address the problem of defining and calculating forward volatility implied by option prices when the underlying asset is driven by a stochastic volatility process. We examine alternative notions of forward implied volatility and the information required to extract these measures from the prices of European options at fixed maturities. We then specialize to the SABR model and show how the asymptotic expansion of the bivariate transition density in Wu (forthcoming) allows calibration of the SABR model with piecewise constant parameters and calculation of forward volatility. We then investigate empirically whether current option prices at multiple maturities contain useful information in predicting future option prices and future implied volatility. We undertake this investigation using data on options on the euro-dollar, sterling-dollar, and dollar-yen exchange rates. We find that prices across maturities do indeed have predictive value. Moreover, we find that model-based forward volatility extracts this predicative information better than a standard "model-free" measure of forward volatility and better than spot implied volatility. The enhancement to out-of-sample forecasting accuracy gained from model-based forward volatility is greatest at longer forecasting horizons.

scholarly journals New trigonometric B-spline approximation for numerical investigation of the regularized long-wave equation

Open Physics ◽  
2021 ◽  
Vol 19 (1) ◽  
pp. 758-769
Author(s):  
Ahmed Hussein Msmali ◽  
Mohammad Tamsir ◽  
Neeraj Dhiman ◽  
Mohammed A. Aiyashi
Keyword(s):  
Applied Sciences ◽  
Spline Approximation ◽  
Spline Functions ◽  
Spline Collocation ◽  
B Spline ◽  
Long Wave ◽  
Collocation Technique ◽  
Regularized Long Wave Equation ◽  
Spatial Derivatives ◽  
The Stability

Abstract The objective of this work is to propose a collocation technique based on new cubic trigonometric B-spline (NCTB-spline) functions to approximate the regularized long-wave (RLW) equation. This equation is used for modelling numerous problems occurring in applied sciences. The NCTB-spline collocation method is used to integrate the spatial derivatives. We use the Rubin–Graves linearization technique to linearize the non-linear term. The accuracy and efficiency of the technique are examined by employing it on three important numerical examples which have three invariants of motion viz. mass, momentum, and energy. It is observed that the error norms of the present method are less than the error norms of the methods available in the literature. The numerical values of these invariants have also been approximated, which remain conserved during the program run which shows that the propagation of the solitary wave is represented perfectly. The propagation of one and two solitary waves and undulations of waves are depicted graphically. The stability analysis shows that the method is unconditionally stable.

Arbitrage-free interpolation of call option prices

2020 ◽  
Vol 37 (1-2) ◽  
pp. 55-78
Author(s):  
Christian Bender ◽  
Matthias Thiel
Keyword(s):  
Implied Volatility ◽  
Reference Model ◽  
Interpolation Method ◽  
Call Option ◽  
Option Prices ◽  
One Dimensional ◽  
Black Scholes Model ◽  
Free Interpolation ◽  
Black Scholes ◽  
Special Case

AbstractIn this paper, we introduce a new interpolation method for call option prices and implied volatilities with respect to the strike, which first generates, for fixed maturity, an implied volatility curve that is smooth and free of static arbitrage. Our interpolation method is based on a distortion of the call price function of an arbitrage-free financial “reference” model of one’s choice. It reproduces the call prices of the reference model if the market data is compatible with the model. Given a set of call prices for different strikes and maturities, we can construct a call price surface by using this one-dimensional interpolation method on every input maturity and interpolating the generated curves in the maturity dimension. We obtain the algorithm of N. Kahalé [An arbitrage-free interpolation of volatilities, Risk 17 2004, 5, 102–106] as a special case, when applying the Black–Scholes model as reference model.

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.

scholarly journals Adaptive Model-Free Coupling Controller Design for Multi-Axis Motion Systems

2020 ◽  
Vol 10 (10) ◽  
pp. 3592
Author(s):  
Bo-Sheng Chen ◽  
Ching-Hung Lee
Keyword(s):  
Controller Design ◽  
Uncertain System ◽  
Contour Error ◽  
Nonlinear Phenomena ◽  
Nonlinear Phenomenon ◽  
Adaptive Model ◽  
Model Free ◽  
Fuzzy Neural ◽  
Adaptive Coupling ◽  
The Stability

In this study, we introduce an adaptive model-free coupling controller while using recurrent fuzzy neural network (RFNN) for multi-axis system to minimize the contour error. The proposed method can be applied to linear or nonlinear multi-axis motion control systems following desired paths. By the concept of cross-coupling control (CCC), multi-axis system is transferred into a nonlinear time-varying system due to the time-dependent coordinate transformation; tangential, normal, and bi-normal components of desired contour. Herein, we propose a model-free adaptive coupling controller design approach for multi-axis linear motor system with uncertainty and nonlinear phenomena. RFNN establishes the corresponding adaptive coupling controller to treat the uncertain system with nonlinear phenomenon. The stability of closed-loop system is guaranteed by the Lyapunov method and the adaptation of RFNN is also obtained. Simulation results are introduced in order to illustrate the effectiveness.

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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