Explaining the Volatility Surface: A Closed-Form Solution to Option Pricing in a Fractional Jump-Diffusion Market

2012 ◽  
Author(s):  
Stefan Rostek
Keyword(s):  
Option Pricing ◽  
Closed Form ◽  
Closed Form Solution ◽  
Jump Diffusion ◽  
Form Solution ◽  
Volatility Surface

An Analytical Approximation for Option Price under the Affine GARCH Model - A Comparison with the Closed-Form Solution of Heston-Nandi

GIS Business ◽  
2017 ◽  
Vol 12 (4) ◽  
pp. 32-46
Author(s):  
Noureddine Lahouel ◽  
Slaheddine Hellara
Keyword(s):  
Option Pricing ◽  
Closed Form ◽  
Empirical Work ◽  
Closed Form Solution ◽  
Garch Model ◽  
Analytical Approximation ◽  
Form Solution ◽  
Option Prices ◽  
European Option ◽  
Comparative Performance

In the option pricing theory, two important approaches have been developed to evaluate the prices of a European option. The first approach develops an almost closed-form option pricing formula under a specific GARCH process (Heston & Nandi, 2000). The second approach develops an analytical approximation for computing European option prices with more widespread NGARCH models (Duan, Gauthier & Simonato, 1999). The analytical approximation was also developed under GJR-GARCH and EGARCH models by Duan, Gauthier, Sasseville & Simonato (2006). However, no empirical work was performed to study the comparative performance of these two formulas (closed-form solution and analytical approximation). Also, it is possible to develop an analytical approximation under the specific GARCH model of Heston & Nandi (2000). In this paper, we have filled up those gaps. We started with the development of an analytical approximation, for computing European option prices, under Heston-Nandis GARCH model. In the second step, we carried out a comparative analysis of the three formulas using CAC 40 index returns from 31 December 1987 to 31 December 2013.

A closed-form approximation formula for pricing European options under a three-factor model

2021 ◽  
pp. 1-27
Author(s):  
Hye-mee Kil ◽  
Jeong-Hoon Kim
Keyword(s):  
Closed Form ◽  
Implied Volatility ◽  
Factor Model ◽  
Closed Form Solution ◽  
Form Solution ◽  
Approximation Formula ◽  
European Options ◽  
Implied Volatility Surface ◽  
Volatility Surface ◽  
Three Factor

Abstract The double-mean-reverting model, introduced by Gatheral [(2008). Consistent modeling of SPX and VIX options. In The Fifth World Congress of the Bachelier Finance Society London, July 18], is known to be a successful three-factor model that can be calibrated to both CBOE Volatility Index (VIX) and S&P 500 Index (SPX) options. However, the calibration of this model may be slow because there is no closed-form solution formula for European options. In this paper, we use a rescaled version of the model developed by Huh et al. [(2018). A scaled version of the double-mean-reverting model for VIX derivatives. Mathematics and Financial Economics 12: 495–515] and obtain explicitly a closed-form pricing formula for European option prices. Our formulas for the first and second-order approximations do not require any complicated calculation of integral. We demonstrate that a faster calibration result of the double-mean revering model is available and yet the practical implied volatility surface of SPX options can be produced. In particular, not only the usual convex behavior of the implied volatility surface but also the unusual concave down behavior as shown in the COVID-19 market can be captured by our formula.

scholarly journals DECOMPOSITION FORMULA FOR JUMP DIFFUSION MODELS

2018 ◽  
Vol 21 (08) ◽  
pp. 1850052
Author(s):  
R. MERINO ◽  
J. POSPÍŠIL ◽  
T. SOBOTKA ◽  
J. VIVES
Keyword(s):  
Option Pricing ◽  
Closed Form ◽  
Stochastic Volatility ◽  
Closed Form Solution ◽  
Form Solution ◽  
Heston Model ◽  
Approximation Formula ◽  
Option Prices ◽  
Currency Options ◽  
Decomposition Formula

In this paper, we derive a generic decomposition of the option pricing formula for models with finite activity jumps in the underlying asset price process (SVJ models). This is an extension of the well-known result by Alòs [(2012) A decomposition formula for option prices in the Heston model and applications to option pricing approximation, Finance and Stochastics 16 (3), 403–422, doi: https://doi.org/10.1007/s00780-012-0177-0 ] for Heston [(1993) A closed-form solution for options with stochastic volatility with applications to bond and currency options, The Review of Financial Studies 6 (2), 327–343, doi: https://doi.org/10.1093/rfs/6.2.327 ] SV model. Moreover, explicit approximation formulas for option prices are introduced for a popular class of SVJ models — models utilizing a variance process postulated by Heston [(1993) A closed-form solution for options with stochastic volatility with applications to bond and currency options, The Review of Financial Studies 6 (2), 327–343, doi: https://doi.org/10.1093/rfs/6.2.327 ]. In particular, we inspect in detail the approximation formula for the Bates [(1996), Jumps and stochastic volatility: Exchange rate processes implicit in Deutsche mark options, The Review of Financial Studies 9 (1), 69–107, doi: https://doi.org/10.1093/rfs/9.1.69 ] model with log-normal jump sizes and we provide a numerical comparison with the industry standard — Fourier transform pricing methodology. For this model, we also reformulate the approximation formula in terms of implied volatilities. The main advantages of the introduced pricing approximations are twofold. Firstly, we are able to significantly improve computation efficiency (while preserving reasonable approximation errors) and secondly, the formula can provide an intuition on the volatility smile behavior under a specific SVJ model.

On a Closed-Form Solution of Prandtl's System of Equations

2013 ◽  
Vol 40 (2) ◽  
pp. 106-114
Author(s):  
J. Venetis ◽  
Aimilios (Preferred name Emilios) Sideridis
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
System Of Equations

scholarly journals Geometric Average Asian Option Pricing with Paying Dividend Yield under Non-Extensive Statistical Mechanics for Time-Varying Model

Entropy ◽  
2018 ◽  
Vol 20 (11) ◽  
pp. 828 ◽  
Author(s):  
Jixia Wang ◽  
Yameng Zhang
Keyword(s):  
Statistical Mechanics ◽  
Option Pricing ◽  
Asset Price ◽  
Closed Form Solution ◽  
Real Data ◽  
Form Solution ◽  
Asian Option ◽  
Time Varying ◽  
Dividend Yield ◽  
Geometric Average

This paper is dedicated to the study of the geometric average Asian call option pricing under non-extensive statistical mechanics for a time-varying coefficient diffusion model. We employed the non-extensive Tsallis entropy distribution, which can describe the leptokurtosis and fat-tail characteristics of returns, to model the motion of the underlying asset price. Considering that economic variables change over time, we allowed the drift and diffusion terms in our model to be time-varying functions. We used the I t o ^ formula, Feynman–Kac formula, and P a d e ´ ansatz to obtain a closed-form solution of geometric average Asian option pricing with a paying dividend yield for a time-varying model. Moreover, the simulation study shows that the results obtained by our method fit the simulation data better than that of Zhao et al. From the analysis of real data, we identify the best value for q which can fit the real stock data, and the result shows that investors underestimate the risk using the Black–Scholes model compared to our model.

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