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 Generalized Mean-Reverting 4/2 Factor Model

2019 ◽  
Vol 12 (4) ◽  
pp. 159 ◽  
Author(s):  
Yuyang Cheng ◽  
Marcos Escobar-Anel ◽  
Zhenxian Gong
Keyword(s):  
Value At Risk ◽  
Implied Volatility ◽  
Factor Model ◽  
Risk Measures ◽  
Risk Measure ◽  
Common Factor ◽  
Principal Component ◽  
Characteristic Functions ◽  
Implied Volatility Surface ◽  
Volatility Surface

This paper proposes and investigates a multivariate 4/2 Factor Model. The name 4/2 comes from the superposition of a CIR term and a 3/2-model component. Our model goes multidimensional along the lines of a principal component and factor covariance decomposition. We find conditions for well-defined changes of measure and we also find two key characteristic functions in closed-form, which help with pricing and risk measure calculations. In a numerical example, we demonstrate the significant impact of the newly added 3/2 component (parameter b) and the common factor (a), both with respect to changes on the implied volatility surface (up to 100%) and on two risk measures: value at risk and expected shortfall where an increase of up to 29% was detected.

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.

A semiparametric factor model for implied volatility surface dynamics

2006 ◽  
Vol 5 (2) ◽  
pp. 189-218 ◽  
Author(s):  
M. R. Fengler ◽  
W. K. Hardle ◽  
E. Mammen
Keyword(s):  
Implied Volatility ◽  
Factor Model ◽  
Surface Dynamics ◽  
Implied Volatility Surface ◽  
Volatility Surface

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
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