Abstract
The volatility of stock return does not follow the classical Brownian motion, but instead it follows a form that is closely related to fractional Brownian motion. Taking advantage of this information, the rough version of classical Heston model also known as rough Heston model has been derived as the macroscopic level of microscopic Hawkes process where it acts as a high-frequency price process. Unlike the pricing of options under the classical Heston model, it is significantly harder to price options under rough Heston model due to the large computational cost needed. Previously, some studies have proposed a few approximation methods to speed up the option computation. In this study, we calibrate five different approximation methods for pricing options under rough Heston model to SPX options, namely a third-order Padé approximant, three variants of fourth-order Padé approximant, and an approximation formula made from decomposing the option price. The main purpose of this study is to fill in the gap on lack of numerical study on real market options. The numerical experiment includes calibration of the mentioned methods to SPX options before and after the Lehman Brothers collapse.
Highlights
Similar to the option price under rough Heston model, its approximation formula has the capability of generating term structure of at-the-money skew of order
The approximation formula shows a great resemblance of the option price under rough Heston model after a calibration [23]
The low Hurst parameter H observed on the OGPadé and OFPadé would mean that they are very consistent to the rough Heston’s option price computed by fractional Adams method [20,22], in other words, it is no coincident that the rest of the parameters
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. The study indicates that stochastic volatility model where the volatility process is driven by fractional Brownian motion (fBm) with Hurst parameter H can generate ATM implied volatility skew of O( T H −1/2 ) when T → 0. This is significant and different from the local-stochastic volatility model [6] as it only possesses an extra. The fact that the volatility is empirically tested to be rough is largely due to Gatheral and his co-authors [7] Their main result of the study indicates that the log-volatility follows a generalization of Brownian motion known as fractional Brownian motion [8].
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