Abstract
Rough volatility models are recently popularized by the need of a consistent model for the observed empirical volatility in the financial market. In this case, it has been shown that the empirical volatility in the financial market is extremely consistent with the rough volatility. Currently, fractional Riccati equation as a part of computation for the characteristic function of rough Heston model is not known in explicit form and therefore, we must rely on numerical methods to obtain a solution. In this paper, we will be giving a short introduction to option pricing theory (Black–Scholes model, classical Heston model and its characteristic function), an overview of the current advancements on the rough Heston model and numerical methods (fractional Adams–Bashforth–Moulton method and multipoint Padé approximation method) for solving the fractional Riccati equation. In addition, we will investigate on the performance of multipoint Padé approximation method for the small u values in D α h ( u − i / 2 , x ) as it plays a huge role in the computation for the option prices. We further confirm that the solution generated by multipoint Padé (3,3) method for the fractional Riccati equation is incredibly consistent with the solution generated by fractional Adams–Bashforth–Moulton method.
Highlights
The classical stochastic volatility diffusion models such as [1,2] have played an important role in volatility modelling to match for the smile or skew effect as displayed in the empirical implied volatility [3]
We have provided the brief history and literature review for the general stochastic volatility model, rough volatility, rough Heston model and ways to obtain solution for the rough
A large part of the literature review is dedicated to numerical methods of solving for the fractional Riccati equation where it can be used to obtain the option prices
Summary
The classical stochastic volatility diffusion models such as [1,2] have played an important role in volatility modelling to match for the smile or skew effect (variation of implied volatility with respect to strike price) as displayed in the empirical implied volatility [3]. We will focus heavily on the work of [29] where it uses multipoint Padé approximants on the asymptotical solutions (t → 0 and t → ∞) of the fractional Riccati equation and applying it to the rough Heston model. This is needed as the typical method (fractional Adams method) requires great computational effort, i.e., not feasible to most practitioners or perhaps even researchers.
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