Rational Approximation of the Rough Heston Solution

Abstract

Pricing in the rough Heston model of Jaisson & M. Rosenbaum [(2016) Rough fractional diffusions as scaling limits of nearly unstable heavy tailed Hawkes processes, The Annals of Applied Probability 26 (5), 2860–2882] requires the solution of a fractional Riccati differential equation, which is not known in explicit form. Though numerical schemes to approximate this solution do exist, they inevitably require significantly more time to compute than the closed-form solution in the classical Heston model. In this paper, we present a simple rational approximation to the solution of the rough Heston Riccati equation valid in a region of its domain relevant to option valuation. Pricing using this approximation is both fast and very accurate.

Series Expansion and Fourth-Order Global Padé Approximation for a Rough Heston Solution

“…This disadvantage is the main cause of us shifting our attention from relying on numerical methods such as the fractional Adams-Bashforth-Moulton method for the computation of the fractional Riccati equation's solution. It is noted that in [19] a large number of steps are required to obtain a stable solution for only the characteristic function of the rough Heston model, not forgetting that the inversion of a characteristic function to the call option price requires additional computational costs.…”

“…While it is noted that the fractional Adams method generally provides extremely consistent results with a large number of time steps, the time complexity of the method is at O(N 2 ), where N is the number of time steps of the fractional Adams method. Nevertheless, it has became a standard method to compare consistency on the computation of call option prices for different methods, such as in [19,24]. By the way, the time complexity O(N 2 ) is not the final computational cost of the option pricing formula, as the time complexity is only for the characteristic function.…”

“…We move on to the discussion of multipoint Padé approximation method from [19]. The multipoint Padé approximation method matches two extreme points or the asymptotic solutions at t → 0 and t → ∞ using the series expansion solutions of the fractional Riccati equation.…”

“…Noticeably, the Padé approximation method has been used in many areas, such as physics, e.g., quantum resonances [40,41], the kinetic electron model [42] and many more noted in [43]. We utilize the literature given in [19]. The author of [44] provided the theory of convergence on the Padé approximants.…”

“…Nevertheless, a theorem called Nutall-Pommerenke's theorem implies the existence of the almost universal convergence for a subsequence of { f (j,j) } ∞ j=1 . As of this condition, the authors of [19] took the initiative to try out different n values of the diagonal Padé approximant on the fractional Riccati equation. See [46] for additional details on the methods.…”