TY - JOUR

T1 - Rough volatility, CGMY jumps with a finite history and the Rough Heston model - small-time asymptotics in the k root t regime

AU - Forde, Martin

AU - Smith, Benjamin

AU - Viitasaari, Lauri

N1 - doi: 10.1080/14697688.2020.1790634

PY - 2021/4

Y1 - 2021/4

N2 - A small-time Edgeworth expansion is established for near-the-money European options under a general rough stochastic volatility (RSV) model driven by a Riemann-Liouville (RL) process plus an additional generalised tempered stable Lévy process with Y∈(1,2) when H∈(1−12Y,2(1−12Y)∧12) (this relaxes the more complicated and restrictive condition which appeared in an earlier version of the article.), in the regime where log-moneyness logKS0∼zt√ as t→0 for z fixed, conditioned on a finite volatility history. This can be viewed as a more practical variant of Theorem 3.1 in Fukasawa [Short-time at-the-money skew and rough fractional volatility. Quant. Finance, 2017, 17(2), 189–198] (Fukasawa [Short-time at-the-money skew and rough fractional volatility. Quant. Finance, 2017, 17(2), 189–198] does not allow for jumps or a finite history and uses the somewhat opaque Muravlev representation for fractional Brownian motion), or if we turn off the rough stochastic volatility, the expansion is a variant of the main result in Mijatovic and Tankov [A new look at short-term implied volatility in asset price models with jumps. Math. Finance, 2016, 26(1), 149–183] and Theorem 3.2 in Figueroa-López et al. [Third-order short-time expansions for close-to-the-money option prices under the CGMY model. J. Appl. Math. Finance, 2017, 24, 547–574]. The zt√ regime is directly applicable to FX options where options are typically quoted in terms of delta (.10,.25 and .50) not absolute strikes, and we also compute a new prediction formula for the Riemann-Liouville process, which allows us to express the history term for the Edgeworth expansion in a more useable form in terms of the volatility process itself. We later relax the assumption of bounded volatility, and we also compute a formal small-time expansion for implied volatility in the Rough Heston model in the same regime (without jumps) which includes an additional at-the-money, convexity and fourth order correction term, and we outline how one can go to even higher order in the three separate cases H>16, H=16 and H<16. (We thank Masaaki Fukasawa and Hongzhong Zhang for fruitful discussions, Adam Hesse for help with Matlab computations and Vian Dinh for help with IT issues.)

AB - A small-time Edgeworth expansion is established for near-the-money European options under a general rough stochastic volatility (RSV) model driven by a Riemann-Liouville (RL) process plus an additional generalised tempered stable Lévy process with Y∈(1,2) when H∈(1−12Y,2(1−12Y)∧12) (this relaxes the more complicated and restrictive condition which appeared in an earlier version of the article.), in the regime where log-moneyness logKS0∼zt√ as t→0 for z fixed, conditioned on a finite volatility history. This can be viewed as a more practical variant of Theorem 3.1 in Fukasawa [Short-time at-the-money skew and rough fractional volatility. Quant. Finance, 2017, 17(2), 189–198] (Fukasawa [Short-time at-the-money skew and rough fractional volatility. Quant. Finance, 2017, 17(2), 189–198] does not allow for jumps or a finite history and uses the somewhat opaque Muravlev representation for fractional Brownian motion), or if we turn off the rough stochastic volatility, the expansion is a variant of the main result in Mijatovic and Tankov [A new look at short-term implied volatility in asset price models with jumps. Math. Finance, 2016, 26(1), 149–183] and Theorem 3.2 in Figueroa-López et al. [Third-order short-time expansions for close-to-the-money option prices under the CGMY model. J. Appl. Math. Finance, 2017, 24, 547–574]. The zt√ regime is directly applicable to FX options where options are typically quoted in terms of delta (.10,.25 and .50) not absolute strikes, and we also compute a new prediction formula for the Riemann-Liouville process, which allows us to express the history term for the Edgeworth expansion in a more useable form in terms of the volatility process itself. We later relax the assumption of bounded volatility, and we also compute a formal small-time expansion for implied volatility in the Rough Heston model in the same regime (without jumps) which includes an additional at-the-money, convexity and fourth order correction term, and we outline how one can go to even higher order in the three separate cases H>16, H=16 and H<16. (We thank Masaaki Fukasawa and Hongzhong Zhang for fruitful discussions, Adam Hesse for help with Matlab computations and Vian Dinh for help with IT issues.)

UR - http://www.scopus.com/inward/record.url?scp=85090167012&partnerID=8YFLogxK

U2 - 10.1080/14697688.2020.1790634

DO - 10.1080/14697688.2020.1790634

M3 - Article

VL - 21

SP - 541

EP - 563

JO - Quantitative Finance

JF - Quantitative Finance

SN - 1469-7688

IS - 4

ER -