Small-time, large-time, and H→0 asymptotics for the Rough Heston model

Martin Forde; Stefan Gerhold; Ben Smith

doi:10.1111/mafi.12290

Small-time, large-time, and H→0 asymptotics for the Rough Heston model

Martin Forde, Stefan Gerhold, Ben Smith

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Abstract

We characterize the behavior of the Rough Heston model introduced by Jaisson and Rosenbaum (2016, Ann. Appl. Probab., 26, 2860–2882) in the small-time, large-time, and (Formula presented.) (i.e., (Formula presented.)) limits. We show that the short-maturity smile scales in qualitatively the same way as a general rough stochastic volatility model, and the rate function is equal to the Fenchel–Legendre transform of a simple transformation of the solution to the same Volterra integral equation (VIE) that appears in El Euch and Rosenbaum (2019, Math. Financ., 29, 3–38), but with the drift and mean reversion terms removed. The solution to this VIE satisfies a space–time scaling property which means we only need to solve this equation for the moment values of (Formula presented.) and (Formula presented.) so the rate function can be efficiently computed using an Adams scheme or a power series, and we compute a power series in the log-moneyness variable for the asymptotic implied volatility which yields tractable expressions for the implied vol skew and convexity which is useful for calibration purposes. We later derive a formal saddle point approximation for call options in the Forde and Zhang (2017) large deviations regime which goes to higher order than previous works for rough models. Our higher-order expansion captures the effect of both drift terms, and at leading order is of qualitatively the same form as the higher-order expansion for a general model which appears in Friz et al. (2018, Math. Financ., 28, 962–988). The limiting asymptotic smile in the large-maturity regime is obtained via a stability analysis of the fixed points of the VIE, and is the same as for the standard Heston model in Forde and Jacquier (2011, Finance Stoch., 15, 755–780). Finally, using Lévy's convergence theorem, we show that the log stock price (Formula presented.) tends weakly to a nonsymmetric random variable (Formula presented.) as (Formula presented.) (i.e., (Formula presented.)) whose moment generating function (MGF) is also the solution to the Rough Heston VIE with (Formula presented.), and we show that (Formula presented.) tends weakly to a nonsymmetric random variable as (Formula presented.), which leads to a nonflat nonsymmetric asymptotic smile in the Edgeworth regime, where the log-moneyness (Formula presented.) as (Formula presented.), and we compute this asymptotic smile numerically. We also show that the third moment of the log stock price tends to a finite constant as (Formula presented.) (in contrast to the Rough Bergomi model discussed in Forde et al. (2020, Preprint) where the skew flattens or blows up) and the (Formula presented.) process converges on pathspace to a random tempered distribution which has the same law as the (Formula presented.) hyper-rough Heston model, discussed in Jusselin and Rosenbaum (2020, Math. Finance, 30, 1309–1336) and Abi Jaber (2019, Ann. Appl. Probab., 29, 3155–3200).

Original language	English
Pages (from-to)	203-241
Number of pages	39
Journal	MATHEMATICAL FINANCE
Volume	31
Issue number	1
Early online date	6 Oct 2020
DOIs	https://doi.org/10.1111/mafi.12290
Publication status	Published - Jan 2021

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10.1111/mafi.12290

Small-time, large-time_FORDE_Accepted 14 Sept 2020_GREEN_AAMAccepted author manuscript, 662 KB

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title = "Small-time, large-time, and H→0 asymptotics for the Rough Heston model",

abstract = "We characterize the behavior of the Rough Heston model introduced by Jaisson and Rosenbaum (2016, Ann. Appl. Probab., 26, 2860–2882) in the small-time, large-time, and (Formula presented.) (i.e., (Formula presented.)) limits. We show that the short-maturity smile scales in qualitatively the same way as a general rough stochastic volatility model, and the rate function is equal to the Fenchel–Legendre transform of a simple transformation of the solution to the same Volterra integral equation (VIE) that appears in El Euch and Rosenbaum (2019, Math. Financ., 29, 3–38), but with the drift and mean reversion terms removed. The solution to this VIE satisfies a space–time scaling property which means we only need to solve this equation for the moment values of (Formula presented.) and (Formula presented.) so the rate function can be efficiently computed using an Adams scheme or a power series, and we compute a power series in the log-moneyness variable for the asymptotic implied volatility which yields tractable expressions for the implied vol skew and convexity which is useful for calibration purposes. We later derive a formal saddle point approximation for call options in the Forde and Zhang (2017) large deviations regime which goes to higher order than previous works for rough models. Our higher-order expansion captures the effect of both drift terms, and at leading order is of qualitatively the same form as the higher-order expansion for a general model which appears in Friz et al. (2018, Math. Financ., 28, 962–988). The limiting asymptotic smile in the large-maturity regime is obtained via a stability analysis of the fixed points of the VIE, and is the same as for the standard Heston model in Forde and Jacquier (2011, Finance Stoch., 15, 755–780). Finally, using L{\'e}vy's convergence theorem, we show that the log stock price (Formula presented.) tends weakly to a nonsymmetric random variable (Formula presented.) as (Formula presented.) (i.e., (Formula presented.)) whose moment generating function (MGF) is also the solution to the Rough Heston VIE with (Formula presented.), and we show that (Formula presented.) tends weakly to a nonsymmetric random variable as (Formula presented.), which leads to a nonflat nonsymmetric asymptotic smile in the Edgeworth regime, where the log-moneyness (Formula presented.) as (Formula presented.), and we compute this asymptotic smile numerically. We also show that the third moment of the log stock price tends to a finite constant as (Formula presented.) (in contrast to the Rough Bergomi model discussed in Forde et al. (2020, Preprint) where the skew flattens or blows up) and the (Formula presented.) process converges on pathspace to a random tempered distribution which has the same law as the (Formula presented.) hyper-rough Heston model, discussed in Jusselin and Rosenbaum (2020, Math. Finance, 30, 1309–1336) and Abi Jaber (2019, Ann. Appl. Probab., 29, 3155–3200).",

author = "Martin Forde and Stefan Gerhold and Ben Smith",

note = "Funding Information: We thank Peter Friz, Eduardo Abi Jaber, Martin Larsson, and Martin Keller‐Ressel for helpful discussions. S. Gerhold acknowledges financial support from the Austrian Science Fund (FWF) under grant P 30750. Publisher Copyright: {\textcopyright} 2020 Wiley Periodicals LLC Copyright: Copyright 2021 Elsevier B.V., All rights reserved.",

year = "2021",

month = jan,

doi = "10.1111/mafi.12290",

language = "English",

volume = "31",

pages = "203--241",

journal = "MATHEMATICAL FINANCE",

issn = "0960-1627",

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T1 - Small-time, large-time, and H→0 asymptotics for the Rough Heston model

AU - Forde, Martin

AU - Gerhold, Stefan

AU - Smith, Ben

N1 - Funding Information: We thank Peter Friz, Eduardo Abi Jaber, Martin Larsson, and Martin Keller‐Ressel for helpful discussions. S. Gerhold acknowledges financial support from the Austrian Science Fund (FWF) under grant P 30750. Publisher Copyright: © 2020 Wiley Periodicals LLC Copyright: Copyright 2021 Elsevier B.V., All rights reserved.

PY - 2021/1

Y1 - 2021/1

N2 - We characterize the behavior of the Rough Heston model introduced by Jaisson and Rosenbaum (2016, Ann. Appl. Probab., 26, 2860–2882) in the small-time, large-time, and (Formula presented.) (i.e., (Formula presented.)) limits. We show that the short-maturity smile scales in qualitatively the same way as a general rough stochastic volatility model, and the rate function is equal to the Fenchel–Legendre transform of a simple transformation of the solution to the same Volterra integral equation (VIE) that appears in El Euch and Rosenbaum (2019, Math. Financ., 29, 3–38), but with the drift and mean reversion terms removed. The solution to this VIE satisfies a space–time scaling property which means we only need to solve this equation for the moment values of (Formula presented.) and (Formula presented.) so the rate function can be efficiently computed using an Adams scheme or a power series, and we compute a power series in the log-moneyness variable for the asymptotic implied volatility which yields tractable expressions for the implied vol skew and convexity which is useful for calibration purposes. We later derive a formal saddle point approximation for call options in the Forde and Zhang (2017) large deviations regime which goes to higher order than previous works for rough models. Our higher-order expansion captures the effect of both drift terms, and at leading order is of qualitatively the same form as the higher-order expansion for a general model which appears in Friz et al. (2018, Math. Financ., 28, 962–988). The limiting asymptotic smile in the large-maturity regime is obtained via a stability analysis of the fixed points of the VIE, and is the same as for the standard Heston model in Forde and Jacquier (2011, Finance Stoch., 15, 755–780). Finally, using Lévy's convergence theorem, we show that the log stock price (Formula presented.) tends weakly to a nonsymmetric random variable (Formula presented.) as (Formula presented.) (i.e., (Formula presented.)) whose moment generating function (MGF) is also the solution to the Rough Heston VIE with (Formula presented.), and we show that (Formula presented.) tends weakly to a nonsymmetric random variable as (Formula presented.), which leads to a nonflat nonsymmetric asymptotic smile in the Edgeworth regime, where the log-moneyness (Formula presented.) as (Formula presented.), and we compute this asymptotic smile numerically. We also show that the third moment of the log stock price tends to a finite constant as (Formula presented.) (in contrast to the Rough Bergomi model discussed in Forde et al. (2020, Preprint) where the skew flattens or blows up) and the (Formula presented.) process converges on pathspace to a random tempered distribution which has the same law as the (Formula presented.) hyper-rough Heston model, discussed in Jusselin and Rosenbaum (2020, Math. Finance, 30, 1309–1336) and Abi Jaber (2019, Ann. Appl. Probab., 29, 3155–3200).

AB - We characterize the behavior of the Rough Heston model introduced by Jaisson and Rosenbaum (2016, Ann. Appl. Probab., 26, 2860–2882) in the small-time, large-time, and (Formula presented.) (i.e., (Formula presented.)) limits. We show that the short-maturity smile scales in qualitatively the same way as a general rough stochastic volatility model, and the rate function is equal to the Fenchel–Legendre transform of a simple transformation of the solution to the same Volterra integral equation (VIE) that appears in El Euch and Rosenbaum (2019, Math. Financ., 29, 3–38), but with the drift and mean reversion terms removed. The solution to this VIE satisfies a space–time scaling property which means we only need to solve this equation for the moment values of (Formula presented.) and (Formula presented.) so the rate function can be efficiently computed using an Adams scheme or a power series, and we compute a power series in the log-moneyness variable for the asymptotic implied volatility which yields tractable expressions for the implied vol skew and convexity which is useful for calibration purposes. We later derive a formal saddle point approximation for call options in the Forde and Zhang (2017) large deviations regime which goes to higher order than previous works for rough models. Our higher-order expansion captures the effect of both drift terms, and at leading order is of qualitatively the same form as the higher-order expansion for a general model which appears in Friz et al. (2018, Math. Financ., 28, 962–988). The limiting asymptotic smile in the large-maturity regime is obtained via a stability analysis of the fixed points of the VIE, and is the same as for the standard Heston model in Forde and Jacquier (2011, Finance Stoch., 15, 755–780). Finally, using Lévy's convergence theorem, we show that the log stock price (Formula presented.) tends weakly to a nonsymmetric random variable (Formula presented.) as (Formula presented.) (i.e., (Formula presented.)) whose moment generating function (MGF) is also the solution to the Rough Heston VIE with (Formula presented.), and we show that (Formula presented.) tends weakly to a nonsymmetric random variable as (Formula presented.), which leads to a nonflat nonsymmetric asymptotic smile in the Edgeworth regime, where the log-moneyness (Formula presented.) as (Formula presented.), and we compute this asymptotic smile numerically. We also show that the third moment of the log stock price tends to a finite constant as (Formula presented.) (in contrast to the Rough Bergomi model discussed in Forde et al. (2020, Preprint) where the skew flattens or blows up) and the (Formula presented.) process converges on pathspace to a random tempered distribution which has the same law as the (Formula presented.) hyper-rough Heston model, discussed in Jusselin and Rosenbaum (2020, Math. Finance, 30, 1309–1336) and Abi Jaber (2019, Ann. Appl. Probab., 29, 3155–3200).

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DO - 10.1111/mafi.12290

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Small-time, large-time, and H→0 asymptotics for the Rough Heston model

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