2019/06/21 by Martin Forde, Forde, Martin, Stefan Gerhold +4
Economics, Econometrics and Finance · #60F10 #91G20 #Complex Systems and Time Series Analysis #FOS: Economics and business #Financial Markets and Investment Strategies #Financial Risk and Volatility Modeling #Pricing of Securities (q-fin.PR) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.1906.09034
openalex publication_date 2019/06/21 · openalex created_date 2022/07/28 · openalex updated_date 2026/07/28
We characterize the behaviour of the Rough Heston model introduced by\nJaisson &Rosenbaum citeJR16 in the small-time, large-time and \α \→\n1/2 (i.e. H\→ 0) limits. We show that the short-maturity smile scales in\nqualitatively the same way as a general rough stochastic volatility model (cf. \n citeFZ17, citeFGP18a et al.), and the rate function is equal to the\nFenchel-Legendre transform of a simple transformation of the solution to the\nsame Volterra integral equation (VIE) that appears in citeER19, but with the\ndrift and mean reversion terms removed. The solution to this VIE satisfies a\nspace-time scaling property which means we only need to solve this equation for\nthe moment values of p=1 and p=-1 so the rate function can be efficiently\ncomputed using an Adams scheme or a power series, and we compute a power series\nin the log-moneyness variable for the asymptotic implied volatility which\nyields tractable expressions for the implied vol skew and convexity. The\nlimiting asymptotic smile in the large-maturity regime is obtained via a\nstability analysis of the fixed points of the VIE, and is the same as for the\nstandard Heston model in citeFJ11. Finally, using L 'evy's convergence\ntheorem, we show that the log stock price Xt tends weakly to a non-symmetric\nrandom variable X(1/2)t as \α \→ 1/2 (i.e. H\→ 0) whose mgf is\nalso the solution to the Rough Heston VIE with \α=1/2, and we show that\nX(1/2)t/\√(t) tends weakly to a non-symmetric random variable as t\→\n0, which leads to a non-flat non-symmetric asymptotic smile in the Edgeworth\nregime. We also show that the third moment of the log stock price tends to a\nfinite constant as H\→ 0 (in contrast to the Rough Bergomi model discussed\nin citeFFGS20 where the skew flattens or blows up) and the V process\nconverges on pathspace to a random tempered distribution.\n