Affine forward variance models

We introduce the class of affine forward variance (AFV) models of which both the conventional Heston model and the rough Heston model are special cases. We show that AFV models can be characterised by the affine form of their cumulant-generating function, which can be obtained as solution of a convolution Riccati equation. We further introduce the class of affine forward order flow intensity (AFI) models, which are structurally similar to AFV models, but driven by jump processes, and which include Hawkes-type models. We show that the cumulant-generating function of an AFI model satisfies a generalised convolution Riccati equation and that a high-frequency limit of AFI models converges in distribution to an AFV model.

[1]  D. Duffie,et al.  Affine Processes and Application in Finance , 2002 .

[2]  D. Filipović.  Term-Structure Models: A Graduate Course , 2009 .

[3]  J. Teichmann,et al.  Generalized Feller processes and Markovian lifts of stochastic Volterra processes: the affine case , 2018, 1804.10450.

[4]  Patrick Billingsley,et al.  Probability and Measure. , 1986 .

[5]  D. Bainov,et al.  Integral Inequalities and Applications , 1992 .

[6]  M. Rosenbaum,et al.  The characteristic function of rough Heston models , 2016, 1609.02108.

[7]  Olof J. Staffans,et al.  Volterra Integral and Functional Equations: Index , 1990 .

[8]  R. Rockafellar.  Convex Analysis: (pms-28) , 1970 .

[9]  Omar El Euch,et al.  Markovian structure of the Volterra Heston model , 2018, Statistics & Probability Letters.

[10]  P. Hartman.  Ordinary Differential Equations , 1965 .

[11]  Olof J. Staffans,et al.  Volterra Integral and Functional Equations , 1990 .

[12]  Mark Veraar,et al.  The stochastic Fubini theorem revisited , 2012 .

[13]  Mathieu Rosenbaum,et al.  Rough fractional diffusions as scaling limits of nearly unstable heavy tailed Hawkes processes , 2015, 1504.03100.

[14]  丸山 徹.  Convex Analysisの二,三の進展について , 1977 .

[15]  S. Heston.  A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options , 1993 .

[16]  I. Podlubny.  Fractional differential equations , 1998 .

[17]  Mathieu Rosenbaum,et al.  Perfect hedging in rough Heston models , 2017, The Annals of Applied Probability.

[18]  Arak M. Mathai,et al.  Mittag-Leffler Functions and Their Applications , 2009, J. Appl. Math..

[19]  Peter E. Rossi,et al.  Stock Prices and Volume , 1992 .

[20]  M. Keller-Ressel,et al.  MOMENT EXPLOSIONS AND LONG‐TERM BEHAVIOR OF AFFINE STOCHASTIC VOLATILITY MODELS , 2008, 0802.1823.

[21]  Mathieu Rosenbaum,et al.  The microstructural foundations of leverage effect and rough volatility , 2018, Finance Stochastics.

[22]  N. G. Parke,et al.  Ordinary Differential Equations. , 1958 .

[23]  P. Billingsley,et al.  Probability and Measure , 1980 .

[24]  Gustaf Gripenberg,et al.  On volterra equations of the first kind , 1980 .

[25]  E. Bacry,et al.  Hawkes Processes in Finance , 2015, 1502.04592.

[26]  Martin Larsson,et al.  Affine Volterra processes , 2017, The Annals of Applied Probability.

[27]  Victor Lapshin,et al.  Term Structure Models , 2009 .

[28]  Mathieu Rosenbaum,et al.  Limit theorems for nearly unstable Hawkes processes , 2013, 1310.2033.