Page 565 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 565
DELING ROUGHNESS AND LONG-RANGE DEPENDENCE WITH FRACTIONAL
PROCESSES (MS-18)

stochastic volatility models. In this paper, we provide a proof of the prediction law for general
Gaussian Volterra processes. The prediction law is then utilized to obtain an adapted projection
of the future squared volatility — a cornerstone of the proposed pricing approximation. Firstly,
a decomposition formula for European option prices under general Volterra volatility models is
introduced. Then we focus on particular models with rough fractional volatility and we derive
an explicit semiclosed approximation formula. Numerical properties of the approximation for
a popular model — the rBergomi model — are studied and we propose a hybrid calibration
scheme which combines the approximation formula alongside MC simulations. This scheme
can significantly speed up the calibration to financial markets as illustrated on a set of AAPL
options.

Approximating expected value of an option with non-Lipschitz payoff in
fractional Heston-type model

Anton Yurchenko Tytarenko, antony@math.uio.no
University of Oslo, Norway
Coauthor: Yuliya Mishura

In this research, we consider option pricing in a framework of the fractional Heston-type model
with stochastic volatility being a fractional modification of the Cox-Ingersoll-Ross process with
H > 1/2. As it is impossible to obtain an explicit formula for the expectation Ef (ST ) in this
case, where ST is the asset price at maturity time and f is a payoff function, we provide a
discretization schemes Yˆ n and Sˆn for volatility and price processes correspondingly and study
convergence Ef (SˆTn) → Ef (ST ) as the mesh of the partition tends to zero. As we allow f to
be non-Lipschitz and/or to have discontinuities of the first kind which can cause errors if ST is
replaced by SˆTn under the expectation straightforwardly, we use Malliavin calculus techniques
to provide an alternative formula for Ef (ST ) with smooth functional under the expectation. In
this case, the rate of convergence is calculated.

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