Electron. J. Differential Equations,
Vol. 2018 (2018), No. 168, pp. 154.
Analytic solutions and complete markets for the
Heston model with stochastic volatility
Benedicte Alziary, Peter Takac
Abstract:
We study the Heston model for pricing European options on stocks
with stochastic volatility.
This is a BlackScholestype equation whose spatial domain
for the logarithmic stock price
and the variance
is the halfplane
.
The volatility is then given by
.
The diffusion equation for the price of the European call option
p = p(x,v,t) at time
is parabolic and degenerates at
the boundary
as
.
The goal is to hedge with this option against
volatility fluctuations, i.e., the function
and its (local) inverse are of particular interest.
We prove that
holds almost everywhere in
by establishing the analyticity of p in both,
space (x,v) and time t variables.
To this end, we are able to show that the BlackScholestype operator,
which appears in the diffusion equation,
generates a holomorphic
semigroup
in a suitable weighted
space over
.
We show that the
semigroup
solution can be extended to
a holomorphic function in a complex domain in
,
by establishing some new a~priori weighted
estimates over
certain complex "shifts" of
for the unique holomorphic extension.
These estimates depend only on
the weighted
norm
of the terminal data over
(at t=T).
Submitted August 22, 2018. Published October 11, 2018.
Math Subject Classifications: 35B65, 91G80, 35K65, 35K15.
Key Words: Heston model; stochastic volatility; BlackScholes equation;
European call option; degenerate parabolic equation;
terminal value problem; holomorphic extension; analytic solution.
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Bénédicte Alziary
Toulouse School of Economics, I.M.T.
Université de Toulouse  Capitole
21 Allées de Brienne
F31000 Toulouse Cedex, France
email: benedicte.alziary@utcapitole.fr


Peter Takác
Universität Rostock
Institut für Mathematik
Ulmenstrase~69, Haus 3
D18057 Rostock, Germany
email: peter.takac@unirostock.de

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