17 research outputs found
Valuation equations for stochastic volatility models
We analyze the valuation partial differential equation for European
contingent claims in a general framework of stochastic volatility models where
the diffusion coefficients may grow faster than linearly and degenerate on the
boundaries of the state space. We allow for various types of model behavior:
the volatility process in our model can potentially reach zero and either stay
there or instantaneously reflect, and the asset-price process may be a strict
local martingale. Our main result is a necessary and sufficient condition on
the uniqueness of classical solutions to the valuation equation: the value
function is the unique nonnegative classical solution to the valuation equation
among functions with at most linear growth if and only if the asset-price is a
martingale.Comment: Keywords: Stochastic volatility models, valuation equations,
Feynman-Kac theorem, strict local martingales, necessary and sufficient
conditions for uniquenes
Outperforming the market portfolio with a given probability
Our goal is to resolve a problem proposed by Fernholz and Karatzas [On
optimal arbitrage (2008) Columbia Univ.]: to characterize the minimum amount of
initial capital with which an investor can beat the market portfolio with a
certain probability, as a function of the market configuration and time to
maturity. We show that this value function is the smallest nonnegative
viscosity supersolution of a nonlinear PDE. As in Fernholz and Karatzas [On
optimal arbitrage (2008) Columbia Univ.], we do not assume the existence of an
equivalent local martingale measure, but merely the existence of a local
martingale deflator.Comment: Published in at http://dx.doi.org/10.1214/11-AAP799 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
On backward stochastic differential equations and strict local martingales
We study a backward stochastic differential equation whose terminal condition
is an integrable function of a local martingale and generator has bounded
growth in . When the local martingale is a strict local martingale, the BSDE
admits at least two different solutions. Other than a solution whose first
component is of class D, there exists another solution whose first component is
not of class D and strictly dominates the class D solution. Both solutions are
integrable for any . These two different BSDE solutions
generate different viscosity solutions to the associated quasi-linear partial
differential equation. On the contrary, when a Lyapunov function exists, the
local martingale is a martingale and the quasi-linear equation admits a unique
viscosity solution of at most linear growth.Comment: Keywords: Backward stochastic differential equation, strict local
martingale, viscosity solution, comparison theore
The Stochastic Solution to a Cauchy Problem for Degenerate Parabolic Equations
We study the stochastic solution to a Cauchy problem for a degenerate
parabolic equation arising from option pricing. When the diffusion coefficient
of the underlying price process is locally H\"older continuous with exponent
, the stochastic solution, which represents the price of a
European option, is shown to be a classical solution to the Cauchy problem.
This improves the standard requirement . Uniqueness results,
including a Feynman-Kac formula and a comparison theorem, are established
without assuming the usual linear growth condition on the diffusion
coefficient. When the stochastic solution is not smooth, it is characterized as
the limit of an approximating smooth stochastic solutions. In deriving the main
results, we discover a new, probabilistic proof of Kotani's criterion for
martingality of a one-dimensional diffusion in natural scale.Comment: Keywords: local martingales, local stochastic solutions, degenerate
Cauchy problems, Feynman-Kac formula, necessary and sufficient condition for
uniqueness, comparison principl