Abstract
We consider the short time behaviour of stochastic systems affected by a stochastic volatility evolving at a faster time scale. We study the asymptotics of a logarithmic functional of the process by methods of the theory of homogenization and singular perturbations for fully nonlinear PDEs. We point out three regimes depending on how fast the volatility oscillates relative to the horizon length. We prove a large deviation principle for each regime and apply it to the asymptotics of option prices near maturity.
Highlights
In this paper we are interested in stochastic differential equations with two small parameters ε > 0 and δ > 0 of the form dXt = εφ(Xt, Yt)d√t + 2εσ(Xt, Yt)dWt dYt
We study the asymptotics as both parameters go to 0 and we expect different limit behaviors depending on the rate ε/δ
We prove that the measures associated to the process Xt in (1.1) satisfy a Large Deviation Principle with good rate function
Summary
In this paper we are interested in stochastic differential equations with two small parameters ε > 0 and δ > 0 of the form. In their model ε represents a short maturity for the options, 1/δ is the rate of mean reversion of Yt, and the asymptotic analysis is performed for δ = εα in the regimes α = 2 and α = 4 Their methods are based on the approach to large deviations developed in [23]. Sharing some motivations with [22] our results are quite different: we treat vector-valued processes under rather general conditions and discuss all the regimes depending on the parameter α; our methods are different, mostly from the theory of viscosity solutions for fully nonlinear PDEs and from the theory of homogenisation and singular perturbations for such equations.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
