Delay Stochastic Models in Finance

The paper is devoted to an overview of delay stochastic models in finance and their applications to modeling and pricing of swaps. The volatility process is an important concept in financial modeling. This process can be stochastic or deterministic. In quantitative finance, we consider the volatility process to be stochastic as it allows to fit the observed market prices under consideration, as well as to model the risk linked with the future evolution of the volatility, which deterministic model cannot. Most stochastic dynamical systems (SDS) in finance describe the state of security (asset or volatility) as a value at time t, -instantaneous or current value at time t. We would like to take into account not only the current value at time t, but also values over some time interval [t −τ, t], where τ is a positive constant and is called the delay Delay . In this way, we incorporate path-dependent history Path-dependent history of the volatility under consideration. In this paper we will mainly focus on newly developed so-called delayed Heston model Delayed Heston model that significantly improve classical Heston model with respect to the market volatility surface fitting by 44 %. Review of some other delay stochastic models in finance will be given as well.