Stochastic Analysis and Particle Filtering of the Volatility

Abstract

AbstractLet \(S = {({S}_{t})}_{t\in {\rm{IR}}_{+}}\) be an \({\rm{IR}}_{+}\)-valued semimartingale based on a filtered probability space \((\Omega,\mathcal{F},{({\mathcal{F}}_{t})}_{t\in {\rm{IR}}_{+}}, \rm{IP})\) which is assumed to be continuous. The process S is interpreted to model the price of a stock. A basic problem arising in Mathematical Finance is to estimate the price volatility, i.e., the square of the parameter σ in the following stochastic differential equation $$d{S}_{t} = \mu {S}_{t}\,dt + \sigma {S}_{t}\,d{W}_{t}$$ where \(W = {({W}_{t})}_{t\in {\rm{IR}}_{+}}\) is a Wiener process. It turns out that the assumption of a constant volatility does not hold in practice. Even to the most casual observer of the market, it should be clear that volatility is a random function of time which we denote σ t 2. Itô’s formula for the return \({y}_{t} =\log ({S}_{t}/{S}_{0})\) yields $$d{y}_{t} = \left (\mu -\frac{{\sigma }_{t}^{2}} {2} \right )\,dt + {\sigma }_{t}\,d{W}_{t}\quad {y}_{0} = 0$$ (11.1) The main objective is to estimate in discrete real-time one particular sample path of the volatility process using one observed sample path of the return. As regards the drift μ, it is constant but unknown. Under the so-called risk-neutral measure, the drift is a riskless rate which is well known; actually one finds that μ does not cancel out, for instance, when calculating conditional expectations in a filtering problem. For this argument no change of measure is required, we work directly in the original measure \(\rm{IP}\), and μ has to be estimated from the observed sample path of the return as well.KeywordsSample PathWiener ProcessStochastic VolatilityStochastic Volatility ModelVolatility ProcessThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.