On the Martingale Property in Stochastic Volatility Models Based on Time‐homogeneous Diffusions

Abstract

Lions and Musiela give sufficient conditions to verify when a stochastic exponential of a continuous local martingale is a martingale or a uniformly integrable martingale. Blei and Engelbert and Mijatović and Urusov give necessary and sufficient conditions in the case of perfect correlation (ρ = 1). For financial applications, such as checking the martingale property of the stock price process in correlated stochastic volatility models, we extend their work to the arbitrary correlation case (−1 ≤ ρ ≤ 1). We give a complete classification of the convergence properties of both perpetual and capped integral functionals of time-homogeneous diffusions and generalize results in Mijatović and Urusov with direct proofs avoiding the use of separating times (concept introduced by Cherny and Urusov and extensively used in the proofs of Mijatović and Urusov).KEY WORDS: Martingale property, local martingale, stochastic volatility, Engelbert Schmidt zero-one law.

Credit Risk, Liquidity, and Bubbles

“…The stochastic volatility case does not lend itself to such a simple and elegant formulation, but partial results exist, and are general enough to be useful. In the one dimensional case we have the results of Andersen and Piterbarg (), Lions and Musiela (), and Bernard et al (). For the multidimensional case the theory becomes difficult.…”

A General Framework for Pricing Asian Options Under Stochastic Volatility on Parallel Architectures

“…The measure change is valid if the discounted asset price is a true martingale. Andersen and Piterbarg (2007, Proposition 2.5) show this for the Heston model, while Bernard et al (2017) study the special case when the Feller condition is satisfied. Cont and Tankov (2004, Lemma 15.2) also show that, in the case of an independent time change, the martingale version of the time changed Lévy process is constructed simply by applying the time change to a martingale version of the Lévy process.…”

No arbitrage in continuous financial markets

“…Our main results are different due to their deterministic nature. The models in [3,14,18,46] are of diffusion-type. In the context of Markov switching models we are only aware of the work of Elliott et al [20,21], where Esscher-type EMMs for a Markov switching Black-Scholes and a Markov switching Heston model are constructed.…”

“…Let us also comment on the approaches in [3,46]. The main idea in [46] is to relate the martingale property of a candidate density process to so-called separation times, which were introduced in [8].…”