Models which hypothesize that returns are pure jump processes with independent increments have been shown to be capable of capturing the observed variation of market prices of vanilla stock options across strike and maturity. In this paper, these models are employed to derive in closed form the...

Stochastic volatility and jumps are viewed as arising from Brownian subordination given here by an independent purely discontinuous process and we inquire into the relation between the realized variance or quadratic variation of the process and the time change. The class of models considered...

We consider the problem of optimal investment in a risky asset, and in derivatives written on the price process of this asset, when the underlying asset price process is a pure jump Lévy process. The duality approach of Karatzas and Shreve is used to derive the optimal consumption and...

In this note, we prove that under some minor conditions on $\sigma$, if a martingale $X_t = \int_0^t \sigma_u dW_u $ satisfies, for every given pair $u \geq 0, \, \xi \geq 0$, $X_{u+\xi}-X_u{\mathop{=}^{\mathrm{(law)}}} X_{\xi},$ then necessarily, $|\sigma_u|$ is a constant and X is a constant...

This article contains the translation into English of the main results found in the Comptes Rendus Volume of December 2000, dedicated to Wolfgang Doeblin's sealed envelope sent to the Académie des Sciences de Paris in February 1940. The genesis of these results - both from human and scientific...

We study a novel pricing operator for complete, local martingale models. The new pricing operator guarantees put-call parity to hold for model prices and the value of a forward contract to match the buy-and-hold strategy, even if the underlying follows strict local martingale dynamics. More...

For a family of functions G, we define the G-variation, which generalizes power variation; G-variation swaps, which pay the G-variation of the returns on an underlying share price F; and share-weighted G-variation swaps, which pay the integral of F with respect to G-variation. For instance, the...