Seminar Continuous Markov asset pricing models with explicit prices

Speaker: Marco Vitelli

Abstract:

We introduce a class of continuous Markovian asset pricing models with closed-form option prices, leading – by construction – to identifiable risk- neutral marginal distributions, and then specialize to a significant instance where the SDE well-posedness can be shown, the generalized beta local volatility (GBLV) model. The GBLV marginal distributions coincide with those of a known discontinuous martingale model that exhibits an at-the- money implied volatility skew divergence. These findings contrast with the commonly accepted wisdom that LV is unsuitable for capturing the implied volatility surface’s singular behav- ior as time-to- maturity approaches zero, and that option prices from jump models cannot be fitted to continuous Markov models. Such claims, typi- cally regarded as valid for the whole local volatility class, ultimately hinge on auxiliary assumptions, most notably, regularity of the diffusion coefficient at initial time. By directly embedding in the risk-neutral distributions the desirable properties an implied volatility surface should have, an LV model is freed from the constraints that make it unsuitable for capturing certain phenomena. As a consequence, the GBLV model does not suffer from several of the commonly exposed drawbacks of continuous Markovian models. With regard to the simulation of the model, the lack of regularity in the diffusion coefficient does not guarantee the convergence of the Euler– Maruyama scheme; therefore, for the simulations of the GBLV model, we introduce a new numerical scheme: the shifted Euler SE scheme. We conduct a numerical study that show the consistency, high accuracy and efficiency of the method.

Organizer: Sabrina Mulinacci