Change of drift in one-dimensional diffusions

It is generally understood that a given one-dimensional diffusion may be transformed by a Cameron–Martin–Girsanov measure change into another one-dimensional diffusion with the same volatility but a different drift. But to achieve this, we have to know that the change-of-measure local martingale that we write down is a true martingale. We provide a complete characterisation of when this happens. This enables us to discuss the absence of arbitrage in a generalised Heston model including the case where the Feller condition for the volatility process is violated.

Forecasting VIX Using Filtered Historical Simulation*

We propose a new VIX forecast method using Generalized Autoregressive Conditional Heteroscedasticity models based on the filtered historical simulation put forward in Barone-Adesi, Engle, and Mancini (2008). The flexible change of measure accommodates for non-normalities such as negative skewness and positive excess kurtosis. We present an application for four well-established volatility indices (VIX9D, VIX, VIX3M, and VIX6M). Our results show that our proposed estimation method outperforms the Normal-VIX model of Hao and Zhang (2013) both in-sample and out-of-sample. Furthermore, the use of volatility indices reduces the computational burden significantly compared to the options-based pricing method.

A NOTE ON REAL-WORLD AND RISK-NEUTRAL DYNAMICS FOR HEATH–JARROW–MORTON FRAMEWORKS

We show that for time-inhomogeneous Markovian Heath–Jarrow–Morton models driven by an infinite-dimensional Brownian motion and a Poisson random measure an equivalent change of measure exists whenever the real-world and the risk-neutral dynamics can be defined uniquely and are related via a drift and a jump condition.

Option pricing formulas under a change of numèraire

We present some formulations of the Cox-Ross-Rubinstein and Black-Scholes formulas for European options obtained through a suitable change of measure, which corresponds to a change of numèraire for the underlying price process. Among other consequences, a closed formula for the price of an European call option at each node of the multi-period binomial tree is achieved, too. Some of the results contained herein, though comparable with analogous ones appearing elsewhere in the financial literature, provide however a supplementary widening and deepening in view of useful applications in the more challenging framework of incomplete markets. This last issue, having the present paper as a preparatory material, will be treated extensively in a forthcoming paper.

STRICT LOCAL MARTINGALES VIA FILTRATION ENLARGEMENT

A strict local martingale is a local martingale that is not a martingale. We investigate how such a process might arise from a true martingale as a result of an enlargement of the filtration and a change of measure. We study and implement a particular type of enlargement, initial expansion of filtration, for stochastic volatility models with and without jumps and provide sufficient conditions in each of these cases such that initial expansion can create a strict local martingale. We provide examples of initial enlargement that effect this change.