Abstracts

We consider a stochastic process for the log-price of a financial asset, recently introduced in [Andreoli et al., Adv. in Appl. Probab. 44 (2012), 1018-1051], which can be defined equivalently as a time-changed Brownian motion or as a stochastic volatility model. This process depends on 3 real parameters $(V, \lambda, D)$ with a precise economical meaning: $V$ is proportional to the average spot volatility, $\lambda$ is the intensity of a Poisson process which models shock times in the market, and $D$ describes the non-linear response of the market to shocks (after each shock time, the volatility increases as $t^{2D} >> t$).
Despite the few parameters, this model was shown to capture in a quantitative fashion some relevant stylized facts of financial series, such as the change in the log-return distribution from power-law tails (small time) to Gaussian (large time), the slow decay in the volatility autocorrelation, and the so-called multiscaling of moments. Here we continue the analysis of this model, with a focus on option pricing.
Our main theoretical result is a large deviations principle for the log-price in the small-time regime, with explicit (and non-trivial) rate and rate functions, based on a related large deviations principle for suitable functionals of a point process, which is of independent interest. This yields asymptotic formulas for the price of European options and the related implied volatility. In particular, despite the price having continuous paths, as the maturity $t$ vanishes the implied volatility of out-of-the-money options explodes as $C [ |x| / (t \sqrt{log(1/t)}) ]^{a}$, where $x$ is the log-moneyness and $C \in (0,\infty)$, $a \in (0,1)$ are constants that are functions of the model parameters.
We also enrich the model introducing jumps in the log-price, using the same Poisson process that drives the evolution of the volatility. This accounts for the so-called leverage effect and produces a skew in the log-return distribution. A Hull-White formula for the price of European options holds also in this correlated framework, averaging the usual Black-Scholes formula with a random volatility and a random initial price. This yields a fast Monte Carlo evaluation of option prices and allows for a numerical calibration of the model on real data, which shows a good agreement.