Multi-scaling of moments in stochastic volatility models

We introduce a class of stochastic volatility models $(X_t)_{t \geq 0}$ for which the absolute moments of the increments exhibit anomalous scaling: \E\left(|X_{t+h} - X_t|^q \right) scales as $h^{q/2}$ for $q < q^*$, but as $h^{A(q)}$ with $A(q) q^*$, for some threshold $q^*$. This multi-scaling phenomenon is observed in time series of financial assets. If the dynamics of the volatility is given by a mean-reverting equation driven by a Levy subordinator and the characteristic measure of the Levy process has power law tails, then multi-scaling occurs if and only if the mean reversion is superlinear

Statistical estimation of the Oscillating Brownian Motion

We study the asymptotic behavior of estimators of a two-valued, discontinuous diffusion coefficient in a Stochastic Differential Equation, called an Oscillating Brownian Motion. Using the relation of the latter process with the Skew Brownian Motion, we propose two natural consistent estimators, which are variants of the integrated volatility estimator and take the occupation times into account. We show the stable convergence of the renormalized errors' estimations toward some Gaussian mixture, possibly corrected by a term that depends on the local time. These limits stem from the lack of ergodicity as well as the behavior of the local time at zero of the process. We test both estimators on simulated processes, finding a complete agreement with the theoretical predictions.Comment: 31 pages, 1 figur

A threshold model for local volatility: evidence of leverage and mean reversion effects on historical data

In financial markets, low prices are generally associated with high volatilities and vice-versa, this well known stylized fact usually being referred to as leverage effect. We propose a local volatility model, given by a stochastic differential equation with piecewise constant coefficients, which accounts of leverage and mean-reversion effects in the dynamics of the prices. This model exhibits a regime switch in the dynamics accordingly to a certain threshold. It can be seen as a continuous-time version of the Self-Exciting Threshold Autoregressive (SETAR) model. We propose an estimation procedure for the volatility and drift coefficients as well as for the threshold level. Parameters estimated on the daily prices of 348 stocks of NYSE and S\&P 500, on different time windows, show consistent empirical evidence for leverageeffects. Mean-reversion effects are also detected, most markedly in crisis periods

Extreme at-the-money skew in a local volatility model

We consider a local volatility model, with volatility taking two possible values, depending on the value of the underlying with respect to a fixed threshold. When the threshold is taken at-the-money, we establish exact pricing formulas and compute short-time asymptotics of the implied volatility surface. We derive an exact formula for the at-the-money implied volatility skew, which explodes as T-1/2, reproducing the empirical "steep short end of the smile". This behavior does not depend on the precise choice of the parameters, but simply follows from the "regime-switch" of the local volatility at-the-money

Extreme at-the-money skew in a local volatility model

We consider a local volatility model, with volatility taking two possible values, depending on the value of the underlying with respect to a fixed threshold. When the threshold is taken at-the-money, we establish exact pricing formulas and compute short-time asymptotics of the implied volatility surface. We derive an exact formula for the at-the-money implied volatility skew, which explodes as T-1/2, reproducing the empirical "steep short end of the smile". This behavior does not depend on the precise choice of the parameters, but simply follows from the "regime-switch" of the local volatility at-the-money

Density estimates and short-time asymptotics for a hypoelliptic diffusion process

We study a system of $n$ differential equations, each in dimension $d$. Only the first equation is forced by a Brownian motion and the dependence structure is such that the noise propagates to the whole system. Supposing a weak H\"ormander condition on the coefficients, we prove upper bounds for the transition density (heat kernel) and its derivatives of any order, Gaussian in the case of bounded diffusion coefficient, log-normal or polynomial in the case of linear-growth diffusion coefficient. Then we give precise short-time asymptotics of the density of the solution at a suitable central limit time scale. Both these results account of the different non-diffusive scales of propagation of the solution in the various components. Finally, we provide an application to valuation of short-maturity at-the-money Asian basket options under multi-asset local volatility dynamics and discuss connections with well known results in the literature

A multivariate model for financial indices and an algorithm for detection of jumps in the volatility

We consider a mean-reverting stochastic volatility model which satisfies some relevant stylized facts of financial markets. We introduce an algorithm for the detection of peaks in the volatility profile, that we apply to the time series of Dow Jones Industrial Average and Financial Times Stock Exchange 100 in the period 1984-2013. Based on empirical results, we propose a bivariate version of the model, for which we find an explicit expression for the decay over time of cross-asset correlations between absolute returns. We compare our theoretical predictions with empirical estimates on the same financial time series, finding an excellent agreement.Comment: 20 pages, 22 figure