Reduction and reconstruction of stochastic differential equations via symmetries

An algorithmic method to exploit a general class of infinitesimal symmetries for reducing stochastic differential equations is presented and a natural definition of reconstruction, inspired by the classical reconstruction by quadratures, is proposed. As a side result the well-known solution formula for linear one-dimensional stochastic differential equations is obtained within this symmetry approach. The complete procedure is applied to several examples with both theoretical and applied relevance

Piecewise Constant Martingales and Lazy Clocks

This paper discusses the possibility to find and construct \textit{piecewise constant martingales}, that is, martingales with piecewise constant sample paths evolving in a connected subset of $\mathbb{R}$. After a brief review of standard possible techniques, we propose a construction based on the sampling of latent martingales $\tilde{Z}$ with \textit{lazy clocks} $\theta$. These $\theta$ are time-change processes staying in arrears of the true time but that can synchronize at random times to the real clock. This specific choice makes the resulting time-changed process $Z_t=\tilde{Z}_{\theta_t}$ a martingale (called a \textit{lazy martingale}) without any assumptions on $\tilde{Z}$, and in most cases, the lazy clock $\theta$ is adapted to the filtration of the lazy martingale $Z$. This would not be the case if the stochastic clock $\theta$ could be ahead of the real clock, as typically the case using standard time-change processes. The proposed approach yields an easy way to construct analytically tractable lazy martingales evolving on (intervals of) $\mathbb{R}$.Comment: 17 pages, 8 figure

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

Imputation, Estimation and Missing Data in Finance

Suppose X is a diffusion process, possibly multivariate, and suppose that there are various segments of the components of X that are missing. This happens, for example, if X is the price of various assets and these prices are only observed at specific discrete trading times. Imputation (or conditional simulation) of the missing pieces of the sample paths of X is discussed in several settings. When X is a Brownian motion the conditioned process is a tied down Brownian motion or a Brownian bridge process. In the special case of Gaussian stochastic processes the problem is simplified since the conditional finite dimensional distributions of the process are multivariate Normal. For more general diffusion processes, including those with jump components, an acceptance-rejection simulation algorithm is introduced which enables one to sample from the exact conditional distribution without appealing to approximate time step methods such as the popular Euler or Milstein schemes. The method is referred to as pathwise imputation. Its practical implementation relies only on the basic elements of simulation while its theoretical justification depends on the pathwise properties of stochastic processes and in particular Girsanov's theorem. The method allows for the complete characterization of the bridge paths of complicated diffusions using only Brownian bridge interpolation. The imputation methods discussed are applied to estimation, variance reduction and exotic option pricing

Exact simulation pricing with Gamma processes and their extensions

Exact path simulation of the underlying state variable is of great practical importance in simulating prices of financial derivatives or their sensitivities when there are no analytical solutions for their pricing formulas. However, in general, the complex dependence structure inherent in most nontrivial stochastic volatility (SV) models makes exact simulation difficult. In this paper, we present a nontrivial SV model that parallels the notable Heston SV model in the sense of admitting exact path simulation as studied by Broadie and Kaya. The instantaneous volatility process of the proposed model is driven by a Gamma process. Extensions to the model including superposition of independent instantaneous volatility processes are studied. Numerical results show that the proposed model outperforms the Heston model and two other L\'evy driven SV models in terms of model fit to the real option data. The ability to exactly simulate some of the path-dependent derivative prices is emphasized. Moreover, this is the first instance where an infinite-activity volatility process can be applied exactly in such pricing contexts.Comment: Forthcoming The Journal of Computational Financ