Stylised facts of financial time series and hidden Markov models in continuous time

Hidden Markov models are often applied in quantitative finance to capture the stylised facts of financial returns. They are usually discrete-time models and the number of states rarely exceeds two because of the quadratic increase in the number of parameters with the number of states. This paper presents an extension to continuous time where it is possible to increase the number of states with a linear rather than quadratic growth in the number of parameters. The possibility of increasing the number of states leads to a better fit to both the distributional and temporal properties of daily returns.     

Performance of information criteria for selection of Hawkes process models of financial data

We test three common information criteria (IC) for selecting the order of a Hawkes process with an intensity kernel that can be expressed as a mixture of exponential terms. These processes find application in high-frequency financial data modelling. The information criteria are Akaike’s information criterion, the Bayesian information criterion and the Hannan–Quinn criterion. Since we work with simulated data, we are able to measure the performance of model selection by the success rate of the IC in selecting the model that was used to generate the data. In particular, we are interested in the relation between correct model selection and underlying sample size. The analysis includes realistic sample sizes and parameter sets from recent literature where parameters were estimated using empirical financial intra-day data. We compare our results to theoretical predictions and similar empirical findings on the asymptotic distribution of model selection for consistent and inconsistent IC.     

Deep learning volatility: a deep neural network perspective on pricing and calibration in (rough) volatility models

We present a neural network-based calibration method that performs the calibration task within a few milliseconds for the full implied volatility surface. The framework is consistently applicable throughout a range of volatility models—including second-generation stochastic volatility models and the rough volatility family—and a range of derivative contracts. Neural networks in this work are used in an off-line approximation of complex pricing functions, which are difficult to represent or time-consuming to evaluate by other means. The form in which information from available data is extracted and used influences network performance: The grid-based algorithm used for calibration is inspired by representing the implied volatility and option prices as a collection of pixels. We highlight how this perspective opens new horizons for quantitative modelling. The calibration bottleneck posed by a slow pricing of derivative contracts is lifted, and stochastic volatility models (classical and rough) can be handled in great generality as the framework also allows taking the forward variance curve as an input. We demonstrate the calibration performance both on simulated and historical data, on different derivative contracts and on a number of example models of increasing complexity, and also showcase some of the potentials of this approach towards model recognition. The algorithm and examples are provided in the Github repository GitHub: NN-StochVol-Calibrations.     

On a general class of one-factor models for the term structure of interest rates

We propose a general one-factor model for the term structure of interest rates which based upon a model for the short rate. The dynamics of the short rate is described by an appropriate function of a time-changed Wiener process. The model allows for perfect fitting of given term structure of interest rates and volatilities, as well as for mean reversion. Moreover, every type of distribution of the short rate can be achieved, in particular, the distribution can be concentrated on an interval. The model includes several popular models such as the generalized Vasicek (or Hull-White) model, the Black-Derman-Toy, Black-Karasinski model, and others. There is a unified numerical approach to the general model based on a simple lattice approximation which, in particular, can be chosen as a binomial or -nomial lattice with branching probabilities .     

Option pricing in the presence of natural boundaries and a quadratic diffusion term

This paper uses a probabilistic change-of-numeraire technique to compute closed-form prices of European options to exchange one asset against another when the relative price of the underlying assets follows a diffusion process with natural boundaries and a quadratic diffusion coefficient. The paper shows in particular how to interpret the option price formula in terms of exercise probabilities which are calculated under the martingale measures associated with two specific numeraire portfolios. An application to the pricing of bond options and certain interest rate derivatives illustrates the main results.