We use the Itô Decomposition Formula (see Alòs (2012)) to express certain conditional expectations as exponentials of iterated integrals. As one application, we compute an exact formal expression for the leverage swap for any stochastic volatility model expressed in forward variance form. As...

Stock market variance-return or price relations are sometimes negative and sometimes positive. We explain these puzzling findings using a model with two ("bad" and "good") variances. In the model, conditional equity premium depends positively on bad variance and negatively on good variance....

This paper presents a new formalism to price European options in all asset classes that fits the market data remarkably well. We use a model-independent representation of European Option prices as path integrals over all of the underlying asset price from inception to maturity. The no arbitrage...

We propose a new methodology for obtaining arbitrage free European option prices from a SABR-like parameterisation. The method consists of specifying the joint distribution of the volatility and underlying at a given expiry and requires the calculation of a simple one-dimensional numerical...

The Local Volatility formula is a precious tool to identify the volatility process that will allow the perfect retrieval of all the market quotes of vanilla options on the market of equity and index derivatives. Since its first inception by Dupire, the local volatility model, applied initially...

We perform a historical analysis of selected rough volatility models to the SPX market. Tailoring the neural network pricing method of [27] to our needs, we train neural networks for the rough Heston model from [14], the rough Bergomi model from [4] as well as an extended version of the latter....

We study the mass at the origin in the uncorrelated SABR stochastic volatility model, and derive several tractable expressions, in particular when time becomes small or large. As an application -- in fact the original motivation for this paper -- we derive small-strike expansions for the implied...

From an analysis of the time series of volatility using recent high frequency data, Gatheral, Jaisson and Rosenbaum previously showed that log-volatility behaves essentially as a fractional Brownian motion with Hurst exponent H of order 0.1, at any reasonable time scale. The resulting Rough...

We introduce a new stochastic volatility model that includes, as special instances, the Heston (1993) and the 3/2 model of Heston (1997) and Platen (1997). Our model exhibits important features: first, instantaneous volatility can be uniformly bounded away from zero, and second, our model is...

This thesis presents our study on using the hybrid stochastic-local volatility model for option pricing. Many researchers have demonstrated that stochastic volatility models cannot capture the whole volatility surface accurately, although the model parameters have been calibrated to replicate...