We present a theory of homogeneous volatility bridge estimators for log-price stochastic processes. The main tool of our theory is the parsimonious encoding of the information contained in the open, high and low prices of incomplete bridge, corresponding to given log-price stochastic process,...

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...

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....

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...

Today there are many equity derivatives that are traded on organized and over-the-counter markets. The models that allow market participants to value them and manage the associated risks on a daily basis are numerous. The idea of this study is, for vanilla equity options, to understand the Black...

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...