5th Barcelona Summer School of Stochastic Analysis and Quantitative Finance

Abstract

The course overviews basic ideas in rough volatility modeling from the viewpoints of i) historical volatility dynamics, ii) implied volatility term structure, and iii) market microstructure. Instead of covering the strongest results up to date, it is meant to bring the understanding of the rough volatility phenomena through the simplest computations. Emphasis will be put on its relations to more traditional models, including local-stochastic volatility models, fast-mean reverting models with long memory, and stable-type jump diffusion models.

Abstract

Path signature were introduced by K. T. Chen in the 50s for smooth paths and later extended to rough paths by T. Lyons in the 90s — in fact, they form the basic building block of Lyons’ theory of rough paths. Signatures are a convenient and efficient way to encode paths, i.e., functions from, say, [0,T] to R^d. Indeed, 
 
1. a path’s signature essentially characterizes the underlying path, and 
 
2. the set of linear functionals of the signature forms a sub-algebra of the continuous functions of paths. 
 
As a consequence, (linear or non-linear functionals of) signatures form a natural set of basis functions for approximation of functions on path-space, similar to the role of polynomials on finite-dimensional Euclidean space. In particular, they are universal approximators. 
 
In this minicourse, we will provide a gentle introduction into signatures, discuss applications of signatures to machine learning (when the input data are time series). Finally, we show how signatures can be used as fundamental building blocks for constructing numerical algorithm for solving optimal stopping or more general stochastic optimal control problems, when the underlying stochastic process lacks the Markov property — such as in rough volatility models. 

Abstract

Weather derivatives are financial contracts written on weather events, and provide a protection against financial risks imposed by undesirable weather events. Weather derivatives contracts are traded on temperature indices and wind. We discuss various weather derivatives and their use, as well as the weather risk exposure in renewable energy markets. In the next step, we present various stochastic processes modelling temperature, wind as well as solar irradiation, suitable for describing the weather risk and analysing weather derivatives. Particular attention is paid to continuous-time autoregressive models, and several case studies are presented where we fit the models to observed weather data. Our focus is then turned to pricing weather fututers, where we introduce so-called pricing measures. We show that the risk premium in temperature markets varies with the temperature. To account for this effect, we introduce pricing meaures being explicitly dependent on the temperature. We finally discuss some hedging aspects concerning basis risk and the use of standardized weather derivatives.