What is a Man, What Has He Got?

This issue Elie Ayache returns to complete his alternative reading of Bergomi which sheds new perspectives on Rough Volatility in relation to the meaning of the Options Market

PORTFOLIO INSURANCE UNDER ROUGH VOLATILITY AND VOLTERRA PROCESSES

Affine Volterra processes have gained more and more interest in recent years. In particular, this class of processes generalizes the classical Heston model and the more recent rough Heston model. The aim of this work is hence to revisit and generalize the constant proportion portfolio insurance (CPPI) under affine Volterra processes. Indeed, existing simulation-based methods for CPPI do not apply easily to this class of processes. We instead propose an approach based on the characteristic function of the log-cushion which appears to be more consistent, stable and particularly efficient in the case of saffine Volterra processes compared with the existing simulation techniques. Using such approach, we describe in this paper several properties of CPPI which naturally result from the form of the log-cushion’s characteristic function under affine Volterra processes. This allows to consider more realistic dynamics for the underlying risky asset in the context of CPPI and hence build portfolio strategies that are more consistent with financial data. In particular, we address the case of the rough Heston model, known to be extremely consistent with past data of volatility. By providing a new estimation procedure for its parameters based on the PMCMC algorithm, we manage to study more accurately the true properties of such CPPI strategy and to better handle the risk associated with it.

From microscopic price dynamics to multidimensional rough volatility models

Rough volatility is a well-established statistical stylized fact of financial assets. This property has led to the design and analysis of various new rough stochastic volatility models. However, most of these developments have been carried out in the mono-asset case. In this work, we show that some specific multivariate rough volatility models arise naturally from microstructural properties of the joint dynamics of asset prices. To do so, we use Hawkes processes to build microscopic models that accurately reproduce high-frequency cross-asset interactions and investigate their long-term scaling limits. We emphasize the relevance of our approach by providing insights on the role of microscopic features such as momentum and mean-reversion in the multidimensional price formation process. In particular, we recover classical properties of high-dimensional stock correlation matrices.

A rough SABR formula

<p style='text-indent:20px;'>Following an approach originally suggested by Balland in the context of the SABR model, we derive an ODE that is satisfied by normalized volatility smiles for short maturities under a rough volatility extension of the SABR model that extends also the rough Bergomi model. We solve this ODE numerically and further present a very accurate approximation to the numerical solution that we dub the <i>rough SABR formula</i>.</p>