Abstracts

This paper is about the risk management of a portfolio of global assets from a practical quantitative finance perspective. The intention is to provide concepts, understanding and possibly guidance to the activity of financial risk control in the real world of investments.
The paper will approach the subject of risk management by three avenues: (1) finance theory, statistics and computing; (2) economic concepts, markets knowledge and intuition and (3) thematic perspectives and historical events.
At level 1 we will cover the main tools, models and applications of finance, statistics and computing in order to be able to quantify the risks of a portfolio on several relevant conditions. At level 2 we will attempt to merge the technical / quantitative knowledge of level 1 with two central fields of investments: markets and economics. Very often the subject of “risk management” does not take into account actual markets and economic conditions.
Finally we have at level 3 the subjective dimension of risk control as complementing economic knowledge we have historical events generating institutions and shaping behavior as well as “themes” that allow us to understand at each point in time the main sources of risk on the portfolio´s investment horizon. This will be illustrated with several results and simulations from a realistic portfolio of international liquid financial assets.

Understanding variance risk is of key importance in mathematical finance since it affects risk management, asset allocation and derivative pricing. Variance risk is priced in financial markets by the so-called variance risk premium (VRP), which refers to the premium demanded for holding assets whose variance is exposed to stochastic shocks. The importance of the VRP is also evident from the proliferation of derivatives products such as the variance swaps, whose market volume has increased steeply over the past few years.
The aim of this paper is to identify a suitable parsimonious model for the stochastic dynamics of equity indices, capable of producing dynamics of the implied VRP in line with the empirical findings that the VRP of equity indices displays stochastic dynamics and jumps, whereas the VRP of individual stocks does not exhibit such stochastic fluctuations.
Existing theoretical and empirical work has so far only advocated univariate models to explain the dynamics of the VRP. However, we argue that dependencies across assets play a key role in explaining the stochastic dynamics of the VRP of stock indices and hence a multivariate stochastic model is needed. This paper presents for the first time explicit analytical formulas for the VRP in a multivariate stochastic volatility framework, which includes multivariate non-Gaussian Ornstein-Uhlenbeck processes and Wishart processes, as well as a new model specification. Moreover, we propose to incorporate self- and mutually exciting multivariate Hawkes processes in the model and find that the resulting dynamics of the VRP represent a convincing alternative to the models studied in the literature up to date.
We show that we can identify the Hawkes intensity process as a possible driver for the stochastic dynamics of the VRP. As a by-product of our work, we also prove useful explicit formulas involving conditional expectations of this popular process.
In addition, we find that our new model can explain the key stylised facts of both equity indices and individual assets and their corresponding VRP, while popular (multivariate) stochastic volatility models, including the Wishart model, fail.
We finally prove the existence of a structure-preserving risk neutral measure for our model, laying the theoretical foundations for the derivations described above. In particular, we establish the class of equivalent probabilities that preserve the self-affecting structure of the Hawkes process.

In the framework of robust mathematical finance, Dolinsky and Soner (2013) showed that there is no duality gap between the robust hedging of path-dependent European options and a martingale optimal transport problem, when option prices for one maturity are given. In this work, we present a duality result in the setup of multiple maturities. The proof proceeds through a discretisation of the problem. Key steps are to relate the robust hedging problem to a probabilistic super-hedging problem, and to use classical duality result to connect the probabilistic super-hedging problem to a discretised martingale optimal transport problem.
Furthermore, in discrete time, we extend a duality result proved by Beiglbock, Henry-Labordere and Penkner (2011) to markets with bubbles. In both continuous and discrete time, motivated by Mykland (2005)’s idea of having a prediction set of paths (i.e. super-replication of a contingent claim required only for paths falling in the prediction set), we add a path restriction into the pricing and hedging framework.

A basic problem in mathematical finance is the problem of an agent wants to maximize his expected utility of terminal wealth. In this paper, we study this problem in a financial market where the risky asset evolves according to the following stochastic volatility model:
\begin{align*}
dS_{t}&=S_{t}[\mu_{t}dt +g(V_{t}) dB_{t}]\\
dV_{t}&=f(\beta_{t},V_{t}) dt + h(V_{t}) dW_{t}
\end{align*}
The special feature of this paper is that the only information available to the investor is the one generated by the asset prices. Especially, the trend $\mu_{t}$ cannot be observed directly but can be modelled by a stochastic process, for example, an Ornstein-Uhlenbeck. $\beta_{t}$ is either a stochastic process or a constant. $B$ and $W$ are two correlated Brownian motions. $g$,$f$,$h$ are Borel mesurable functions such that a unique solution of the above dynamics exist.
We are in the context of a portfolio optimization problem under partial information. In order to solve it, we need first to reduce this problem to one with complete information. This step can be done with the change of probability method and filtering theory. The idea is to exploit all the information coming from the market, in order to update the knowledge of the not fully know quantities, precisely, we need to replace the unknown risk premia processes of the models by their filter estimates, which are given by the conditional expectations of these variables with respect to the filtration generated by the price process. Secondly, we aim to find an explicit computations of these filters. We show that in the uncorrelated case ($B$ and $W$ are independent), we can use the Bayesian framework to deduce an explicit computations of these filters. But in the correlated case, we use the non-linear filtering theory to show that these processes satisfy a Kushner-Stratonovich equations.
Finally, according to the aboves steps, we reduced our problem to a full observation optimization problem. We solve it with the martingale duality approach in the decorrelated case, and with the dynamic programming approach in the correlated case. We show that the dynamic approach leads to a characterization of the value function as viscosity solution of a nonlinear PDE (Hamilton-Jaccobi-Bellman equation). But, by logarithm transformation, we show that the value function and the optimal portfolio can be expressed in terms of a smooth solution of a semilinear parabolic equation.

After Lehman default (credit crisis which started in 2007), practitioners considered the default risk as a major risk. The Industry began to charge for the default risk of any derivatives. In this article we defined a methodology in order to fully adjusted the close out premium used to compute the CVA according to the Liquidity Risk. It is the first time that the CVA is adjusted taking into account the Liquidity Market Risk. We innovate market liquidity risk methodologies, in order to quantify this risk in the future to the close out premium. Therefore the CVA or Exposure will be function of our risk view or apetite.