Elenco seminari del ciclo di seminari
“SEMINARI DI FINANZA MATEMATICA”

Many optimization problems in theoretical and applied science are difficult to solve: they exhibit multiple local optima or are not ’well-behaved’ in other ways (e.g., have discontinuities in the objective function). The still-prevalent approach to handling such difficulties -- other than ignoring them -- is to adjust or reformulate the problem until it can be solved with standard numerical methods. Unfortunately, this often involves simplifications of the original problem; thus we obtain solutions to a model that may or may not reflect our initial problem. But there is yet another approach: the application of optimization heuristics like Simulated Annealing or Genetic Algorithms. These methods have been shown to be capable of handling non-convex optimization problems with all kinds of constraints, and should thus be ideal candidates for many optimization problems. In this talk we motivate the use of such methods by first presenting some examples from finance for which optimization is required, and where standard methods often fail. We briefly review some heuristics, and look into their application to finance problems. We will also discuss the stochastics of the solutions obtained from heuristics, in particular we compare the randomness generated by the optimization methods with the randomness inherent to the problem.

We review some statistical arbitrage strategies applied to stock markets and investigate their use for trading in commodity markets. A general methodology to develop a statistical arbitrage trading strategy is devised. Cointegration techniques are used to find a long run relationship among commodities and build a portfolio whose value is represented by the deviation from this long run equilibrium. We then verify that the portfolio dynamics contain some predictable components and test trading rules that rely on these predictability properties of the portfolio process. Finally, several profit indicators are proposed in order to measure the performance of the strategy.

In Foreign Exchange Options markets first generation exotics like barrier and touch options have become vanilla-like commoditized derivatives traded in a liquid market. We review some of the traditional vanna-volga models frequently used by practitioners and software vendors, demonstrate their pros and cons, determine consistency and design requirements as well as limitations. We show how recent trends try to overcome the inconsistencies with stochastic-local volatility hybrid models (SLV). We provide an overview of such models used in the market and show by case studies how they relate to vanna-volga based approaches.

A financial market in the case where there are restrictions on the available is considered. We provide the Galtchouk-Kunita-Watanabe representation for a contingent claim under restricted information and, as a consequence, we deduce existence and uniqueness for the solution of linear backward stochastic differential equations (BSDEs) driven by a general càdlàg martingale in a partial information setting. Next, this result is extended to non linear BSDEs with Lipschitz driver and we provide the Föllmer-Schweizer decomposition (in the restricted information framework) with respect to an underlying risky asset price process described by a semimartingale. We discuss an application to risk-minimization. First, in the case where the risky asset price process is directly modeled under a martingale measure. Second, in the more general semimartingale case by introducing the concept of locally risk-minimizing strategies and characterizing the optimal strategy via the Föllmer Schweizer decomposition under restricted information. Finally, an example in the martingale case shows how to compute the risk-minimizing hedging strategy in terms of the filter when the risky asset price is described by a jump-diffusion process. More precisely, we assume that the behavior of the risky asset price depends on an unobservable stochastic factor and that the investors can only observe the prices but not the stochastic factor which affects their dynamics.

After the deregulation process observed in the electricity sector,producers and consumers, and in general all market participants,faced different types of risks as market and volume risks. Hence, practitioners and scholars have been continuously involved in monitoring electricity prices. Over more than a 10-year history,power markets changed according to new technologies and energy regulations. Therefore, we investigate the “new stylized facts” of these prices. Comparing alternative methods and controlling for the price fundamentals of demand and fuels, we show how electricity prices have been affected by the stochastic nature of wind generation.

We present a fast and accurate pricing techniques based on the Spitzer identity and the Wiener-Hopf factorization. We apply it to barrier and lookback options when the monitoring is discrete and the underlying evolves according to an exponential Lévy process. The numerical implementation exploits the fast Fourier transform and the Euler summation. The computational cost is independent of the number of monitoring dates; the error decays exponentially with the number of grid points.

Longevity risk, i.e. the risk of unexpected changes in the survivorship of policyholders, is perceived as one of the major threats to the long-run solvency of annuity providers, such as pension funds. Continuous-time models represent a useful tool in the modelling of stochastic mortality. Non-mean reverting affine cohort processes (Luciano and Vigna, 2008)provide a parsimonious but accurate description of mortality tables. They are particularly suited to pricing and hedging purposes, due to their analytical tractability. I present applications of such models to the management of insurance portfolios. I focus on longevity risk hedging techniques, such as natural hedging, and reinsurance strategies. I will discuss the implementation and the effectiveness of such strategies, as well as the effects of different risk sources (interest-rate risk, investment risk) - along with longevity risk - on the solvency of insurers.

* The interest rate market after the credit crunch * Modern derivatives modelling with multiple funding sources* Funding Value Adjustment (FVA)

A generalization of the Marshall-Olkin distribution is presented in order to allow for dependence among the systemic shock and the idiosyncratic shocks inducing defaults in a system. This model is used to incorporate contagion in the analysis of a set of obligors. The task will be to identify the infectious elements, to measure the degree of contagion and to allow for it in the estimation of the systemic risk.

We discuss the main properties of the expectiles, a one parameter family of coherent risk measures introduced in the statistical literature by Newey and Powell (1987). Expectiles are the only coherent risk measures that are also elicitable, in the sense that they can be equivalently defined as the minimizers of an expected loss; this property provides a natural methodology to perform a consistent backtesting. In this talk we explore the potential applicability of expectiles in a capital regulation framework, as an alternative to VaR and CVaR.

A dynamic factor model for the cross section for italian electricity forward prices is presented. This market is characterised by a small number of quotations for each forward contract and by a large bid-ask spread. Moreover, it is not uncommon to observe inconsistencies in the cross section of forward quoted prices. The aim of the proposed model is twofold. Firstly, this approach allows to filter out noise and to help in identifying inconsistencies. Secondly, it provide a tool to fill-in missing quotations and to unpack forward contract with long delivery into smaller ones (i.e. a quarter or a calendar into its monthly components). The model is built using monthly contracts are used as basic entities and their dynamics is modeled as follows. The cross section of those contracts is decomposed in two set of factors: the first set, which depends on the time-to-delivery, allows to financial structure of the market. The second set, which depends on the actual delivery month, will capture the seasonal component. The parameters of those factors are allowed to be slowly varying to achieve maximal flexibility.

The talk is about a recently introduced methodology in stochastic optimal control theory, known as randomization method, firstly developed for classical Markovian control problem in the paper: I. Kharroubi and H. Pham "Feynman-Kac representation for Hamilton-Jacobi-Bellman IPDE", Ann. Probab., 2015. The randomization method consists, in a first step, in replacing the control by an exogenous process independent of the driving noise and in formulating an auxiliary (“randomized”) control problem where optimization is performed over changes of equivalent probability measures affecting the characteristics of the exogenous process. We will discuss the main features of this approach, showing that the randomization method allows for greater generality beyond the Markovian case. In particular, we may consider stochastic control problems with path-dependence in the coefficients (with respect to both state and control), without requiring any non-degeneracy condition on the controlled equation. The talk is based on joint works with E. Bandini, M. Fuhrman, H. Pham.

We provide analytical approximations for the law of the solutions to a certain class of scalar McKean-Vlasov stochastic differential equations (MKV-SDEs) with random initial datum. "Propagation of chaos" results (Sznitman 1991) connect this class of SDEs with the macroscopic limiting behavior of a particle, evolving within a mean-field interaction particle system, as the total number of particles tends to infinity. Here we assume the mean-field interaction only acting on the drift of each particle, this giving rise to a MKV-SDE where the drift coefficient depends on the law of the unknown solution. By perturbing the non-linear forward Kolmogorov equation associated to the MKV-SDE, we perform a two-steps approximating procedure that decouples the McKean-Vlasov interaction from the standard dependence on the state-variables. The first step yields an expansion for the marginal distribution at a given time, whereas the second yields an expansion for the transition density. Both the approximating series turn out to be asymptotically convergent in the limit of short times and small noise, the convergence order for the latter expansion being higher than for the former. The resulting approximation formulas are expressed in semi-closed form and can be then regarded as a viable alternative to the numerical simulation of the large-particle system, which can be computationally very expensive. Moreover, these results pave the way for further extensions of this approach to more general dynamics and to high-dimensional settings.

In financial markets, the introduction of inside information can lead to profitable trading opportunities and, in particular, to arbitrage possibilities. In the context of stochastic finance, this issue can be addressed by relying on the theory of enlargement of filtrations. We present some simple examples where informational arbitrage is possible and study the absence of arbitrage under additional information in the context of general semimartingale models. Finally, we try to determine the value of a private information which allows to realize arbitrage opportunities.