Francesca Biagini

Shifting martingale measures and the birth of a bubble as a submartingale

In an incomplete financial market model, we study a flow in the space of equivalent martingale measures and the corresponding shifting perception of the fundamental value of a given asset. This allows us to capture the birth of a perceived bubble and to describe it as an initial submartingale which then turns into a supermartingale before it falls back to its initial value zero.

A unified approach to systemic risk measures via acceptance sets

The financial crisis has dramatically demonstrated that the traditional approach to apply univariate monetary risk measures to single institutions does not capture sufficiently the perilous systemic risk that is generated by the interconnectedness of the system entities and the corresponding contagion effects. This has brought awareness of the urgent need for novel approaches that capture systemic riskiness. The purpose of this paper is to specify a general methodological framework that is flexible enough to cover a wide range of possibilities to design systemic risk measures via multi-dimensional acceptance sets and aggregation functions, and to study corresponding examples. Existing systemic risk measures can usually be interpreted as the minimal capital needed to secure the system after aggregating individual risks. In contrast, our approach also includes systemic risk measures that can be interpreted as the minimal capital funds that secure the aggregated system by allocating capital to the single institutions before aggregating the individual risks. This allows for a possible ranking of the institutions in terms of systemic riskiness measured by the optimal allocations. Moreover, we also allow for the possibility of allocating the funds according to the future state of the system (random allocation). We provide conditions which ensure monotonicity, convexity, or quasi-convexity properties of our systemic risk measures.

Local Risk-Minimization under the Benchmark Approach

We study the pricing and hedging of derivatives in incomplete financial markets by considering the local risk-minimization method in the context of the benchmark approach, which will be called benchmarked local risk-minimization. We show that the proposed benchmarked local risk-minimization allows to handle under extremely weak assumptions a much richer modeling world than the classical methodology.

MINIMAL VARIANCE HEDGING FOR INSIDER TRADING

In this paper, we first study the problem of minimal hedging for an insider trader in incomplete markets. We use the forward integral in order to model the insider portfolio and consider a general larger filtration. We characterize the optimal strategy in terms of a martingale condition. In the second part we focus on a problem of mean-variance hedging where the insider tries to minimize the variance of his wealth at time T given that this wealth has a fixed expected value A. We solve this problem for an initial enlargement of filtration by providing an explicit solution.

Asymptotics for Operational Risk Quantified with Expected Shortfall

In this paper we estimate operational risk by using the convex risk measure Expected Shortfall (ES) and provide an approximation as the confidence level converges to 100% in the univariate case. Then we extend this approach to the multivariate case, where we represent the dependence structure by using a Levy copula as in Bocker and Kluppelberg (2006) and Bocker and Kluppelberg, C. (2008). We compare our results to the ones obtained in Bocker and Kluppelberg (2006) and (2008) for Operational VaR and discuss their practical relevance.

Risk-Minimization for Life Insurance Liabilities

In this paper we study the pricing and hedging of a very general class of life insurance liabilities by means of the risk-minimization approach. We find the price and risk-minimizing strategy in two cases, first in the case when the financial market consists only of one risky asset, e.g., a stock, and a bank account, and second in an extended financial market, allowing for investments in two additional traded assets, representing the systematic and unsystematic mortality risk. We also provide an application in the case of a unit-linked term insurance contract in a jump-diffusion model for the stock price and affine stochastic mortality intensity. The main novelties of this work are that we allow for hedging of the risk inherent in the insurance liabilities by investing not only in the stock and money market account but also in a longevity bond, representing the systematic mortality risk, and a pure endowment contract, accounting for the unsystematic mortality risk. Besides that we consider a very general ...

BEHAVIOR OF LONG-TERM YIELDS IN A LÉVY TERM STRUCTURE

We study the behavior of the long-term yield in a HJM setting for forward rates driven by Levy processes. The long-term rates are investigated by examining continuously compounded spot rate yields with maturity going to infinity. In this paper, we generalize the model of Karoui et al. (1997) by using Levy processes instead of Brownian motions as driving processes of the forward rate dynamics, and analyze the behavior of the long-term yield under certain conditions which encompass the asymptotic behavior of the interest rate models volatility function as well as the variation of the paths of the Levy process. One of the main results is that the long-term volatility has to vanish except in the case of a Levy process with only negative jumps and paths of finite variation serving as random driver. Furthermore, we study the required asymptotic behavior of the volatility function so that the long-term drift exists.

Evaluating Hybrid Products: The Interplay Between Financial and Insurance Markets

A current issue in the theory and practice of insurance and reinsurance markets is to find alternative ways of securitizing risks. Insurance companies have the possibility of investing in financial markets and therefore hedge against their risks with financial instruments. Furthermore they can sell part of their insurance risk by introducing insurance linked products on financial markets. Hence insurance and financial markets may no longer be considered as disjoint objects, but can be viewed as one arbitrage-free market. Here we provide an introduction to how mathematical methods for pricing and hedging financial claims such as the benchmark approach and local risk minimization can be applied to the valuation of hybrid financial insurance products, as well as to premium determination, risk mitigation and claim reserve management.

FORWARD INTEGRALS AND AN ITÔ FORMULA FOR FRACTIONAL BROWNIAN MOTION

We consider the forward integral with respect to fractional Brownian motion B(H)(t) and relate this to the Wick–Ito–Skorohod integral by using the M-operator introduced by Ref. 10 and the Malliavin derivative . Using this connection we obtain a general Ito formula for the Wick–Ito–Skorohod integralswith respect to B(H)(t), valid for .

PRICING OF UNEMPLOYMENT INSURANCE PRODUCTS WITH DOUBLY STOCHASTIC MARKOV CHAINS

This paper provides a new approach for modeling and calculating premiums for unemployment insurance products. The innovative modeling concept consists of combining the benchmark approach with its real-world pricing formula and Markov chain techniques in a doubly stochastic setting. We describe individual insurance claims based on a special type of unemployment insurance contracts, which are offered on the private insurance market. The pricing formulas are first given in a general setting and then specified under the assumption that the individual employment-unemployment process of an employee follows a time-homogeneous doubly stochastic Markov chain. In this framework, formulas for the premiums are provided depending on the ℙ-numeraire portfolio of the benchmark approach. Under a simple assumption on the ℙ-numeraire portfolio, the model is tested on its sensitivities to several parameters. With the same specification the models employment and unemployment intensities are estimated on public data of the Federal Employment Office in Germany.