Robustness of quadratic hedging strategies in finance via Fourier transforms

In this paper we investigate the consequences of the choice of the model to partial hedging in incomplete markets in finance. In fact we consider two models for the stock price process. The first model is a geometric Lévy process in which the small jumps might have infinite activity. The second model is a geometric Lévy process where the small jumps are truncated or replaced by a Brownian motion which is appropriately scaled. To prove the robustness of the quadratic hedging strategies we use pricing and hedging formulas based on Fourier transform techniques. We compute convergence rates and motivate the applicability of our results with examples

Mortality/longevity risk-minimization with or without securitization

This paper addresses the risk-minimization problem, with and without mortality securitization, a la Follmer-Sondermann for a large class of equity-linked mortality contracts when no model for the death time is specified. This framework includes situations in which the correlation between the market model and the time of death is arbitrary general, and hence leads to the case of a market model where there are two levels of information-the public information, which is generated by the financial assets, and a larger flow of information that contains additional knowledge about the death time of an insured. By enlarging the filtration, the death uncertainty and its entailed risk are fully considered without any mathematical restriction. Our key tool lies in our optional martingale representation, which states that any martingale in the large filtration stopped at the death time can be decomposed into precise orthogonal local martingales. This allows us to derive the dynamics of the value processes of the mortality/longevity securities used for the securitization, and to decompose any mortality/longevity liability into the sum of orthogonal risks by means of a risk basis. The first main contribution of this paper resides in quantifying, as explicitly as possible, the effect of mortality on the risk-minimizing strategy by determining the optimal strategy in the enlarged filtration in terms of strategies in the smaller filtration. Our second main contribution consists of finding risk-minimizing strategies with insurance securitization by investing in stocks and one (or more) mortality/longevity derivatives such as longevity bonds. This generalizes the existing literature on risk-minimization using mortality securitization in many directions

A martingale representation theorem and valuation of defaultable securities

We consider a financial framework with two levels of information: the public information generated by the financial assets, and a larger flow of information that contains additional knowledge about a random time. This random time can represent many economic and financial settings, such as the default time of a firm for credit risk, and the death time of an insured for life insurance. As the random time cannot be seen before its occurrence, the progressive enlargement of filtration seems tailor-fit to model the larger flow of information that incorporates both the public flow and the information about the random time. In this context, our interest focuses on the following challenges: (a) How to single out the various risks coming from the financial assets, the random time, and their correlations? (b) How these risks interplay and lead to the formation of any risk in the larger flow of information? It is clear that understanding how risks build-up and interact, when one enlarges the flow of information, is vital for an efficient risk management and derivatives' evaluation in those informational markets. Our answers to these challenges are full and complete no matter what the model for the random time is and no matter how the random time is related to the public flow. In fact, we introduce "pure default" risks, and quantify and classify these risks afterward. Then we elaborate our martingale representation results, which state that any martingale in the large filtration stopped at the random time can be decomposed into orthogonal local martingales (i.e., local martingales whose product remains a local martingale). This constitutes our first principal contribution, while our second contribution consists in evaluating various defaultable securities according to the recovery policy, within our financial setting that encompasses any default model, using a martingale "basis." Our pricing formulas explain the impact of various recovery policies on securities and determine the types of pure default risk they entail