Working Papers and Lectures 
 
 

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Abstract

A maintained assumption which is typically inspiring the risk-management practice is that the pricing functional of the economy is unique and is perfectly known by the agents. The major implication of this assumption is that when we compute the marked-to-market value of a position we come up with a unique value. Distribution risk, i.e. the case in which the probability measure used to price contingent claims is not known precisely, is instead paramount in corporate claims evaluation where the dynamics of the firm’s assets value is opaque to the investors. In this paper we use fuzzy measure theory to represent distribution risk by the upper and lower bound of  the price of the contingent claim computed as the upper and lower Choquet integral with respect to a subadditive function. In this paper we present a fuzzified version of Merton model. Sensitivity analysis shows that both the level and the difference between upper and lower bounds of credit spreads are positively related to the «quasi debt to firm value ratio» and to the volatility of the firm value. This finding may be read as correlation between credit risk and liquidity risk, a result which is particularly useful in concrete risk-management  applications.
 
 

Abstract

Value-at-Risk is typically thought of as the amount of equity to provide insurance against an abnormal fall of the value of the position. We show that this is about equivalent to considering the  Value-at-Risk as the difference between the the current forward value of the risky asset involved and the strike price of a protective put which is far out-of-the-money. So, in a complete market in which the amount of capital and the  protective put could be continuously rebalanced, this option based Value-at-Risk would effectively provide protection against a shortfall in value of the asset. While this approach has the crucial advantage of being inherently non-parametric, it is not well suited to accommodate the effects of market illiquidity and incompleteness. We show that, by using the fuzzy measure approach, we may easily generalize this approach to an incomplete market setting, so  providing a methodology  to compute a Value-at-Risk measure which is economically founded, non-parametric, and well suited to represent the component of uncertainty due to the choice of the right pricing kernel for illiquid assets.
 
 

Abstract

We investigate a methodology to set up consistent scenarios for stress testing analysis in financial  risk control and management. The method, based on the Black and Litterman bayesian approach to portfolio optimization, enables to mix historic and implied or private information, accounting for the co-movement among the markets. By tuning the mean values chosen for the scenarios and the degree of precision attached to them we are able to devise a whole range of mean loss and maximum probable loss, or Value-at-Risk measures. In particular, by setting a very precise scenario the mean and maximum probable loss converge toward similar values, while for very imprecise scenarios the mean loss figure is found to converge to zero, and the maximum probable loss collapses to the standard Value-at-Risk figure computed using historical information. As for options, we show that tuning the precision of the scenarios allows for the effects of changes in volatility on the option value, under each different scenarios. Finally, for more complex positions, such as those involving credit risk exposures, or more generally exposures to different markets, we suggest a tree methodology to report the scenarios and to pinpoint the key sources of risk.