# "Nicola Bruti Liberati" Scholarship for Postgraduate Studies in Quantitative Finance

Project location: ITALY

Project start date: September 2009 -
Project end date: August 2010

Project number: 2008-50

Beneficiary: POLITECNICO DI MILANO

This report summarizes Claudio Fontana's scientific activities in the period between the beginning of March 2010 and the end of August 2010.

Report by Claudio Fontana.

I have spent this period at ETH Zürich, in the first part, and at the University of Padova thereafter. The present report is structured as follows. The first part deals with my study and research activities at ETH Zürich, for both the Master of Advanced Studies in Finance program and an ongoing research project. The second part briefly describes the results obtained in Fontana (2010b), while the third part discusses the topics studied in Fontana (2010c) together with some recent related open problems. Finally, I conclude by listing my talks, seminars, conference presentations, workshops and schools in the period under analysis.

[1]

The first part of the present period has been mostly spent at ETH Zürich, in order to conclude the Master of Advanced Studies in Finance program, offered jointly by ETH Zürich and University of Zürich. A preliminary version of the MAS Thesis, entitled "Mean-variance Problems with Applications to Credit Risk Models" has been already sent together with the previous report. Such version of the Thesis has been significantly improved and updated, with the help of several useful remarks by my supervisor Prof. Martin Schweizer (ETH Zürich). The Thesis was then successfully defended with a final mark of 6.00/6.00. I have thus been awarded the "Master of Advanced Studies in Finance" diploma by ETH Zürich and University of Zürich with the final grade of 5.94/6.00 summa cum laude. As already described in the previous report, the first chapter of my Thesis deals with a general and abstract formulation of mean-variance portfolio choice problems and mean-variance indifference pricing. In this direction, there is an ongoing research project with Prof. Martin Schweizer aiming at a generalization of the results obtained in the Thesis, also on the basis of the abstract framework originally introduced in Schweizer (2001). In particular, the main goal is to arrive at a consistent formulation of a general quadratic utility maximization problem in an abstract setting from which one can easily deduce the solutions to the classical Markowitz problems and the mean-variance indifference price of a given contingent claim. In this way one can provide clear connections between several seemingly different optimization problems and also clarify some previous results from the literature. To this effect, one needs to modify some of the techniques used in Fontana (2010a) and this is currently under investigation. Major advantages of the proposed approach include solvability under a minimal no-arbitrage condition and explicit model-free characterizations of the optimal solutions. A further issue concerns the behavior of the optimal mean-variance solutions under a change of numéraire.

[2]

In the present period, the second main area of research has been represented by the prosecution of the analysis of credit risk models under incomplete information, leading to the paper Fontana (2010b) (currently under review). This research activity has been carried out at the University of Padova, under the supervision of Prof. Wolfgang J. Runggaldier. The paper Fontana (2010b) aims at extending the framework previously introduced in Fontana & Runggaldier (2010). In particular, the proposed model explicitly considers a stochastic not fully observable market price of risk process, thus characterizing the transition between the pricing and the physical probability measures. A rather flexible and general functional form is assumed for the market price of risk process, which is also correlated with the interest rates and the default intensities of the firms operating on the market. The formulation of the model under the real-world probability measure allows to consider, besides the information coming from financial market data, also the information embedded in the rating scores, due to a suitable modification of the filtering framework introduced in Fontana & Runggaldier (2010). The proposed approach has a number of advantages. First, it allows for a dynamic and continuous updating of the market price of risk, thus allowing the model to track the actual latent "market sentiment". Second, the filtering framework is consistent with both forward-looking information (i.e. financial market data) as well as backward-looking information (i.e. rating scores). Finally, the model allows for a coherent and integrated approach to both pricing and risk-management. The results of this paper will be presented by myself (as invited plenary speaker) at the upcoming AMaMeF workshop in Berlin (27-30 September 2010). As partially explained at the end of the previous report, I have an ongoing research project related to these issues, in collaboration with Juan Miguel Montes (PhD student at the University of Padova). More specifically, we aim at developing a model which can meaningfully link the structure of the random default event to the pattern of the implied volatility skew extracted from equity option prices. To this effect, one needs an explicit model describing the stochastic evolution of the prices of the risky assets which can also take account of the occurrence of random default events (see e.g. Carr & Linetsky (2006), Campi et al. (2009), Carr & Madan (2010)).

[3]

In accordance with one of the research projects outlined in the previous report, the last main area of research during the present period has been represented by the study of financial market models which may not necessarily admit an Equivalent (Local) Martingale Measure. In the last years there has been significant interest in such models, maybe due to the dramatic turbulences raging over financial markets. A related phenomenon concerns also the mathematical modeling of financial bubbles, which are however coherent with the classical no-arbitrage theory as developed in Delbaen & Schachermayer (1994) and Delbaen & Schachermayer (1998). However, when a martingale measure does not exist most of the classical results of mathematical finance seem to break down and one is led to ask whether there is any meaningful way to proceed in order to solve the crucial problems of pricing and hedging contingent claims. It is a remarkable results that the answer to such a question is positive, as documented for instance by the so-called benchmark approach developed by Eckhard Platen and several co-authors in a series of papers (see Platen & Heath (2006) for a textbook account). On the basis of this literature, the paper Fontana (2010c) studies rather general financial market models based on a diffusion structure for the price processes of the risky assets and deals with the problem of valuing and hedging general contingent claims under the absence of an ELMM. The paper starts with some preliminary results on the no-arbitrage properties of the model. In fact, it is shown that one can replace the classical condition of No Free Lunch with Vanishing Risk (NFLVR) with the weaker condition of No Unbounded Profit with Bounded Risk (NUPBR), originally introduced in Karatzas & Kardaras (2007). It is shown that (NUPBR) can be equivalently hold if and only if the market price of risk process exists and satisfies a suitable integrability property (a similar result can also be found in the recent paper Hulley & Schweizer (2010)). The same assumption ensures in turn the existence of the Growth Optimal Portfolio (GOP), which plays a fundamental role in the benchmark approach. The paper contains several extensions and clarifications of previous results from the literature. In particular, an interesting extension concerns market incompleteness (for example, most of the results in Platen & Heath (2006) are derived in the context of complete financial market models). The paper discusses also some applications to interest rate modeling and to arbitrage relations under the absence of an ELMM. The main results of the present paper have been presented by myself at the XXXIV A.M.A.S.E.S. conference in Macerata (1-4 September 2010). Together with Prof. Wolfgang J. Runggaldier there is an ongoing research project which aims at the development of a real-world reduced-form credit risk model, thus applying the concept of real-world pricing (which forms the essence of the benchmark approach) to defaultable markets. Due to the inherent incompleteness of defaultable markets, this raises also the interest in studying hedging approaches when no ELMM exists (compare e.g. the very recent paper Biagini et al. (2010)). Futhermore, the analysis of defaultable securities requires significant extensions of the methodologies studied so far to the more general case of asset price processes with discontinuous paths.

Scientific Activities (seminars, conference presentations, workshops and schools):

• Invited seminar at the Department of Mathematics (institute for Quantitative Finance) of Politecnico di Milano, 17 March 2010.

• Colloquium talk at the Department of Mathematics of ETH Zürich, 23 March 2010.

• Participation (with oral presentation) at the international conference on "Mathematical and Statistical Methods for Actuarial Sciences and Finance", Ravello, 7-9 April 2010.

• Participation (with oral presentation) at the "Fifth General AMaMeF Conference", Bled, 4-8 May 2010.

• Participation (with poster presentation) at the "45th Scientific Meeting of the Italian Statistical Society", Padova, 16-18 June 2010.

• Participation to the course on "Monte Carlo Statistical Methods" (Prof. G. Casella, University of Florida), Padova, Department of Statistical Sciences, 16-18 June 2010.

• Participation to the international conference "Contemporary Issues and New Directions in Quantitative Finance", Oxford-Man Institute of Quantitative Finance, University of Oxford, 9-11 July 2010.

References

Biagini, F., Cretarola, A. & Platen, E. (2010), Local Risk-minimization under the Benchmark Approach, preprint.

Campi, L., Polbennikov, S. & Sbuelz, A. (2009), Systematic Equity-based Credit Risk: A CEV Model with Jump to Default, Journal of Economic Dynamics and Control, 33: 93-108.

Carr, P. & Linetsky, V. (2006), A Jump to Default Extended CEV Model: an Application of Bessel Processes, Finance and Stochastics, 10: 303-330.

Carr, P. & Madan, D. (2010), Local Volatility Enhanced by a Jump to Default, SIAM Journal of Financial Mathematics, 1: 2-15.

Delbaen, F. & Schachermayer, W. (1994), A General Version of the Fundamental Theorem of Asset Pricing, Mathematische Annalen, 300: 463-520.

Delbaen, F. & Schachermayer, W. (1998), The Fundamental Theorem of Asset Pricing for Unbounded Stochastic Processes, Mathematische Annalen, 312: 215-250.

Fontana, C. (2010a), Mean-variance Problems with Applications to Credit Risk Models, MAS Thesis, ETH Zürich and University of Zürich.

Fontana, C. (2010b), Credit Risk and Incomplete Information: A Filtering Framework for Pricing and Risk Management, submitted.

Fontana, C. (2010c), Diffusion-based Financial Market Models without Martingale Measures: An Overview, preprint, Department of Pure and Applied Mathematics, University of Padova.

Fontana, C. & Runggaldier, W. (2010), Credit Risk and Incomplete Information: Filtering and EM Parameter Estimation, International Journal of Theoretical and Applied Finance, 13(5): 683-715.

Heath, D. & Platen, E. (2006), A Benchmark Approach to Quantitative Finance, Springer.

Hulley, H. & Schweizer, M. (2010), M6 - On Minimal Market Models and Minimal Martingale Measures, in: Chiarella, C. & Novikov, A. (eds.), Contemporary Quantitative Finance: Essays in Honour of Eckhard Platen, Springer.

Karatzas, I. & Kardaras, K. (2007), The Numéraire Portfolio in Semimartingale Financial Models, Finance and Stochastics, 11: 447-493.

Schweizer, M. (2001), From Actuarial to Financial Valuation Principles, Insurance: Mathematics and Economics, 28: 31-47.