Computational Mathematical Finance

Research grant PTDC/MAT/64838/2006 funded by FCT.

May 2007 - April 2010

Standard financial equity market models exclusively model the price process of the stock prices. The prices of liquid derivatives, like plain vanilla calls, are only used to calibrate the parameters of the model. A more natural approach is to model the evolution of stock price and some liquid instruments simultaneously. The motivation comes from the so-called market models on implied volatility surfaces. In these models, liquidly traded plain vanilla options and the underlying asset are modeled simultaneously and the ultimate goal is to find the arbitrage free dynamics of the implied volatility surface.

The application of robust optimization to finance is fairly recent. The goal of robust optimization is to find a solution for an optimization model under a variety of different scenarios and/or a range of uncertain or unknown input parameters. Robust optimization solutions are typically less sensitive to input variations. There are several open questions concerning the stability of robust optimization models in asset allocation and static hedging of exotic and path dependent options.

Another topic of intensive current research are pricing models for portfolio credit risk derivative products, such as CDOs (Collateralized Debt Obligations) and FtDS (First to Default Swaps).

The first task and perhaps the only relatively classic in mathematical finance is related to the estimation of volatility surfaces. Here we will exploit the connection between the forward price term structure models and the implied volatility term structure. In a first step, the analysis will focus at at-the-money options and will be mainly analytical. A second step considers different strike prices and will require a numerical nonlinear optimization approach. A number of different sub products will be developed, including numerical packages for estimating risk-neutral densities and volatilities functions based on vanilla options for different strike prices and maturity dates.

The second component of this project is devoted to the study of stability in robust portfolio selection. Robust optimization is a recent area of research in mathematical programming. Its application to portfolio selection or asset allocation is even more recent. Despite some progress, there are still many open questions. Due to transaction costs and conservative trends, asset allocation managers prefer portfolio solutions which are the least sensitive to perturbations of the market data estimators. We will focus both on Markowitz portfolio selection and on portfolio replication for static hedging of exotic and path dependent options (which offers attractive alternatives to the prohibitive dynamic hedging approaches). In both cases we will study the stability of the solution and the corresponding robust models and develop more efficient and robust algorithms.

Finally, we will analyze credit derivatives, in particular we will develop
models that will allow us to price portfolio credit derivatives, such as
Collateralized Debt Obligations (CDOs) or First-to-Default Swaps (FtDS).
CDOs are becoming increasingly popular in the credit derivatives market but
up to now the existing models are far from satisfying the industry needs.
We will use the reduced-form model approach to study portfolio credit
derivatives both from the traditional botton-up and the new top-down points
of view. We start by adapting some recently developed reduced-form models to
the portfolio setup and show that new features such as common shot-noise
processes in default intensities are able to overcome the traditional
reduced-form model limitation and produce realistic default correlations
across firms. In addition to being fundamental and realistic, these models can,
in principle, produce closed-form solutions for pricing CDOs and will therefore
compete with the ad hoc copula approach in terms of industry applicability.
For the top-down approach we will develop a framework to price CDOs based on
modeling forward looking loss probabilities simultaneously for all possible
time-horizons and loss levels. For calibration purposes we will use data on
credit indices, such as CDX or i-Traxx and then check model performance.

**Papers:**

[1] T. Björk and R.M. Gaspar, Interest rate theory and geometry, Portugaliae Mathematica, 67 (2010) 321-367.

[2] A. Egídio dos Reis, R.M. Gaspar, and A.T. Vicente, Solvency II - an important case in applied VaR, to appear in The VaR Modeling Handbook: Practical Applications in Alternative Investments, Banking, Insurance and Portfolio Management, G.N. Gregoriou, ed., McGraw-Hill, 2009.

[3] R.M. Gaspar, Implied volatility and forward prices term structures, submitted to publication.

[4] R.M. Gaspar and A. Murgoci, Convexity adjustments, to appear in Encyclopedia of Quantitative Finance, R. Cont, ed., Wiley Finance, 2009.

[5] R.M. Gaspar, A. Murgoci, Convexity Adjustments for ATS models, submitted.

[6] R.M. Gaspar and T. Schmidt, On the pricing of collateralized debt obligations, Credit Derivatives Handbook, P.U. Ali and G.N. Gregoriou, eds., Ch. 11, pp. 229-258, McGraw-Hill, 2008.

[7] R.M. Gaspar and T. Schmidt, On the pricing of CDOs, in Credit Derivatives Handbook, P. U. Ali and G. Gregouriou, eds., Ch. 11, pp. 229-258, McGraw-Hill, 2008.

[8] R.M. Gaspar and T. Schmidt, CDOs in the light of the current crisis, to appear in Economica.

[9] R.M. Gaspar and T. Schmidt, Term structure models with shot-noise effects, submitted to publication.

[10] R.M. Gaspar and I. Slinko, On recovery and intensity correlation - a new class of credit risk models, The Journal of Credit Risk, 4 (2008) 1-33.

[11] A.M. Monteiro, R.H. Tütüncü, and L.N. Vicente, Recovering risk-neutral probability density functions from options prices using cubic splines and ensuring nonnegativity, European Journal of Operational Research, 187 (2008) 525-542. PDF

[12] A.M. Monteiro, R.H. Tütüncü, and L. N. Vicente, Estimation of risk-neutral density surfaces, Computational Management Science, 8 (2011) 387-414. PDF

[13] R. Silva, J. Soares, and L.N. Vicente, Local analysis of the feasible primal-dual interior-point method, Computational Optimization and Applications, 40 (2008) 41-57. PDF

[14] R. Silva, M. Ulbrich, S. Ulbrich, and L.N. Vicente, A globally convergent primal-dual interior-point filter method for nonlinear programming: new filter optimality measures and computational results, preprint 08-49, Dept. of Mathematics, Univ. Coimbra. PDF

[15] L. N. Vicente, Implicitly and densely discrete black-box optimization
problems, Optimization Letters, 3 (2009) 475-482.
PDF

**Software:**

[1] Recovering risk-neutral
probability density functions from options prices using cubic splines
and ensuring nonnegativity (MATLAB) - available under request.