Mathematical Methods in Finance at IMPA

Description

The use of sophisticated mathematical tools in financial engineering ranging from partial differential equations to stochastic analysis and numerical methods has been growing steadily during the past few decades. On the one hand, the mathematical tools and results have impacted the way financial phenomena are modeled and understood, and how risk is assessed and managed. On the other hand, the financial industry has been presenting a number of mathematical and computational challenges to researchers. 

The research on mathematical methods in finance at IMPA is directed towards:

•  Inverse Problems in Finance and Model Calibration
• Asymptotics of Stochastic Volatility Models        
• Real Options

IMPA Group on Math Finance

        Jorge P. Zubelli  (PI)
        Edgardo Brigatti (Collaborator)
        Steven Lillywhite (Former Post Doc)
        Luca Mertens   (Former Post Doc)
        (see also collaborators below)

        Students and Former Students
        Ph.D. Students
                Lucia Chiappara (2015 expected)
                Vinicius Albani (2012 expected)
                Leonardo Muller (2009)
                Cesar A. Gomez Velez (2007)
               

        M.Sc. Students
               Diogo Duarte (2010)
               Guillermo Gomez (2010)
               Ana Luiza Abrão Roriz (2009)
               Cassio Alves (2008)
               Bernardo Meres (2008)
               Sérgio V. Bruno (2008)

Pictures

Collaborators and Co-organizers

Events and Seminars

        IMPA Conferences

        Minisimposia and Minicourses

        Talks
          Fourth Brazilian Conference on Statistical Modelling in Insurance and Finance:  Maresias,               April 4 - 8, 2009, Brazil
          Chemnitz Symposium on Inverse Problems 2008: Chemnitz, September 25 - 27, 2008

Recent Papers and Publications on Math Finance
Robust Management and Pricing of LNG Contracts with Cancellation Options. by V. Guigues, C. Sagastizabal & J.P. Zubelli Abstract: The management of Liquefied Natural Gas contracts with cancellation options is a stochastic multi-commodity flow problem that can be modelled as a multistage stochastic linear program with mixed-binary variables. For this general type of problems we propose a rolling horizon robust policy that is feasible and can be used in simulations as a selection and pricing mechanism. The approach is assessed by numerical results on a realistic data set for a large company owing a network of pipelines and storages that desires to price several Liquefied Natural Gas contracts with cancellation options.	Submitted	article (pdf)
A Convex-Regularization Framework for Local-Volatility Calibration in Derivative Markets. 6th World Congress of the Bachelier Finance Society by A. de Cezaro, O. Scherzer & J.P. Zubelli Abstract: We present a unified framework for the calibration of local volatility models that makes use of recent tools of convex regularization of ill-posed Inverse Problems.  The unique aspect of the present approach is that it address in a general and rigorous way the key issue of convergence and sensitivity of the regularized solution when the noise level of the observed prices goes to zero. In particular, we present convergence results that include convergence rates with respect to noise level in fairly general contexts and go well beyond the classical quadratic regularization. Our approach directly relates to many of the different techniques that have been used in volatility surface estimation. In particular, it directly connects with the Statistical concept of exponential families and entropy-based estimation. Finally, we also show that our framework connects with the Financial concept of Convex Risk Measures.		article (pdf)
Evaluation of Optional Cancellation Contracts. International Annual Real Options Conference 2010 by L.E. Muller, M. Souza & J.P. Zubelli We consider the problem of evaluating the cost of the optionality to cancel a future delivery of a commodity when the seller has a number of markets to choose from. The technique has potential applications to contracts of Liquefied Natural Gas loads and re- quires solving certain diffusion problems in a multi-variable context.		article(pdf)
Real Option Pricing with Mean-Reverting Investment and Project Value. To appear: European Journal of Finance by S. Jaimungal, M. Souza & J.P. Zubelli Abstract: In this work we are concerned with valuing the option to invest in a project when the project value and the investment value are both mean-reverting. Previous works which dealt with stochastic project and investment value concentrate on geometric Brownian motions for driving the values. However, when the project involved is linked to commodities, mean-reverting assumptions are more meaningful. Here, we introduce a model and prove that the optimal exercise strategy is not a function of ratio of project value to investment V/I -- as it is in the Brownian case. We further apply the Fourier space time-stepping algorithm of Jaimungal and Surkov (2009) to numerically investigate the option to invest. The optimal exercise policies are found to be approximately linear in $V/I$; however, the intercept is not zero.		article(pdf)
"Strategic Investment Decisions under Fast Mean-Reversion Stochastic Volatility" Published in Applied Stochastic Models in Business and Industry by  M. O. Souza & Jorge P. Zubelli. Abstract: We are concerned with investment decisions when the spanning asset that correlates with the investment value undergoes a stochastic volatility dynamics. The project value in this case corresponds to the value of an American call with dividends, which can be priced by solving a generalized Black-Scholes free boundary value problem. Following ideas of Fouque et al., under the hypothesis of fast mean reversion, we obtain the formal asymptotic expansion of the project value and compute the adjustment of the price due to the stochastic volatility. We show that the presence of the stochastic volatility can alter the optimal time investment curve in a significative way, which in turn implies that caution should be taken with the assumption of constant volatility prevalent in many real option models. We also indicate how to calibrate to market data the model in the asymptotic regime.		article(pdf)
Real Option Pricing with Mean-Reverting Investment and Project Value. Real Options: Theory Meets Practice. 2009. by S. Jaimungal, M. Souza & J.P. Zubelli Abstract: In this work we are concerned with real option prices when the project value V_t and the investment value I_t undergo a mean-reverting stochastic dynamics. We consider the question of finding the dynamics for which an investment trigger curve, based on the ratio V_t /I_t, can be determined.    For a particular class of mean-reverting processes, we show that the investment frontier can be represented by such a ratio. In particular, the dynamics of the ratio is also mean-reverting.    For more general dynamics, which might include jumps, the above reductions do not seem to be possible, and a Fast Fourier Stepping Method, developed by Jackson, Jaimungal, and Surkov (2008) and Jaimungal and Surkov (2009), is discussed instead.		article(pdf)
Towards a Generalization of Dupire's Equation for Several Assets Journal of Mathematical Analysis and Applications Vol. 355, No. 1, 170-179 (2009) by P. Amster, P. de Napoli & Jorge P. Zubelli. Abstract: We pose the problem of generalizing Dupire's equation for the price of call options on a basket of underlying assets.  We present an analogue of Dupire's equation that holds in the case of several underlying assets  provided the volatility is time dependent but not asset-price dependent. We deduce it from a relation that seems to be of interest on its own.		article(pdf)
Inverse problems and regularization techniques in option pricing PAMM Volume 7 Issue 1, Pages 1042403 - 1042404 (2008) by M. O. Souza and J. P. Zubelli Option-price based calibration of stochastic volatility models under fast mean reversion poses quite challenging inverse problems. Nevertheless, in this note we remark that by an appropriate multi-scale asymptotic analysis, one can calibrate the models in a stable way for a number of different asymptotic regimes. These regimes include, but are not restricted to, those studied by Fouque et al.		link (html)
Real Options under Fast Mean Reversion Stochastic Volatility Real Options: Theory Meets Practice. 2008. by  M. Souza & J.P. Zubelli Abstract: In this paper, we study the McDonald-Siegel (MS) model for real options under the assumption that the spanning asset undergoes a stochastic volatility dynamics that reverts to a historical value according to an Ornstein-Uhlenbeck process driven by a second source of uncertainty. In this case, the market is not complete, and valuation, even for a perfectly correlated asset, is not as straightforward as in the MS model. Nevertheless, it is possible to derive a pricing equation by risk-neutral arguments that depends on the so-called market risk premium. Under the further assumption that the driving volatility process is fast-mean reverting, we derive an asymptotic approximation for the value of a real-option. In such case, the model becomes very parsimonious and can be calibrated to real data.		article(pdf)
On the Asymptotics of Fast Mean-Reversion Stochastic Volatility Models International Journal of Theoretical and Applied Finance (IJTAF) Page. 817 - 835. August 2007 by M. O. Souza & Jorge P. Zubelli Abstract: We consider the asymptotic behavior of options under stochastic volatility models for which the volatility process fluctuates on a much faster time scale than that defined by the riskless interest rate. We identify the distinguished asymptotic limits and, in contrast with previous studies, we deal with small volatility-variance (vol-vol) regimes. We derive the corresponding asymptotic formulae for option prices, and find that the first order correction displays a dependence on the hidden state and a non-diffusive terminal layer. Furthermore, this correction cannot be obtained as the small variance limit of the previous calculations. Our analysis also includes the behavior of the asymptotic expansion, when the hidden state is far from the mean. In this case, under suitable hypothesis, we show that the solution behaves as a constant volatility Black–Scholes model to all orders. In addition, we derive an asymptotic expansion for the implied volatility that is uniform in time. It turns out thatthe fast scale plays an important role in such uniformity. The theory thus obtained yields a more complete picture of the different asymptotics involved under stochastic volatility. It also clarifies the remarkable independence on the state of the volatility in the correction term obtained by previous authors.		article(pdf)
"Multiple Scale Asymptotics of Fast Mean Reversion Stochastic Volatility Models." Proceedings of the Third Brazilian Confererence: on Statistical Modelling in Insurance and Finance. Sao Paulo : Institute of Mathematics and Statistics, USP. pp. 248-253. (2007).  by  M. O. Souza & Jorge P. Zubelli.		article(pdf)
"Asymptotic Behavior of Stochastic Volatility Models." Proceedings of the Second Brazilian Confererence: on Statistical Modelling in Insurance and Finance. Sao Paulo: Institute of Mathematics and Statistics, USP. pp. 222-227. (2005). by M. O. Souza & Jorge P. Zubelli		article(pdf)
"Inverse Problems in Finance: A Short Survey on Calibration Techniques." Proceedings of the Second Brazilian Conference on Statistical Modelling in Insurance and Finance. pp. 64-76 (2005). by Jorge P. Zubelli.		article(ps)
"Discrete-Time Mathematical Modeling in Quantitative Finance" (in portuguese) - Minicourse textbook for the XXX CNMAC. by Max O. de Souza & Jorge P. Zubelli.		table of contents (in portuguese)

Master's Progam on Mathematical Methods in Finance:

Teaching (links to courses on Math Finance at IMPA)

        Mathematical Methods in Finance - 2005 - 2006 - 2007
        Computational Methods in Finance - 2004 - 2005
        Derivatives - 2005 - 2006
        Partial Differential Equations in Finance - 2004 - 2005
        Risk - 2007

Useful Links

       
        Courant Institute Financial Mathematics M Sc Program
        Columbia University
        Frontières en Finance
        Probabilités et Finance
        Center for Research in Financial Mathematics Santa Barbara
        Differences between Finance and Economics: Please click here
        Math Finance Blog by Matheus Grasselli.