Mathematical Methods in Finance at IMPA
Mathematical Methods in Finance
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
Xu Yang (Post Doc)
(see also collaborators below)
Students and Former Students
Vinicius Albani (2012)
Leonardo Muller (2009)
Cesar A. Gomez Velez (2007)
Diogo Duarte (2010)
Guillermo Gomez (2010)
Ana Luiza Abrão Roriz (2009)
Cassio Alves (2008)
Bernardo Meres (2008)
Sérgio V. Bruno (2008)
P. Amster (UBA, Argentina)
M. Avellaneda (Courant Institute, USA)
B. Dupire (Bloomberg, USA)
M. Grasselli (McMaster University, Canada)
G. Iori (City University, London, UK)
S. Jaimungal (Toronto, Canada)
B. Hofmann (Chemnitz, Germany)
S. Lillywhite (Former Post Doc)
A. Meucci (Bloomberg, USA)
Leonardo Muller (J.P. Morgan, Sao Paulo, Brazil)
P. de Napoli (UBA, Argentina)
C. Sagastizabal (IMPA, Brazil and INRIA, France)
Max O. de Souza (UFF, Brazil)
2015 - Mathematics and Finance: Research in Options RIO2015
2014 - Mathematics and Finance: Research in Options RIO2014
2013 - Mathematics and Finance: Research in Options RIO2013
2012 - Mathematics and Finance: Research in Options RIO2012
2011 - Mathematics and Finance: Research in Options RIO2011
2010 - Mathematics and Finance: Research in Options RIO2010
2009 - Mathematics and Finance: Research in Options RIO2009
2008 - Mathematics and Finance: Research in Options RIO2008
2007 - Mathematics and Finance: Research in Options RIO2007
2006 - Mathematics and Finance: From Theory to Practice
2004 - Modelagem Matemática e Computacional em Finanças Quantitativas
Minisimposia and Minicourses
III Bienal Meeting of the Brazilian Math Society - Minicourse Part 1 - Minicourse Part 2
De Finetti Risk Seminar: Milano, September 15th, 2015, Italy.
Department of Statistics and Operations Research, University of Vienna: Colloquium Talk, March 23rd, 2015, Vienna, Austria.
Seventh Conference on Multivariate Distributions with Applications: Maresias August 8th-13th, 2010, Brazil
Chemnitz Symposium on Inverse Problems 2008: Chemnitz, September 25 - 27, 2008
International Congress of Mathematicians, Madrid 2006: Applications of Mathematics in the Sciences.
SIAM Conference on Financial Engineering 2006: Session on Volatility and Simulation
Second Brazilian Conference on Statistical Modelling in Insurance and Finance: Maresias, August 28 - September 3, 2005
Tenth CLAPEM - Latin American Congress of Probability and Mathematical Statistics - Session on Stochastic Caluculus and Finance
Third ERPEM - Session on Stochastica Calculus, Finance and Actuarial Science
|Convex Regularization of Local Volatility Estimation.
Vinicius Albani, Adriano De Cezaro, and Jorge P. Zubelli
TO APPEAR: International Journal of Theoretical and Applied Finance
We apply convex regularization techniques to the problem of calibrating Dupire's local volatility surface model taking into account the practical requirement of discrete grids and noisy data. Such requirements are the consequence of bid and ask spreads, quantization of the quoted prices and lack of liquidity of option prices for strikes far way from the at-the-money level. We obtain convergence rates and results comparable to those obtained in the idealized continuous setting. Our results allow us to take into account separately the uncertainties due to the price noise and those due to discretization errors. Thus allowing estimating better discretization levels both in the domain and in the image of the parameter to solution operator by a Morozov's discrepancy principle. We illustrate the results with simulated as well as real market data. We also validate the results by comparing the implied volatility prices of market data with the computed prices of the calibrated model.
|Click here for a preprint from SSRN
Local Volatility Models in Commodity Markets and Online Calibration.
Vinicius Albani, Uri M. Ascher, and Jorge P. Zubelli.
TO APPEAR: Journal of Computational Finance.
We introduce a local volatility model for the valuation of options on commodity futures by using European vanilla option prices. The corresponding calibration problem is addressed within an online framework, allowing the use of multiple price surfaces. Since uncertainty in the observation of the underlying future prices translates to uncertainty in data locations, we propose a model-based adjustment of such prices that improves reconstructions and smile adherence.
In order to tackle the ill-posedness of the calibration problem we incorporate a priori information through a judiciously designed Tikhonov-type regularization. Extensive empirical tests with market as well as synthetic data are used to demonstrate the effectiveness of the methodology and algorithms.
|Click here for the arxiv link|
|A pairs trading strategy based on linear state space models and the Kalman filter.
Carlos Eduardo de Moura, Adrian Pizzinga & Jorge Zubelli
Quantitative Finance, 2016,
Abstract: Among many strategies for financial trading, pairs trading has played an important role in practical and academic frameworks. Loosely speaking, it involves a statistical arbitrage tool for identifying and exploiting the inefficiencies of two long-term, related financial assets. When a significant deviation from this equilibrium is observed, a profit might result.
In this paper, we propose a pairs trading strategy entirely based on linear state space models designed for modelling the spread formed with a pair of assets. Once an adequate state space model for the spread is estimated, we use the Kalman filter to calculate conditional probabilities that the spread will return to its long-term mean. The strategy is activated upon large values of these conditional probabilities: the spread is bought or sold accordingly.
Two applications with real data from the US and Brazilian markets are offered, and even though they probably rely on limited evidence, they already indicate that a very basic portfolio consisting of a sole spread outperforms some of the main market benchmarks.
|A Hedged Monte Carlo Approach to Real Option Pricing
Edgardo Brigatti, Felipe Macias, Max O. Souza, Jorge P. Zubelli
Fields Institute Communications,
Volume 74, Commodities, Energy and Environmental Finance. Rene Aid, Michael Ludkovski,Ronnie Sircar (Editors)
Abstract: In this work we are concerned with valuing optionalities associated to invest or to delay investment in a project when the available information provided to the manager comes from simulated data of cash flows under historical (or subjective) measure in a possibly incomplete market. Our approach is suitable also to incorporating subjective views from management or market experts and to stochastic investment costs. It is based on the Hedged Monte Carlo strategy proposed by Potters et al (2001) where options are priced simultaneously with the determination of the corresponding hedging. The approach is particularly well-suited to the evaluation of commodity related projects whereby the availability of pricing formulae is very rare, the scenario simulations are usually available only in the historical measure, and the cash flows can be highly nonlinear functions of the prices.
|On the choice of the Tikhonov regularization parameter and the discretization level: A discrepancy-based strategy.
Inverse Problems in Imaging. Pages: 1 - 25, Volume 10, Issue 1, February 2016 doi:10.3934/ipi.2016.10.1
by Vinicius Albani, Adriano De Cezaro and Jorge P. Zubelli
Abstract: We address the classical issue of appropriate choice of the regularization and discretization level for the Tikhonov regularization of an inverse problem with imperfectly measured data. We focus on the fact that the proper choice of the discretization level in the domain together with the regularization parameter is a key feature in adequate regularization. We propose a discrepancy-based choice for these quantities by applying a relaxed version of Morozov's discrepancy principle. Indeed, we prove the existence of the discretization level and the regularization parameter satisfying such discrepancy. We also prove associated regularizing properties concerning the Tikhonov minimizers. We conclude by presenting some numerical examples of interest.
|Online local volatility
calibration by convex regularization. Applicable Analysis and Discrete
Mathematics, v. 8, p. 243-268, 2014.
by Albani, V. and Zubelli, J.P.
Abstract: We address the inverse problem of local volatility surface calibration from market given option prices. We integrate the ever-increasing flow of option price information into the well-accepted local volatility model of Dupire. This leads to considering both the local volatility surfaces and their corresponding prices as indexed by the observed underlying stock price as time goes by in appropriate function spaces. The resulting parameter to data map is defined in appropriate Bochner-Sobolev spaces. Under this framework, we prove key regularity properties. This enable us to build a calibration technique that combines online methods with convex Tikhonov regularization tools. Such procedure is used to solve the inverse problem of local volatility identification. As a result, we prove convergence rates with respect to noise and a corresponding discrepancy-based choice for the regularization parameter. We conclude by illustrating the theoretical results by means of numerical tests.
|The tangential cone condition for the
iterative calibration of local volatility surfaces. IMA Journal of Applied Mathematics, v. online, p. 1-21, 2013.
by De Cezaro, A. and Zubelli, J.P.
Abstract: In this paper, we prove that the parameter-to-solution map associated to the inverse problem of determining the diffusion coefficient in a parabolic partial differential equation satisfies the local tangential cone condition. In particular, we show stability and convergence of the regularized solutions by means of Landweber iteration. Our result has an immediate application to the local volatility calibration problem of the Black–Scholes model for European call options. We present a numerical validation based on simulated data to this calibration problem and discuss the results. We also prove convergence and stability of Kaczmarz-type strategies of the local volatility calibration problem by transforming the problem into a system of non-linear ill-posed equations.
|Real Option Pricing with Mean-Reverting Investment and Project Value. In: Real Options:
Theory Meets Practice. 13th Annual International Conference, 2009, Minho e Santiago. Real Options: Theory Meets Practice. 13th Annual International Conference, 2009.
by Jaimungal, S. ; Souza, Max O. ; Zubelli, J. P.
In this work we are concerned with real option prices when the project value and the investment value undergo a mean-reverting stochastic dynamics.
We consider the question of finding the dynamics for which an investment trigger curve, based on the ratio, 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 is discussed instead.
Multiscale Stochastic Volatility Model for Derivatives on Futures
by Jean-Pierre Fouque, Yuri Saporito, and Jorge P. Zubelli
The goal of this article is to present a new method to extend the singular and
regular perturbation techniques developed in the book by Fouque, Papanicolaou, Sircar and Sølna (2011, CUP) to price derivatives on futures when the asset presents mean reversion. We also consider the calibration procedure of the proposed model. We apply it to options on the Henry Hub natural gas futures and to options on stock and its volatility index in a consistent way. The main contribution of our work is a general method to compute the first-order approximation for the price of general compound derivatives such that no additional hypothesis on the regularity of the payoff function must be assumed. The only pre-requisite is the first-order approximation for the underlying derivative. In other words, the method proposed here allows us to derive the first-order approximation for compound derivatives keeping the hypotheses of the original approximation given in Fouque et al. (2011, CUP). Furthermore, this method maintains another desirable feature of the perturbation method: the direct calibration of the market group parameters.
|Convex Regularization of Local Volatility Models from Option Prices: Convergence Analysis and Rates
Nonlinear Analysis: Theory, Methods & Applications. Volume 75, Issue 4, March 2012, Pages 2398–2415
by A. de Cezaro, O. Scherzer & J.P. Zubelli
Abstract: We study a convex regularization of the local volatility surface identification problem for the Black-Scholes partial differential equation from prices of European call options. This is a highly nonlinear ill-posed problem which in practice is subject to different noise levels associated to bid-ask spreads and sampling errors. We analyze, in appropriate function spaces, different properties of the parameter-to-solution map that assigns to a given volatility surface the corresponding option prices. Using such properties, we show stability and convergence of the regularized solutions in terms of the Bregman distance with respect to a class of convex regularization functionals when the noise level goes to zero.
We improve convergence rates available in the literature for the volatility identification problem. Furthermore, in the present context, we relate convex regularization with the notion of exponential families in Statistics. Finally, we connect convex regularization functionals with convex risk measures through Fenchel conjugation. We do this by showing that if the source condition for the regularization functional is satisfied, then convex risk measures can be constructed.
|Robust Management and Pricing of LNG Contracts with Cancellation Options.
To appear in Journal of Optimization Theory and Applications
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.
|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.
|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.
|Real Option Pricing with Mean-Reverting Investment and
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.
|"Strategic Investment Decisions under Fast Mean-Reversion
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.
|Real Option Pricing with Mean-Reverting Investment and
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.
|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.
|Inverse problems and regularization techniques in
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.
|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.
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.
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.
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
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.
Modeling in Quantitative Finance" (in
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:
Link to the 2003 year book (in portuguese)
Link to the 2002 year book (in portuguese)
2005 - 2006 - 2007 - 2012
Computational Methods in Finance - 2004 - 2005
Derivatives - 2005 - 2006
Partial Differential Equations in Finance - 2004 - 2005 - 2012
Risk - 2007
Courant Institute Financial Mathematics M Sc Program
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.