Mathematical Methods in Finance at IMPA

Mathematical Methods in Financeat 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)Xu Yang (Post Doc)
(see also collaborators below)
Students and Former Students
Ph.D. Students
Vinicius Albani (2012)
Edgardo Brigatti
Ewan Mackie
Luca Mertens
Maria Nogueiras
Leonardo Vicchi
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 Coorganizers
Cesar A. Gomez Velez (Colombia)
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)
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)
Events and Seminars
IMPA Conferences2016  Mathematics and Finance: Research in
Options RIO2016
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
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
Inverse Problems in
Finance  Zurich  ICIAM 2007
III Bienal Meeting of the Brazilian Math Society  Minicourse Part 1  Minicourse Part 2
III Bienal Meeting of the Brazilian Math Society  Minicourse Part 1  Minicourse Part 2
Talks
Vienna Congress on Mathematical Finance: Vienna, September 2016, Austria.
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 8th13th, 2010, Brazil
Fourth Brazilian Conference
on Statistical Modelling in Insurance and
Finance:
Maresias,
April 4  8, 2009, BrazilDe 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 8th13th, 2010, Brazil
Chemnitz Symposium on Inverse Problems 2008: Chemnitz, September 25  27, 2008
Third Brazilian Conference
on Statistical Modelling in Insurance and
Finance: Maresias,
March 25  30, 2007
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
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
Recent Papers and Publications on Math Finance 

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 atthemoney 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
modelbased adjustment of such prices that improves reconstructions and
smile adherence.
In order to tackle the illposedness of the calibration problem we incorporate a priori information through a judiciously designed Tikhonovtype 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 longterm, 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 longterm 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. 
Article (doi) 

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 wellsuited 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. 
preprint (arxiv) 

On the choice of the Tikhonov regularization parameter and the discretization level: A discrepancybased 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 discrepancybased 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.

Article (doi) 

Online local volatility
calibration by convex regularization. Applicable Analysis and Discrete
Mathematics, v. 8, p. 243268, 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 everincreasing flow of option
price information into the wellaccepted 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
BochnerSobolev 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 discrepancybased choice for the
regularization parameter. We conclude by illustrating the theoretical
results by means of numerical tests.

article(pdf) 

The tangential cone condition for the iterative calibration of local volatility surfaces. IMA Journal of Applied Mathematics, v. online, p. 121, 2013. by De Cezaro, A. and Zubelli, J.P. Abstract: In this paper, we
prove that the parametertosolution 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
Kaczmarztype strategies of the local volatility calibration problem by
transforming the problem into a system of nonlinear illposed
equations.

article(pdf) 

Real Option Pricing with MeanReverting 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 meanreverting 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 meanreverting processes, we show that the investment frontier can be represented by such a ratio. In particular, the dynamics of the ratio is also meanreverting. 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. 
article(pdf) 

Multiscale Stochastic Volatility Model for Derivatives on Futures by JeanPierre 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 firstorder 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 prerequisite is the firstorder approximation for the underlying derivative. In other words, the method proposed here allows us to derive the firstorder 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. 
article (pdf) 

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 BlackScholes partial differential equation from prices of
European call options. This is a highly nonlinear illposed problem
which in practice is subject to different noise levels associated
to bidask spreads and sampling errors. We analyze, in appropriate
function spaces, different properties of the
parametertosolution 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. 
article (pdf)  
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 multicommodity flow problem that can be modelled as a
multistage stochastic linear program with mixedbinary 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.

article (pdf) 

A ConvexRegularization Framework for LocalVolatility 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 illposed 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 entropybased 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
multivariable context.

article(pdf)  
Real Option Pricing with MeanReverting 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
meanreverting. 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, meanreverting 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 timestepping
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 MeanReversion
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 BlackScholes 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 MeanReverting 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 meanreverting
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 meanreverting processes, we show that the investment frontier can be represented by such a ratio. In particular, the dynamics of the ratio is also meanreverting. 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, 170179 (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
assetprice
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 Optionprice
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 multiscale 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 McDonaldSiegel (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
OrnsteinUhlenbeck 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
riskneutral arguments that depends on the socalled market risk
premium. Under the further assumption that the driving volatility
process is fastmean reverting, we derive an asymptotic approximation
for the value of a realoption. In such case, the model becomes very
parsimonious and can be calibrated to real data.

article(pdf)  
On
the Asymptotics of Fast MeanReversion 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
volatilityvariance (volvol) 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
nondiffusive 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. 248253.
(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. 222227.
(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. 6476 (2005).
by Jorge P. Zubelli. 
article(ps)  
"DiscreteTime 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:
Description
of the Program (in portuguese)
Link to the 2003 year book (in portuguese)
Link to the 2002 year book (in portuguese)
Link to the 2003 year book (in portuguese)
Link to the 2002 year book (in portuguese)
Teaching (links to courses on Math Finance at IMPA)
Mathematical Methods in Finance  2005  2006  2007  2012Computational Methods in Finance  2004  2005
Derivatives  2005  2006
Partial Differential Equations in Finance  2004  2005  2012
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.