This volume contains four selected survey papers on some aspects of mathemat-
ical ﬁnance presented and discussed at the “Lluis A. Santal´ o Summer School” that
was held in Santander (Spain) on July 2007 as part of the activities of the Uni-
versidad Internacional Men´ endez Pelayo (UIMP), in collaboration with the Real
Sociedad Matem´ atica Espa˜nola (RSME).
The role of stochastic diﬀerential equations in ﬁnance is well known and so
is the need for solving numerically these kinds of equations. In computational
ﬁnance, path-wise approximation of solutions for stochastic diﬀerential equations
is required, for example, when evaluating strongly path-dependent products, such
as Asian options or range accrual products. The paper by L.G. Gyurko and T.
Lyons presents a general framework for deriving high-order, stable and tractable
path-wise approximations of Stratonovich stochastic diﬀerential equations.
Hedge funds have been one of the hot topics in ﬁnance in the last twelve years
or more. At various times, some of them have been the focus of attention of the
mass media. The paper by L. Seco and F. Chen presents a systematic survey on this
topic from a mathematical point of view. The authors analyze, among other issues,
the diﬀerent classes of hedge funds and the models used both for management and
Still in the world of hedge funds, the paper by M. Escobar, S. Kr¨ amer, F.
Scheibl, L. Seco and R. Zagst introduces a new theoretical framework to price
hedge funds’ equity, inspired by the framework of Black and Cox for the valuation
of companies’ equity as call options. The proposed approach is able to ﬁt quite well
the ﬁrst four moments of the distribution of real returns when used with a sample
of over a thousand hedge funds.
Credit derivatives have been among the most demanded products in the ﬁnan-
cial markets. The subprime crisis has placed this kind of product in the spotlight.
R. Zagst and M. Scherer present a survey of the most widely used credit deriva-
tives and analyze the most relevant approaches and models used in the ﬁnancial
sector: structural-default, intensity-based, reduced-form and hybrid models which
combine the advantages of structural and intensity based models. They also focus
on the modeling of joint defaults, one of the key issues for the pricing and risk
measurement of this class of derivatives.
The editors wish to thank the Real Sociedad Matem´ atica Espa˜nola for giving
them the opportunity to organize the Summer School. Our thanks also go to
Universidad Internacional Men´ endez Pelayo, a nice place with excellent organization
and outstanding facilities, which made it easy for us to organize the lectures series
within the School. Finally, the speakers at the Summer School and authors of this
volume deserve our heartfelt gratitude both for the excellent lectures delivered at