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## Models and Numerics in Financial Mathematics |

In the wake of the financial crisis, which
has brought several weaknesses in the global financial system to light, there
is also a reevaluation of the assumptions that have been standard in the
underlying financial mathematical models. It is now well understood that the
usual paradigm, in which financial risks can be mitigated, spread, or even
hedged away perfectly, is too simplistic for markets under stressed conditions. In this workshop we aim to focus on topics
in financial mathematics that have emerged from the financial crisis. We wish
to bring together academic researchers in financial mathematics and researchers
in financial institutions, interact and discuss, for example, risk measures
that have recently come in place. Moreover, we would like to discuss about
recent financial product valuation approaches, as a way to improve and better understand
hedging in incomplete financial markets. We are interested in the underlying mathematical
models, as well as in advanced numerical solution techniques used for pricing
and risk measurement. The crisis has provided important
information on appropriate directions for the required mathematical
improvements. As regards to hedging and risk mitigation, which are important
steps in the risk management chain, nowadays even the hedging of basic financial
derivatives has become a complicated task. Since the 2008 crisis, the
historically stable relationship between a bank's funding rate, government rates,
and inter-bank offered rates is no longer valid, which can be explained by
credit risk, liquidity risk, and related bid-ask spreads. Therefore, more
sophisticated models for these market imperfections are needed if hedging
programs are to remain effective under financial stress. It is not always clear
which techniques should preferably be used for specific financial applications.
In this workshop we wish to focus on the following three relevant, modern
mathematical themes to cover the different aspects. (A) The public demand for simpler financial
products is observed throughout financial practice nowadays. However, for each
product a careful consideration of all risk involved is taking place, making
the valuation of the basic products far from trivial, especially from a
modeling point of view. As an example, in the over-the-counter (OTC) market, trades
are settled directly between two parties and there is no third party to cover a
possible huge loss because of a defaulting party. When a counterparty defaults
before the contract's maturity, the investment in the OTC option will be lost
and the payoff will not be paid out; in other words, counterparty credit risk
is highly relevant. Within the context of risk management, we will focus on the
mathematical aspects of a recent concept called Credit Valuation Adjustment (CVA).
Credit Value Adjustment is the difference between a bank's portfolio value
without counterparty credit risk and the value which takes into account the
possible default of a counterparty. Aspects like the volatility smile and its
impact on CVA, consistent treatment of the (time-dependent) dependency between
the relevant risk factors, or fast estimation of the sensitivities making use
of novel numerical algorithms and mapping on massively parallel systems like
the multicore Graphics Processing Units may also appear in the discussion. When computing CVA, the so-called Expected
Exposure (EE) is an important building block, and Potential Future Exposure
(PFE) is the loss given a fixed confidence interval. Numerical methods to keep
track of the option values and their distributions during the life of the
option contracts contain essentially two elements, a forward Monte Carlo step for
generating future scenarios and a backward sweep to calculate exposures along
the generated asset paths. Along the paths, option values are determined at
each exercise time, for which efficient computation of the option prices is
required. Numerical integration methods, discretization schemes for partial
differential equations (PDEs) as well as Monte Carlo simulation methods seem
suitable candidates. An important issue for accurate CVA, is the modeling of wrong-way
risk for large portfolios. Wrong-way risk is the risk if exposure to a
counterparty is adversely correlated with the credit quality of that counterparty,
i.e., when default risk and credit exposure increase together. Especially for large
portfolios it is nontrivial to accurately model wrong-way risk within a
portfolio. (B) When valuing and risk-managing financial
products, practitioners demand fast and accurate prices and sensitivities, and
efficient methods have to be developed to cope with the pertinent mathematical
models. The choice of stochastic model for the underlying asset prices has a
crucial impact on the techniques required for calibration and for the pricing
of exotic financial products. In this workshop we aim at a deeper
understanding, from a mathematical perspective, of the strong and weak points
of the various models advocated. Stochastic Local Volatility models have
recently gained a lot of support in practice as well as in academics. The
calibration of such models to European options and first-generation exotics
enables a fairly complete coverage of the price dynamics of many financial
options, in particular in the foreign exchange market. For numerically solving
the obtained inverse problems, one can again distinguish Monte Carlo simulation,
numerical solution of (multidimensional) PDEs, and numerical integration.
Numerical techniques of the highest efficiency are however still lacking, and
closed-form analytical solutions facilitating the calibration are only
available for limited ranges of parameters. (C) The third theme concerns recent
insights in stochastic models based on (systems of) Backward Stochastic Differential
Equations (BSDEs). The well-known Feynman-Kac theorem
gives a probabilistic representation for the solution of a linear PDE by means
of the corresponding forward SDE and a conditional expectation. The solution of
a BSDE provides a probabilistic representation for semi-linear parabolic PDEs,
which forms a generalization of the Feynman-Kac
theorem. This connection enables us to solve a semi-linear PDE by probabilistic
numerical methods, like Monte Carlo simulation techniques. Probabilistic numerical methods to solve
BSDEs rely on a time discretization of the stochastic process and accurate
approximations for the appearing conditional expectations. BSDEs are thus
tightly connected to the numerical solution of the (nonlinear) Hamilton-Jacobi-Bellman
PDEs, and we wish to interact with researchers in HJB equations as well as in
the modeling with, and numerical solution of, BSDEs. As one of the simplest
examples, the Black-Scholes formula for pricing options can be represented by a
system of decoupled forward and backward SDEs. Market imperfections can be
incorporated, such as different lending and borrowing rates for money, the presence
of transaction costs or short sales constraints. These imperfections give rise
to more involved nonlinear BSDEs. If the asset price follows a jump diffusion
process, then the option cannot perfectly be replicated by assets and cash,
i.e., the market is not complete. A way to value and hedge options in this
setting is by means of a BSDE with jumps. Pricing and hedging of products
exposed to equity and interest rate risk by means of BSDEs has generated
interest in the novel concept of second-order BSDEs (2BSDEs). This
mathematically challenging topic will lead to the incorporation of volatility
model risk into pricing, hedging and mitigation issues. Little is known at the
moment about numerical schemes and their asymptotic properties. [Back] |