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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the historically stable relationship between a bank's funding rate, government rates, and inter-bank offered rates is no longer valid, which can be explained by credit risk, liquidity risk, and related bid-ask spreads.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.