Contributions published at Financial Engineering (Markus Leippold)

We introduce an adaptive importance sampling method for the loss distribution of credit portfolios based on the Robbins-Monro stochastic approximation procedure. After presenting the subtle construction of the algorithm, we apply our adaptive scheme for calculating the risk figures of a typical medium-sized credit risk portfolio with 2000 obligors. Simulating the tail of the loss distribution, we can improve significantly the variance reduction and outperform other recently proposed importance sampling approaches that are based on deterministic methods providing asymptotically optimal importance sampling distributions. Furthermore, the simple structure of the algorithm not only allows a straightforward implementation, but also offers a lot of flexibility for extensions to more complex models. Therefore, our numerical results motivate interesting future research paths for the application of stochastic approximation methods in risk management.

This paper develops a framework for the quantification of operational risk based on a network with functional dependencies that represent work flows for business activities. The functioning of each node depends on stochastic risk factors driven by inputs such as human resources, data and inputs from other nodes. Using analytical and numerical methods, we obtain answers concerning capital allocation, stability, risk figures, the effect of different network structures (called “topological diversification”) and dynamic diversification. Interpreting the results shows that the usual intuition gained from market and credit risk does not apply to the quantification of operational risk.

Belief that a single number can capture the degree of risk being taken within a bank or an investment is mistaken - especially when that number is value at risk. Markus Leippold explains why the measure is flawed, points to the dangers of its widespread acceptance by regulators and investors, and suggests an alternative.

We present a geometric approach to discrete time multiperiod mean variance portfolio optimization that largely simplifies the mathematical analysis and the economic interpretation of such model settings. We show that multiperiod mean variance optimal policies can be decomposed in an orthogonal set of basis strategies, each having a clear economic interpretation. This implies that the corresponding multiperiod mean variance frontiers are spanned by an orthogonal basis of dynamic returns. Specifically, in a k-period model the optimal strategy is a linear combination of a single k-period global minimum second moment strategy and a sequence of k local excess return strategies which expose the dynamic portfolio optimally to each single-period asset excess return. This decomposition is a multi period version of Hansen and Richard (Econometrica (1987)) orthogonal representation of single-period mean variance frontiers and naturally extends the basic economic intuition of the static Markowitz model to the multiperiod context. Using the geometric approach to dynamic mean variance optimization we obtain closed form solutions in the i.i.d. setting for portfolios consisting of both assets and liabilities (AL), each modelled by a distinct state variable. As a special case, the solution of the mean variance problem for the asset only case in Li and Ng (Mathematical Finance 10 (2000)) follows directly and can be represented in terms of simple products of some single period orthogonal returns. We illustrate the usefulness of our geometric representation of multiperiods optimal policies and mean variance frontiers by discussing specific issues related to AL portfolios: The impact of taking liabilities into account on the implied mean variance frontiers, the quantification of the impact of the rebalancing frequency and the determination of the optimal initial funding ratio.

In this article we discuss the implementation of general one-factor short rate models with a trinomial tree. Taking the Hull-White model as a starting point, our contribution is threefold. First, we show how trees can be spanned using a set of general branching processes. Secondly, we improve Hull-White's procedure to calibrate the tree to bond prices by a much more efficient approach. This approach is applicable to a wide range of term structure models. Finally, we show how the tree can be adjusted to the volatility structure. The proposed approach leads to an efficient and exible construction method for trinomial trees, which can be easily implemented and calibrated to both prices and volatilities.

We consider the design and estimation of quadratic term structure models. We start with a list of stylized facts on interest rates and interest rate derivatives, classified into three layers: (1) general statistical properties, (2) forecasting relations, and (3) conditional dynamics. We then investigate the implications of each layer of property on model design and strive to establish a mapping between evidence and model structures. We calibrate a two-factor model that approximates these three layers of properties well, and show that a flexible specification for the market price of risk is important in capturing the stylized evidence in forecasting relations while factor interactions are indispensable in generating the hump-shaped dynamics of bond yields.

The financial industry puts the Basle Committee under strain to align regulatory capital with economic capital. This could be reached by allowing more flexibility in the choice of risk measure for regulatory reporting. Markus Leippold and Paolo Vanini show that if banks could use the theoretically more sound risk measure of Expected Shortfall, the three cheers of Stahl (1997) would be reduced to exactly half as many cheers. This would substantially decrease the regulatory capital in most cases. There is no dispute on the necessity of regulating the financial industry. Ideally, a regulation aims at maintaining and improving the safety of the financial industry. At the root of every regulation framework lies the basic idea of calculating a capital reserve as a function of the firm’s total capital and its risk. While the calculation of the firm’s capital is tedious but feasible, the measurement of risk is a difficult task. Consequently, regulators advocate normative and simplified rules applicable to all financial institutions.

We identify and characterize a class of term structure models where bond yields are quadratic functions of the state vector. We label this class the quadratic class and aim to lay a solid theoretical foundation for its future empirical application. We consider asset pricing in general and derivative pricing in particular under the quadratic class. We provide two general transform methods in pricing a wide variety of fixed income derivatives in closed or semi-closed form. We further illustrate how the quadratic model and the transform methods can be applied to more general settings.

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