Markus Leippold

Asset Pricing under the Quadratic Class

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.

The Term Structure of Variance Swap Rates and Optimal Variance Swap Investments

This paper performs specification analysis on the term structure of variance swap rates on the S&P 500 index and studies the optimal investment decision on the variance swaps and the stock index. The analysis identifies 2 stochastic variance risk factors, which govern the short and long end of the variance swap term structure variation, respectively. The highly negative estimate for the market price of variance risk makes it optimal for an investor to take short positions in a short-term variance swap contract, long positions in a long-term variance swap contract, and short positions in the stock index.

A Geometric Approach To Multiperiod Mean Variance Optimization of Assets and Liabilities

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 multi period 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 (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 (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 issued related to AL portfolios: The impact of taking liabilities into account on the implied mean variance frontiers, the quantification of the impact of the investment horizon and the determination of the optimal initial funding ratio.

A Simple Model of Credit Contagion

We propose a simple and implementable model of credit contagion where we include macro- and microstructural dependencies among the debtors within a credit portfolio. We show that, even for diversified portfolios, moderate microstructural dependencies already have a significant impact on the tails of the loss distribution. This impact increases dramatically for less diversified microstructures. Since the inclusion of microstructural dependencies acts on the tails, the choice of an appropriate risk measure for credit risk management is a delicate task.

Design and Estimation of Multi-Currency Quadratic Models

To simultaneously account for the properties of interest-rate term structure and foreign exchange rates within an arbitrage-free framework, we propose a multi-currency quadratic model with an (m+n) factor structure. The m factors model the term structure of interest rates in both countries. The n factors capture the portion of the exchange rate movement that is independent of the term structure of either country. We estimate a series of multi-currency quadratic models using U.S. and Japanese LIBOR and swap rates and the exchange rate between the two countries.

Quantile Estimation with Adaptive Importance Sampling

We introduce new quantile estimators with adaptive importance sampling. The adaptive estimators are based on weighted samples that are neither independent nor identically distributed. Using the law of iterated logarithm for martingales, we prove the convergence of the adaptive quantile estimators for general distributions with non-unique quantiles, thereby extending the work of Feldman and Tucker (1966). We illustrate the algorithm with an example from credit portfolio risk analysis.

Multiperiod mean-variance efficient portfolios with endogenous liabilities

We study the optimal policies and mean-variance frontiers (MVF) of a multiperiod mean-variance optimization of assets and liabilities (AL). This makes the analysis more challenging than for a setting based on purely exogenous liabilities, in which the optimization is only performed on the assets while keeping liabilities fixed. We show that, under general conditions for the joint AL dynamics, the optimal policies and the MVF can be decomposed into an orthogonal set of basis returns using exterior algebra. This formalism, novel to financial applications, allows us to study analytically the structure of optimal policies and MVF representations under endogenous liabilities in a multidimensional and multiperiod setting. Using a numerical example, we illustrate our methodology by analysing the impact of the rebalancing frequency on the MVF and by highlighting the main differences between exogenous and endogenous liabilities.

A New Goodness-of-Fit Test for Event Forecasting and Its Application to Credit Defaults

We develop a new goodness-of-fit test for validating the performance of probability forecasts. Our test statistic is particularly powerful under sparseness and dependence in the observed data. To build our test statistic, we start from a formal definition of calibrated forecasts, which we operationalize by introducing two components. The first component tests the level of the estimated probabilities; the second validates the shape, measuring the differentiation between high and low probability events. After constructing test statistics for both level and shape, we provide a global goodness-of-fit statistic, which is asymptotically χ2 distributed. In a simulation exercise, we find that our approach is correctly sized and more powerful than alternative statistics. In particular, our shape statistic is significantly more powerful than the Kolmogorov--Smirnov test. Under independence, our global test has significantly greater power than the popular Hosmer--Lemeshows χ2 test. Moreover, even under dependence, our global test remains correctly sized and consistent. As a timely and important empirical application of our method, we study the validation of a forecasting model for credit default events. This paper was accepted by Wei Xiong, finance.

What's Beneath the Surface? Option Pricing with Multifrequency Latent States

We introduce a tractable class of multi-factor price processes with regime-switching stochastic volatility and jumps, which flexibly adapt to changing market conditions and permit fast option pricing. A small set of structural parameters, whose dimension is invariant to the number of factors, fully specifies the joint dynamics of the underlying asset and options implied volatility surface. We develop a novel particle filter for efficiently extracting the latent state from joint S&P 500 returns and options data. The model outperforms standard benchmarks in- and out-of-sample, and remains robust even in the wake of seemingly large discontinuities such as the recent financial crisis.

Are Ratings the Worst Form of Credit Assessment Apart from All the Others

We present a prediction model to forecast corporate defaults. In a theoretical model, under incomplete information in a market with publicly traded equity, we show that our approach must outperform ratings, Altman’s Z-score, and Merton’s distance to default. We reconcile the statistical and structural approaches under a common framework, i.e., our approach nests Altman’s and Merton’s approaches as special cases. Empirically, we cannot reject the superiority of our approach.Furthermore, the numbers of observed defaults align well with the estimated probabilities. Finally, with rank transforms, we obtain cycle-adjusted forecasts that still outperform ratings.