Conference Agenda

This presentation addresses the calibration of financial mathematical capital market models, highlighting the practical challenges involved and focusing on improving the accuracy and consistency of these models.

Using the so-called PIA basic model, a market model for interest rate and share price development that is recognized as the industry standard for capital market simulations in Germany, we examine each step of the standard calibration procedure in detail.

We particularly focus on the calibration of the interest rate model, specifically a two-factor Hull-White model with a perfect fit to the initial market yield curve. We demonstrate how the quality of calibration can be improved by selecting appropriate underlying yield curves and introduce a framework for measuring calibration quality, including metrics such as fit errors and stability over time.

Furthermore, we study the influence of the choice of interest rate derivatives, namely interest rate caps and swaptions, on the calibration process. Given the tendency of the two stochastic factors within the interest rate model to be highly negatively correlated, the necessity and advantages of a two-factor model are critically discussed by comparing the calibration results to those of a one-factor model.

Finally, the market data used for calibration, including yield curves, interest rate derivatives, and forecasts, are thoroughly examined regarding their construction and consistency. Overall, we aim to provide practical insights for improving the robustness of capital market models.

The Nelson-Siegel and the Svensson family are parametric interpolation families for yield curves and forward curves, which are widely used by national banks and other financial institutions. The Nelson-Siegel family expresses the forward curve as a linear combination of three basis functions, commonly associated to level, slope and curvature in the form $$\beta_0 + \beta_1\exp\left(-\frac{x}{\tau}\right) + \frac{\beta_2}{\tau}x\exp\left(\frac{x}{\tau}\right)$$ Svensson argues that this family is not flexible enough to reproduce more complex shapes with multiple humps and dips, as they are frequently encountered in the market, and adds another curvature term with a different time-scale $\tau_2 \neq \tau_1$, resulting in $$\beta_0 + \beta_1\exp\left(-\frac{x}{\tau_1}\right) + \frac{\beta_2}{\tau_1}x\exp\left(\frac{x}{\tau_1}\right)+ \frac{\beta_3}{\tau_2}x\exp\left(\frac{x}{\tau_2}\right)$$ A further interpolation family, due to Bliss, is obtained by setting $\beta_2 = 0$ in the Svensson parametrization. We are interested in the term structure shapes that can be represented by the Svensson family of curves and by its subfamilies (Nelson-Siegel, Bliss). The shape of the term structure is a fundamental economic indicator and it encodes important information on market preferences for short-term vs. long-term investments, on expectations of central bank decisions and on the general economic outlook. Decreasing (inverse) shapes of the term structure, for example, have been identified as signals of economic recessions. On the other hand, complex shapes with multiple local extrema have frequently been observed in both US and Euro area markets.

We provide a complete classification of all attainable shapes and partition the parameter space of each family according to these shapes. Building upon these results, we then examine the consistent dynamic evolution of the Svensson family under absence of arbitrage. Our analysis shows that consistent dynamics further restrict the set of attainable shapes, and we demonstrate that certain complex shapes can no longer appear after a deterministic time horizon. Moreover a single shape (either inverse or normal curves) must dominate in the long-run.

Equity-indexed annuities (EIAs) with investment guarantees are pension products sensitive to changes in the interest rate environment. A flexible and common choice for modelling this risk factor is a Hull-White model in its G2++ variant. We investigate the valuation of EIAs in this model setting and extend the literature by introducing a more efficient framework for Monte-Carlo simulation. In addition, we build on previous work by adapting an approach based on scenario matrices to a two-factor G2++ model. This method does not rely on simulations or on Fourier transformations. In numerical studies, we demonstrate its fast convergence and the limitations of techniques relying on the independence of annual returns and the central limit theorem.

Model calibration is a complicated yet fundamental task in financial engineering. By exploiting sequential Monte Carlo methods, we turn the non-convex optimization problem into a Bayesian estimation task based on the construction of a sequence of distributions from the prior to the posterior. This allows to compute any statistic of the estimated parameters, to overcome the strong dependence on the starting point, and to avoid troublesome local minima, all of which are typical plagues of the standard calibration. To highlight the strength of our approach, we consider the calibration of an affine stochastic volatility model with price-volatility co-jumps on both simulated and real implied volatility surfaces and find that our Bayesian approach largely outperforms the standard calibration approach in terms of run-time/accuracy, option pricing errors, and statistical fit. We further accelerate the computations by using Markov Chain Monte Carlo methods with delayed-acceptance and a neural network approach to option pricing that exploits the risk-neutral cumulants of the log returns as additional highly informative features.