Model Selection

Which framework should we choose for simulating mortgage interest rate paths for the purpose of this research? Before answering this question, we must clarify which qualities we seek in an interest rate model.

• Since modeling mortgage rate is only a small part of this thesis, the resources to do so are fairly limited. Therefore, we seek an interest model that is generally simple to understand and implement.

• Second, we are looking for a framework that enables us to interpret and control its parameters intuitively, because we will simulate several interest rate scenarios.

• Third, we would like to achieve a certain degree of accuracy. However, the objective for high accuracy is not our primary concern: As mentioned previously, mortgage rate depends on a multitude of exogenous factors and the assessment of many such factors are out of our capabilities. For example, we will make very limited use of historical data to estimate interest rate parameters. The ability to control scenarios is in our case more important than whether or not these scenarios accurately match future reality.

Table 3.1 shows a brief comparison between the four models that were intro- duced in the previous paragraphs. Although the random walk is by far the easiest framework to use, it is not suitable for our purposes, since it is not possible to con- trol the long-term trend at all.

3.3. INTEREST RATE MODEL 23 Table 3.1: Comparison of interest rate models. Here,RW is the ran- dom walk andHW is Hull-White.

RW CIR Vasicek HW Easy to use

Control No arbitrage Mean-reverting Negative rate possible

Variability adjusts to interest rate level Aims to match real market rate

Since Hull-White has the potential to match actual market prices well and is furthermore unbiased in terms of historical mortgage rate, it first seems like a good choice for our purposes. However, fitting the Hull-White model to the current mortgage yield curve might result in over-fitting parameters. Furthermore, the pa- rameters are dynamically intertwined in a way that is not easy to understand. For instance, a scenario with a certain assumed long-term mean - say three percent - is not easily implemented, because the long-term mean under Hull-White depends on the mean reversion rate and its development over time is not easy to control. This does not fit into our objective of conducting a scenario analysis. Furthermore, unlike the other models, Hull-White cannot be represented by a simple normal distribution. In conclusion, we decide against the high complexity and accuracy ambitions of Hull and White.

By now we have narrowed down the range of choices for appropriate interest rate models to CIR and Vasicek. Both are equilibrium models whose parameters are simple to understand and control. Furthermore, both have one critical charac- teristic that the other one lacks, respectively: Under CIR, volatility dynamically adjusts to the current interest rate level. However, negative interest rates are not possible under CIR, as the volatility simply approaches zero as interest rate ap- proaches zero. Volatility under Vasicek on the other hand is constant and negative interest rates are possible. To choose between the two models, we must examine their two key differences more closely:

Negative interest rates: Until recent years it was common to assume that interest rates can generally not be negative, but today we know that rates below zero are in fact possible.2 However, can mortgage rates become negative as well? To answer this question, we recall our knowledge about the two main com- ponents of the mortgage rate: the risk-free component and a spread. As we

2

Depositing at the European Central Bank (ECB) currently yields−40basis points and the rates are expected to remain on this level until at least mid 2019:https://www.ecb.europa.eu/ press/pr/date/2018/html/ecb.mp180726.en.htmlAccessed: 26-09-2018

24 CHAPTER 3. THEORETICAL MODEL

have just established, the risk-free rate can be negative, but we do not know where the floor value is. We further assume that a negative spread on top of the risk-free rate is not possible: In case of a spread below zero, it is always be preferable to invest in a risk-free asset instead of a mortgage. Thus, we draw two conclusions: First, the risk-free rate can be negative. Second, the mortgage rate is always above the risk-free rate. As we do not know how low the risk-free rate can sink and we cannot assess the minimum spread size other than making the assumption that it must have a positive value, we cannot find any evidence that negative mortgage rates are impossible. Dynamic volatility: It is a common assumption that low interest rates are less

volatile than high rates and the CIR model accounts for this circumstance, while Vasicek has no such mechanism.

The description of the two aspects shows that both CIR and Vasicek model have characteristics that we require to fit mortgage rate development to our view of reality. Therefore, we propose to use two interest rate domains which are modeled in a different manner, respectively. If interest rates are low, they follow a Vasicek process with sufficiently low volatility, while high rates are modeled using the CIR approach. As a consequence, volatility dynamically adjusts to the current level of mortgage rate and at the same time negative rates are not excluded. Formally:

dr=κ(θ−r)dt+σpmax(r, ζ)dz (3.7) Here,ζ is the threshold below which the interest rate is described in terms of the Vasicek model and above which it is modeled using CIR. Consequently,ζbecomes an additional parameter in the framework of the interest rate model. The values of the four parameters κ, θ, σ, and ζ will be determined in Chapter 5, where we present the methodology for our simulation study. Furthermore, Appendix A.1 provides an in-depth technical analysis of the model induced by Equation 3.7.