Short-Rate Models

The temporal development of the mortgage short rate depends on a larger number of exogenous variables. For example, different indicators for the economic status of Europe such as unemployment-rate, economic output, population growth or in- flation rate can provide an incentive to the European Central Bank to make changes to the risk-free rate. For background information on the drivers of interest rate in

1_{Source: https://www.hypotheekrente.nl/rente/30-jaar-rentevast/100/#overzicht , accessed: 10-}

20 CHAPTER 3. THEORETICAL MODEL

the European economic area, the reader is referred to Bonam et al. (2018).

Even attempting to develop a statistical framework that models all those vari- ables for the purpose of determining mortgage rate is far out of scope of this thesis. Instead, we assume that interest rate moves up and down in a random manner and may be described in terms of a stochastic differential equation (SDE). We define the one-factor process that describes interest rate behavior with the SDE

dr=µ(r, t)dt+δ(r, t)dz (3.1) which is known as Markov diffusion or Itˆo process. Here, r is the interest rate which depends on time,µ(r, t)is the drift, andδ(r, t)is the standard deviation of the rate. Finally,dzdenotes the Wiener process. The functionsµandδmay, but do not have to, depend on interest rate and time. In fact, equilibrium models, which make up a large portion of popular interest models for option pricing, assume that these variables are independent of time. In the following sections, we discuss some specific choices forµ(r, t)andδ(r, t)for the sake of modeling the interest rate. Random Walk

The simplest possible process constitutes a plain random walk. The process is described by the SDE

dr=σdz

whereσ is the volatility of the interest rate. However, it is not common to use a simple random walk for option pricing. Researchers usually assume that the real interest rate behavior is more accurately described by a mean-reverting process. The frameworks in the following subsections are all frequently used for the mean- reversion modeling of interest rates.

Cox-Ingersoll-Ross Model

The Cox-Ingersoll-Ross (CIR) model (Cox et al., 1985) assumes that interest rate change behaves according to a square root mean-reversion process. That is, the interest rate follows

dr=κ(θ−r)dt+σ√rdz (3.2) Here,κis the mean-reversion rate, which is the speed at whichrreverses towards the long-term equilibrium meanθ and is a measure of sensitivity of the interest rate with respect to time. While a random walk approaches (plus or minus) infinity when running for a long period of time, mean reversion processes - as their name suggests - revert around their equilibriumθ.

3.3. INTEREST RATE MODEL 21 The special characteristic of the CIR model is that the rate’s volatility depends on the square root of the interest rate. As consequence, the standard deviation of the stochastic process becomes smaller as the interest rate decreases. Furthermore, interest rate cannot be negative under this model, because the square root of the rate would not yield a real solution. To ensure thatrdoes not jump to a negative value when using a discrete version of this model, it is common to set2κθ ≥σ2. The larger the difference between 2κθ andσ2, the lower the probability that interest rates comes close to zero.

Vasicek Model

Another popular mean-reversion process for modeling interest rates was first pro- posed by Vasicek (1997). It is similar to the CIR model, but the square root factor of the random process is omitted. Therefore, volatility does not decrease when the interest rate becomes smaller, which makes negative interest rates possible. The following SDE describes the corresponding process:

dr=κ(θ−r)dt+σdz (3.3) Hull-White Model

To understand the Hull-White model, we must first take a look at the term structure of interest ratesR(t, T), which is also known as yield curve. The term structure describes the relationship between returns of the same underlying security for dif- ferent times to maturity. In the case of risk-neutral mortgage valuation, it is defined as the rate of return on a mortgage with maturityT at timet. That means that the risk-neutral price of the mortgage is given bye−R(t,T)(T−t)and therefore the term structure is:

R(t, T) =− 1

T−tln E[e

−˜r(T−t)_{] (3.4)}

Here,r˜is the average interest rate during the term T −t andE[e−˜r(T−t)]is the expected value of the expected return at timet.

The CIR and Vasicek SDEs are equilibrium frameworks, which means that they attempt to model market supply and demand of the underlying security. Therefore, equilibrium models do not necessarily fit the actual market prices and are hence not arbitrage-free. In the CIR and Vasicek SDEs, both drift and volatility depend only on the interest rate and not on time, which means that the term structureR(t, T)

merely is an output of the model. For the Hull-White model on the other hand, the long-term mean does depend on time which makes today’s term structure an input for calculating the interest rate. In other words, Hull and White (1990) suggest a no-arbitrage extension of both the CIR and the Vasicek model. However the Vasicek extension is more commonly used in literature on mortgage valuation, and

22 CHAPTER 3. THEORETICAL MODEL

it is defined as follows:

dr=κ(θ(t)

κ −r)dt+σdz (3.5)

The main difference to Vasicek’s original model is the definition of the long-term mean, which now depends on time and is defined as

θ(t) = ∂F

∂t (0, T) +κF(0, T) + σ2

2κ(1−e

−2κt_{) (3.6)} where F(0, T) is the instantaneous forward rate. In Hull and White’s original Vasicek extension, the mean-reversion rate also depends on time, but this version of the framework is barely used in practice.

Other Models

There are many other processes such as log-normal, multi-factor or multi-regime frameworks that potentially describe interest rate movements, but we omit review- ing all of them.