9.2 Optimally Prepaid Mortgage


9.2.1 Financial Characteristics and Cash Flow Analysis

For the sake of simplicity, all cash flows are assumed to be paid continuously in time. Given a maturity $ T,$ the mortgage is defined by a fixed actuarial coupon rate $ c$ and a principal $ N$. If the mortgagor chooses not to prepay, he refunds a continuous flow $ \phi dt,$ related to the maturity $ T$ and the coupon rate $ c$ through the initial parity condition

$\displaystyle N=\int_{0}^{T}\phi \exp \left( -cs\right) ds,$ (9.1)

where

$\displaystyle \phi =N\dfrac{c}{1-\exp \left( -cT\right) }.$    

As opposed to in fine bonds where intermediary cashflows are only made of interest and the principal is fully redeemed at maturity, this flow includes payments of both interest and principal. At time $ t\in \left[ 0,T
\index{interest}
\right] $ the remaining principal $ K_{t}$ is contractually defined as the forthcoming cash flows discounted at the initial actuarial coupon rate
$\displaystyle K_{t}$ $\displaystyle \stackrel{\mathrm{def}}{=}$ $\displaystyle \int_{t}^{T}\phi \exp \left\{ -c\left( s-t\right)
\right\} ds$  
  $\displaystyle =$ $\displaystyle \dfrac{\phi }{c}\left[ 1-\exp \left\{ -c\left( T-t\right) \right\} \right]$  
  $\displaystyle =$ $\displaystyle N\frac{1-\exp \left\{ -c\left( T-t\right) \right\} }{1-\exp \left(
-cT\right) }$  

Early prepayment at date $ t$ means paying $ K_{t}$ to the bank. In financial terms, the mortgagor owns an American prepayment option with strike $ K_{t}$. The varying proportion between interest and capital in the flow $ \phi$ is displayed in Figure 9.2.

Figure 9.2: The proportion between interest and principal varying in time.
\includegraphics[width=0.89\defpicwidth]{principinterest.ps}


9.2.2 Optimal Behavior and Price

9.2.2.1 The financial model

Given its callability feature, the mortgage is a fixed income derivative product. Its valuation must therefore be grounded on the definition of a particular interest rate model. Since many models can be seen as good candidates, we need to specify some additional features. First, this model should be arbitrage free and consistent with the observed forward term structure. This amounts to selecting a standard Heath-Jarrow-Morton (HJM) type approach. Second, we specify an additional Markovian structure for tractability purposes. While our theoretical analysis is valid for any Markovian HJM model, all (numerical) results will be presented, for simplicity, using a one factor enhanced Vasicek model (Priaulet, 2000); see Martellini and Priaulet (2000) for practical uses or Björk (1998) for more details on theoretical grounds. Let us quickly recap its characteristics.

Assumption A The short rate process $ r_{t}$ is defined via an Ornstein-Uhlenbeck process:

$\displaystyle dr_{t}=\lambda \left\{ \theta \left( t\right) -r_{t}\right\} dt+\sigma dW_{t},$ (9.2)

with

$\displaystyle \theta \left( t\right) =\dfrac{\partial }{\partial t}f\left( 0,t\...
...( 0,t\right) +\sigma ^{2}\dfrac{1-\exp \left( -2\lambda t\right) }{ 2\lambda },$    

and $ f\left( 0,t\right) $ being the initial instantaneous forward term curve. The parameters $ \sigma$ and $ \lambda $ control the volatility $ \sigma \left( \tau \right) $ of forward rates of maturity $ \tau $

$\displaystyle \sigma \left( \tau \right) =\frac{\sigma }{\lambda }\left\{ 1-\exp \left( -\lambda \tau \right) \right\} ,$    

and allow for a rough calibration to derivative prices. Note that in this enhanced Vasicek framework, all bond prices can be written in closed form, Martellini and Priaulet (2000).

9.2.2.2 The optimal stopping problem

The theory of optimal stopping is well known, Pham (2003). It is widely used in mathematical finance for the valuation of American contracts, Musiela and Rutkowski (1997). In the sequel, the optimally prepaid mortgage price is explicitly calculated as a solution of an optimal stopping problem.

Let $ \tau \in \left[ t,T\right] $ be the stopping time at which mortgagors choose to prepay. Cash flows are of two kinds. If $ \tau <T,$ mortgagors keep on paying continuously $ \phi dt$ at any time, with discounted (random) value equal to

$\displaystyle \int_{t}^{\min \left( \tau ,T\right) }\phi \exp \left( -\int_{t}^{s}r_{u}du\right) ds.$    

At date $ \tau ,$ if $ \tau <T,$ the remaining capital $ K_{t}$ must be paid, implying a discounted cash flow equal to

$\displaystyle I\left( \tau <T\right) \exp \left( -\int_{t}^{\tau }r_{u}du\right) K_{\tau },$    

The mortgagor will choose his prepayment time $ \tau $ in order to minimize the risk neutral expected value of these future discounted cashflows. The value of the optimally prepaid mortgage is then obtained as
$\displaystyle V_{t}$ $\displaystyle =$ $\displaystyle \inf\limits_{t<\tau <T} \mathrm{E} \Biggl[
\Biggl\{ \int_{t}^{\min \left( \tau ,T\right) }\phi \exp \left(
-\int_{t}^{s}r_{u}du\right) ds$ (9.3)
    $\displaystyle \qquad\qquad\quad \left. +I{\left( \tau <T\right) }\exp \left(
-\...
...t}^{\tau }r_{u}du\right) K_{\tau }\Biggr\} \right\vert \mathcal{F}_{t}
\Biggr],$  

where $ \mathcal{F}_{t}$ is the relevant filtration. Since $ r_{t}$ is Markovian, $ V_{t}$ can be expressed as a function of the current level of the state variables and reduces to $ V\left( t,r_{t}\right) .$ The problem in (9.3) is therefore a standard Markovian optimal stopping problem (Pham, 2003).

At time $ t,$ the mortgagor's decision whether to prepay or not is made on the following arbitrage: the cost of prepaying immediately $ \left( \tau
=t\right) $ is equal to the current value of the remaining mortgage principal $ K_{t}.$ This cost has to be compared to the expected cost $ V\left( t,r_{t}\right) $ of going on refunding the continuous flow $ \phi dt$ and keeping the option to prepay until later$ .$ Obviously, the optimal mortgagor should opt for prepayment if

$\displaystyle V\left( t,r_{t}\right) \geqslant K_{t}.$ (9.4)

Conversely, within the non-prepayment region, the mortgage can be sold or bought: its price must be the solution of the standard Black-Scholes partial differential equation. The following proposition sums up these intuitions. Its proof uses the link between conditional expectation and partial differential equations called the ``Feynman-Kac analysis.''

PROPOSITION 9.1   Under Assumption A, $ V\left( t,r_{t}\right) $ is solution of the partial differential equation :

$\displaystyle \max \left\{ \begin{array}{c} \dfrac{\partial V\left( t,r\right) ...
...( t,r\right) +\phi , \\ [3mm] V\left( t,r\right) -K_{t} \end{array} \right\} =0$ (9.5)
$\displaystyle V\left( T,r\right) =0$ (9.6)

where $ \mu \left( t,r\right) \stackrel{{\mathrm{def}}}{=}\lambda \left( \theta \left(
t\right) -r\right) $ and $ \sigma$ are fixed by Assumption A.

Proof: We only give a sketch for constructing a solution. The optimal stopping time problem at time $ t$ is given by

$\displaystyle V_{t}$ $\displaystyle =$ $\displaystyle \left\{ \inf\limits_{\tau <T}\mathrm{E}\int_{t}^{\min \left( \tau ,T\right)
}\phi \exp \left( -\int_{t}^{s}r_{u}du\right) ds\right.$ (9.7)
    $\displaystyle \left. \left. +I{\left( \tau <T\right) }\exp \left(
-\int_{t}^{\m...
...t( \tau ,T\right) }r_{u}du\right) K_{\tau }\right\}
\right\vert \mathcal{F}_{t}$ (9.8)

The Markovian property allows to change the conditioning by $ \mathcal{F}_{t}$ by a conditioning by $ r_{t}$. Thus, $ V_{t}$ is a function of $ \left(
t,r_{t}\right) $. If the mortgagor does not prepay during the time interval $ \left[ t,t+h\right] ,$ $ h>0$, the discounted cashflows refunded in the interval $ \left[ t,t+h\right] $ equal to

$\displaystyle \int\limits_{t}^{T}\exp \left( -\int_{t}^{s}r_{u}du\right) \phi ds$    

The value at time $ t+h$ of the remaining cash flows to be paid by the mortgagor is equal to $ V\left( t+h,r_{t+h}\right) .$ Its discounted value, at time $ t$ is:

$\displaystyle \exp \left( -\int_{t}^{t+h}r_{u}du\right) V\left( t+h,r_{t+h}\right) .$    

Finally, the expected value of the cash flows to be paid for a mortgage not prepaid on the interval $ \left[ t,t+h\right] $ equals to
    $\displaystyle \mathrm{E}\left\{ \int\limits_{t}^{T}\exp \left( -\int_{t}^{s}r_{...
...\exp \left( -\int_{t}^{t+h}r_{u}du\right) V\left(
t+h,r_{t+h}\right) \right\} .$  

Not prepaying on the time interval $ \left[ t,t+h\right] $ may not be optimal so that

$\displaystyle %%\begin{eqnarray*}
V\!\left( t,r_{t}\right) \leq \mathrm{E}\!\le...
...eft( -\int_{t}^{t+h}r_{u}du\right) V\!\left(
t+h,r_{t+h}\right)\! \right\}\! .
$

Assuming regularity conditions on $ V$, classical Taylor expansion yields

$\displaystyle 0\leq \dfrac{\partial V\left( t,r_{t}\right) }{\partial t}+\mu \l...
...l ^{2}V\left( t,r_{t}\right) }{\partial r^{2}} -rV\left( t,r_{t}\right) +\phi .$ (9.9)

Furthermore, using the definition (9.7), the inequality

$\displaystyle V\left( t,r_{t}\right) \leq K_{t}$    

is satisfied. Assuming this inequality to be strictly satisfied, the stopping time $ \tau $ is defined by

$\displaystyle \tau =\inf \left\{ s\geq t:V\left( s,r_{s}\right) =K_{s}\right\} .$    

On the time interval $ \left[ t,\min \{ t+h ,\tau \}
\right] ,$ the non-prepayment strategy is optimal since $ V\left(
s,r_{s}\right) <K_{s}.$ As a consequence:

$\displaystyle %%\begin{eqnarray*}
V\!\left( t,r_{t}\right) =\mathrm{E}\!\left\{...
...!\left( -\int_{t}^{t+h}r_{u}du\right) V\!\left(
t+h,r_{t+h}\right) \right\}\!.
$

Letting $ h\rightarrow 0$ and applying Itô's lemma, as previously yields

$\displaystyle 0=\dfrac{\partial V\left( t,r_{t}\right) }{\partial t}+\mu \left(...
...tial ^{2}V\left( t,r_{t}\right) }{\partial r^{2}}-rV\left( t,r_{t}\right) +\phi$ (9.10)

as long as $ V\left( t,r_{t}\right) <K_{t}.$

Formula (9.9) combined with (9.10) implies

$\displaystyle \max \left\{ \dfrac{\partial V_{t}}{\partial t}+\mu \left( t,r\ri...
...dfrac{\partial ^{2}V_{t} }{\partial r^{2}}-rV_{t}+\phi ,V_{t}-K_{t}\right\} =0.$    

$ \Box$

Figure 9.3: The sensitivity of the optimal prepayment-frontier to forward-rates slope: steeper forward-rate curve leads to the dotted frontier, less steep forward-rate curve to solid frontier.
\includegraphics[width=0.91\defpicwidth]{frontieroptsensislope2.ps}

Figure 9.4: The sensitivity of the optimal prepayment frontier to interest-rates volatility: volatilities of the 1-year and 10-year bonds are 90 bps and 37 bps (solid line) and 135 bps and 55 bps (dotted line), respectively.
\includegraphics[width=0.91\defpicwidth]{frontiersensivol.ps}

9.2.2.3 Discussion and visualization

In this one-dimensional framework, the prepayment condition (9.4) defines a two-dimensional no prepayment region

$\displaystyle D=\left\{ \left( t,r\right) :V_{t}<K_{t}\right\} .$    

In particular, it includes the set

$\displaystyle \left\{ \left( t,r\right) :r_{t}\geq c\right\}.$    

The optimal stopping theory provides characterization of $ D,$ Pham (2003). In fact, there exists an optimal, time-dependent, stopping frontier $ r_{t}^{opt}$ such that

$\displaystyle D=\left\{ \left( t,r\right) :r_{t}>r_{t}^{opt}\right\}.$    

The price $ V_{t}$ and the optimal frontier $ r_{t}^{opt}$ are jointly determined: this is a so-called free boundary problem. It can only be calculated via a standard finite difference approach, Wilmott (2000). An example is displayed in Figure 9.3. Interestingly enough, the optimal frontier heavily depends on the time to maturity and it may be far away from the mortgage coupon $ c$. Both its shape and its level $ r_{t}^{opt}$ strongly depend on market conditions.

Figure 9.5: The sensitivity of the time value of the embbeded option to interest-rate volatility: volatilities of the 1-year and 10-year bonds are 90 bps and 37 bps (solid line) and 135 bps and 55 bps (dotted line), respectively.
\includegraphics[width=0.91\defpicwidth]{timevalueopti.ps}

Figure 9.3 illustrates the positive impact of the slope of the curve on to the slope of the optimal frontier. The influence of implicit market volatility on the optimal prepayment frontier is displayed in Figure 9.4. As expected, the more randomness $ \sigma$ around future rates moves, the stronger the incentive for mortgagors to delay their prepayment in time. In the language of derivatives, the time value of the embedded option increases, see Figure 9.5. All these effects are summed up in one key indicator: the duration of the optimally prepaid mortgage. Defined as the sensitivity to the variation of interest rates, this indicator has two interesting interpretations. From an actuarial point of view, it represents the average expected maturity of the future discounted cash flows. From a hedging point of view, duration may be interpreted as the ``delta'' of the mortgage with respect to interest rates.

Figure 9.6: The sensitivity of the duration to interest-rate volatility: volatilities of the 1-year and 10-year bonds are 90 bps and 37 bps (solid line) and 135 bps and 55 bps (dotted line), respectively.
\includegraphics[width=0.91\defpicwidth]{durationMBSsensivol.ps}

If interest rate is deep inside the continuing region, the expected time before prepayment is large and the duration increases. As displayed in Figure 9.6, the higher the volatility, the higher the duration. The preceeding discussion indicates that the optimally prepaid mortgage can be understood as a standard interest rate derivative, allowing one to get asymmetric exposure to future interest rates shifts.