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
the mortgage is defined by a
fixed actuarial coupon rate
and a principal
. If the mortgagor
chooses not to prepay, he refunds a continuous flow
related to
the maturity
and the coupon rate
through the initial parity
condition
 |
(9.1) |
where
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
the remaining principal
is contractually defined as the
forthcoming cash flows discounted at the initial actuarial coupon rate
Early prepayment at date
means paying
to the bank. In financial
terms, the mortgagor owns an American prepayment option with strike
.
The varying proportion between interest and capital in the flow
is
displayed in Figure 9.2.
Figure 9.2:
The proportion between interest and principal varying in time.
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9.2.2 Optimal Behavior and Price
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
is defined via an
Ornstein-Uhlenbeck process:
 |
(9.2) |
with
and
being the initial instantaneous forward term
curve. The parameters
and
control the volatility
of forward rates of maturity
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).
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
be the stopping time at which mortgagors
choose to prepay. Cash flows are of two kinds. If
mortgagors keep
on paying continuously
at any time, with discounted (random) value
equal to
At date
if
the remaining capital
must be paid,
implying a discounted cash flow equal to
The mortgagor will choose his prepayment time
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
where
is the relevant filtration. Since
is
Markovian,
can be expressed as a function of the current level of
the state variables and reduces to
The problem in
(9.3) is therefore a standard
Markovian optimal stopping problem (Pham, 2003).
At time
the mortgagor's decision whether to prepay or not is made on
the following arbitrage: the cost of prepaying immediately
is equal to the current value of the remaining mortgage
principal
This cost has to be compared to the expected cost
of going on refunding the continuous flow
and keeping the option to prepay until later
Obviously, the optimal
mortgagor should opt for prepayment if
 |
(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,

is solution of the partial
differential equation :
where

and

are fixed by Assumption A.
Proof:
We only give a sketch for constructing a solution.
The optimal stopping time problem at time
is given by
The Markovian property allows to change the conditioning by
by a conditioning by
. Thus,
is a function of
. If the mortgagor does not prepay during the time interval
, the discounted cashflows refunded in the
interval
equal to
The value at time
of the remaining cash flows to be paid by the
mortgagor is equal to
Its discounted value,
at time
is:
Finally, the expected value of the cash flows to be paid for a mortgage not
prepaid on the interval
equals to
Not prepaying on the time interval
may not be optimal
so that
Assuming regularity conditions on
, classical Taylor expansion yields
 |
(9.9) |
Furthermore, using the definition (9.7), the inequality
is satisfied. Assuming this inequality to be strictly satisfied, the
stopping time
is defined by
On the time interval
the non-prepayment strategy is optimal since
As a consequence:
Letting
and applying Itô's lemma, as previously yields
 |
(9.10) |
as long as
Formula (9.9) combined with (9.10) implies
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.
|
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.
|
In this one-dimensional framework, the prepayment condition
(9.4) defines a two-dimensional no prepayment
region
In particular, it includes the set
The optimal stopping theory provides characterization of
Pham (2003). In fact, there exists an optimal, time-dependent, stopping
frontier
such that
The price
and the optimal frontier
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
. Both its shape
and its level
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.
|
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
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.
|
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.