A New View of Mortgages
(and life)
Scene 1
• A farmer owns a horse farm outside
Lexington on Richmond Road.
• Demographic trends indicate that this part
of Lexington is booming and is projected to
continue to grow.
• Problem: Current local government is
hostile to development.
Scene 2
• Local developer notices the horse farm and
thinks that the site is an excellent candidate
for a new shopping mall.
• Developer knows that the local mayor is up
for re-election next year. Outcome of
election is uncertain, but has potential to
install new mayor with pro-growth views.
Scene 3
• What can the developer do to take
advantage of this opportunity?
• Approach farmer with an offer to buy an
option to purchase the horse farm.
The Option
• Developer pays the farmer $X for the right
to purchase the horse farm after the election
for $Y.
• If pro-growth mayor wins, then horse farm
will be worth $Z1 (where E[Z1] > Y).
– developer exercises the right to purchase the
land for $Y and either develops the shopping
mall or sells to another for $Z1 (profit = Z1-Y).
The Option
• If current mayor wins, horse farm will be
worth $Z2 where $Z2 < $Y.
– Can assume that Z2 is probably the value of the
land as a farm.
– Developer lets option expire without
purchasing land
– Farmer keeps the payment $X.
Next Example
• An insurance company has a large real
estate portfolio.
• The insurance company projects that it will
need $1 million next year to fund possible
claims.
• What can it do to protect itself from
changes in value to its real estate portfolio
between now and when the claims will have
to be paid.
Answer
• Purchase an option to sell one of its
properties for $1 million.
• If prices go down then protected
• If prices go up, will lose the appreciation
but still locked in with enough funds to pay
the claims.
Options
• In the first example, the developer
purchased a CALL option.
– the right to buy an asset
• In the second example, the insurance
company purchased a PUT option.
– the right to sell an asset.
Call Option
• Contract giving its owner the right to
purchase a fixed number of shares of a
specified common stock at a fixed price by
a certain date
•
•
•
•
•
•
Stock = underlying security (ST = market price)
price = strike price (K)
date = expiration date
writer = person who issues the call (the seller)
buyer = person who purchases the call
call price = market price of the call, (CT)
Types of Call Options
• European Call = exercise only at maturity
• American Call = exercise at any time up to
maturity
Call Option Payoff at Maturity
ST  K if ST  K
CT  
if ST  K
0
CT  max 0, ST  K 
Put Option
• Contract giving its owner the right to sell a
fixed number of shares of a specified stock
at a fixed price at any time by a certain date.
Put Option Payoff at Maturity
0
PT  
 K  ST
if ST  K
if ST  K
PT  max0, K  ST 
Mortgages as Options
• A mortgage is a promise to repay a debt
secured by property.
– property = collateral = underlying security =
stock
• However, a mortgage is much more
complex than a simple stock option.
– mortgage is a contract with several options
Mortgages as Options
• Default Option
– right of borrower to stop making payments in
exchange for the property
– default = exercise of a PUT option
Mortgages as Options
• Prepayment Option
– right of borrower to prepay the mortgage at any
time
– prepayment = exercise of a CALL option
Default
• Mortgage Default is defined as a failure to
fulfill a contract
– Technical default = breech of any provision of
the mortgage contract
• 1 day late on payment
• failure to pay property taxes
• failure to pay insurance premiums
Default
• Industry Standards:
– Delinquency: missed payment
– Default = 90 days delinquent
• (3 missed payments)
– Foreclosure: process of selling the property to
pay off the debt
• takes many months to foreclosure
Default
• Default is considered a European put
option.
– Borrowers will only default when a payment is
due
– Thus, the mortgage can be thought of as a string
of default options. Every time you make a
payment, you are purchasing a put option
giving you the right to sell the house to the
lender for the mortgage balance next month.
Mortgage Default
Simplistic Default Example
• Assume the following:
– a house has a current value of $100.
– The standard deviation of the return to housing
is 0.22314355
– The risk-free interest rate is 4% per annum.
– In order to purchase the house, we promise to
repay a lender $95 in 2 years.
• Note: This is a zero-coupon bond – no monthly or
yearly payments are made.
• Given the previous assumptions, we assume
that the house value will either rise to $125
or fall to $80 by the end of the first year
(with equal probability).
• By the end of the second year, the value of
the house will be $156.25, $100, or $64.
House Price Paths
Year 0
Year 1
Year 2
$156.25
$125.00
$100.00
$100.00
$80.00
$64.00
Binomial Model
• Cox, Ross, and Rubinstein (CRR) –
(discrete time version)
E[ H ]  pHu  1  pHd
ad
p
, u  e
ud
t
, d e
 t
, a  1  r 
t
Default Values
• At end of year 2, we owe $95 to lender.
– If house value = $156.25, then our equity is $61.25 and
we should repay the loan (not default).
• ($156.25 - $95 = $61.25)
– If house value = $64.00, then our equity is $-31.00 and
we should default (lender gets to keep house).
• ($64.00 - $95 = $-31)
– D = min[K,H]
Mortgage Value
• Starting with the terminal payoffs, we need
to calculate the present value of the
mortgage.
– Thus, we need to calculate the pseudoprobability of a change in house prices.
a  d 1.04  0.80
p

 0.533333
u  d 1.25  0.80
u  a 1.25  1.04
1 p 

 0.466666
u  d 1.25  0.80
Mortgage Value
• At the end of year 1, the present values of
the terminal pay-offs are calculated as:
Du   pDuu  1 p Dud  / r
Dd   pDdu  1 p Ddd / r
Mortgage Value
• Finally, at mortgage origination, the present
value of the loan is calculated as:
D   pDu  1 p Dd  / r
Mortgage Value
Year 0
Year 1
Year 2
$95.00
$91.35
$81.59
$95.00
$77.44
$64.00
Mortgage Value
• Note: Based on our assumptions of changes
in house prices, the lender will originate a
mortgage of $81.59 at Year 0.
– We borrower $81.59 and promise to repay $95
at the end of Year 2.
• What is our effective interest rate on this mortgage?
Mortgage Value
• Interest Rate Answer:
$81.59  $95 / 1  r 
2
• r = 7.9054%
• Note: since the risk-free rate is 4% this
implies that the default risk premium for
this mortgage is 3.9054%
rc  rrf  rd
Prepayment
• Paying off mortgage early (prior to maturity
date)
– Financial = when interest rates fall below
contract rate
– Non-financial = borrower moves, divorce, (not
optimal with respect to interest rates)
– prepayment is considered to be an American
option
• borrower may prepay at any time prior to maturity
Default and Prepayment
• Default and Prepayment are substitutes.
– If borrower prepays the mortgage, then he can’t
default
• implies that default has no value
– If borrower defaults on the mortgage, then she
can’t prepay
• implies that prepayment has no value
Mortgage Pricing
• Ten years ago Enterprise S&L made a 30
year mortgage for $100,000 at an annual
interest rate of 8%. The current market rate
for an equivalent loan is 12%. What is the
market value of this loan?
Mortgage Pricing
• Simplistic Answer: Price = $66,640
• More Complex (realistic)
– Price = PV of Payments - Value of Default
Option - Value of Prepayment Option