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A New View of Mortgages (and life)

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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.

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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.

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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.

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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).

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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.

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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.

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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.

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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.

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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 (S T = 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, (C T )

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Types of Call Options European Call = exercise only at maturity American Call = exercise at any time up to maturity

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Call Option Payoff at Maturity

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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.

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Put Option Payoff at Maturity

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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

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Mortgages as Options Default Option –right of borrower to stop making payments in exchange for the property –default = exercise of a PUT option

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Mortgages as Options Prepayment Option –right of borrower to prepay the mortgage at any time –prepayment = exercise of a CALL option

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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

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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

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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.

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Mortgage Default

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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.

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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.

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House Price Paths Year 0Year 1Year 2 $156.25 $125.00 $100.00 $80.00 $64.00

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Binomial Model Cox, Ross, and Rubinstein (CRR) – (discrete time version)

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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]

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Mortgage Value Starting with the terminal payoffs, we need to calculate the present value of the mortgage. –Thus, we need to calculate the pseudo- probability of a change in house prices.

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Mortgage Value At the end of year 1, the present values of the terminal pay-offs are calculated as:

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Mortgage Value Finally, at mortgage origination, the present value of the loan is calculated as:

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Mortgage Value Year 0Year 1Year 2 $95.00 $91.35 $81.59$95.00 $77.44 $64.00

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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?

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Mortgage Value Interest Rate Answer: r = 7.9054%

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Note: since the risk-free rate is 4% this implies that the default risk premium for this mortgage is 3.9054%

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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

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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

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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?

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Mortgage Pricing Simplistic Answer: Price = $66,640 More Complex (realistic) –Price = PV of Payments - Value of Default Option - Value of Prepayment Option

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Properties of Stock Option Prices Chapter 9. Notation c : European call option price p :European put option price S 0 :Stock price today K :Strike price.

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