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Collateralized Loans
Many loan arrangements require that the borrower put up collateral to guarantee the loan
will be paid back. In the event of default, the lender takes possession of the collateral. A
nonrecourse loan gives the lender no recourse beyond the right to the collateral. That is,
the lender may not sue the borrower for further payment if the collateral turns out not to be
valuable enough to repay the loan.
This arrangement gives an implicit call option to the borrower. Assume the borrower is
obligated to pay back L dollars at the maturity of the loan. The collateral will be worth S
T
dollars at maturity. (Its value today is S
0
.) The borrower has the option to wait until loan
maturity and repay the loan only if the collateral is worth more than the L dollars necessary
to satisfy the loan. If the collateral is worth less than L, the borrower can default on the
loan, discharging the obligation by forfeiting the collateral, which is worth only S
T
.
4
Another way of describing such a loan is to view the borrower as turning over the col-
lateral to the lender but retaining the right to reclaim it by paying off the loan. The transfer
of the collateral with the right to reclaim it is equivalent to a payment of S
0
dollars, less a
simultaneous recovery of a sum that resembles a call option with exercise price L. In effect,
the borrower turns over collateral but keeps an option to “repurchase” it for L dollars at the
maturity of the loan if L turns out to be less than S
T
. This is a call option.
A third way to look at a collateralized loan is to assume that the borrower will repay the
L dollars with certainty but also retain the option to sell the collateral to the lender for L
dollars, even if S
T
is less than L. In this case, the sale of the collateral would generate the
cash necessary to satisfy the loan. The ability to “sell” the collateral for a price of L dollars
represents a put option, which guarantees the borrower can raise enough money to satisfy
the loan simply by turning over the collateral.
It is perhaps surprising to realize that we can describe the same loan as involving either
a put option or a call option, as the payoffs to calls and puts are so different. Yet the equiva-
lence of the two approaches is nothing more than a reflection of the put-call parity rela-
tionship. In our call-option description of the loan, the value of the borrower’s liability is
S
0
2 C: The borrower turns over the asset, which is a transfer of S
0
dollars, but retains
a call worth C dollars. In the put-option description, the borrower is obligated to pay L
dollars but retains the put, which is worth P: The present value of this net obligation is
L /(1 1 r
f
)
T
2 P. Because these alternative descriptions are equivalent ways of viewing the
same loan, the value of the obligations must be equal:
S
0
2 C 5
L
(1
1 r
f
)
T
2 P
(20.3)
Treating L as the exercise price of the option, Equation 20.3 is simply the put-call parity
relationship.
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In reality, of course, defaulting on a loan is not so simple. There are losses of reputation involved as well as
considerations of ethical behavior. This is a description of a pure nonrecourse loan where both parties agree from
the outset that only the collateral backs the loan and that default is not to be taken as a sign of bad faith if the col-
lateral is insufficient to repay the loan.
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P A R T V I
Options, Futures, and Other Derivatives
Figure 20.13 illustrates this fact. Figure 20.13, panel A is the value of the payment to
be received by the lender, which equals the minimum of S
T
or L. Panel B shows that this
amount can be expressed as S
T
minus the payoff of the call implicitly written by the lender
and held by the borrower. Panel C shows it also can be viewed as a receipt of L dollars
minus the proceeds of a put option.
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