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5.2.1 Vanilla swaps
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Remember from sect.2.2.3 that a swap is a contract derived from
a loan, where the payments from a fixed interest rate
are exchanged
for the payments from a floating interest rate.
In a plain vanilla swap, the floating rate is evaluated at the end of
every accrual period
when a payment is made in
compensation for the difference in rates.
To avoid difficulties with a floating rate that is only known at the
end of the accrual period, the calculation proceeds backwards in time
and evaluates the price (of what is sometimes called FRA) for each
period separately using the same procedure as for bonds.
The equilibrium swap rate
is chosen so as to make the contract
worthless at the onset
and the mismatch between the spot
rate
and the swap rate
is accumulated over the accrual period
to calculate the price of a swap having a finite lifetime
.
For a unit Notional principal, the incremental change in swap value
is the difference of interest rates multiplied by the time interval
.
As for any asset with an investment value, the accumulated earnings or
losses from the swap can themselves be viewed as bonds with a positive
or negative value and can therefore be described using the Vasicek
model from the previous section.
In fact, a swap can be understood as a bond that starts with zero as
initial value and pays a continuously compounded annual coupon
. Only one parameter is required in addition to those that
have been defined in sect.5.1.2:
 the fixed swap rate (
or
StrikePrice ) is
expressed as the relative annual return in the fixed leg,
e.g. 0.04 for a predetermined swap rate of 4%.
The VMARKET applet below shows how the
value of a swap with a fixed rate of 8% evolves as a function of the
spot rate for an increasing time to the maturity.
VMARKET applet: press Start/Stop
to simulate the price of a swap with a fixed rate of 8%
(StrikePrice=0.08) as a function of the spot rate V(r)
for up to 10 years to the maturity.

Immediately after the start of the contract, the swap acquires a
negative value if the spot rate is below the swap rate because the
holder of the swap has the obligation to pay the fixed rate: this
is indeed more than the market is asking for and the swap holder
is therefore loosing money to the writer of the swap.
On the contrary, the swap acquires a positive value if the spot rate
is above the swap rate: the swap holder has the right to pay only
a fixed rate and is therefore earning money at the expense of the
writer.
Think of a swap as a coupon paying bond: the downward curvature of the
price (
) is the results of the exponential growth
at the spot rate, which is expected for any risk free investment when
the time runs forward. The opposite happens when the time is reversed
and the exponential decrease of the swap price with the spot rate
results in a downward curvature in the same manner as previously seen
for the discount function.
The same models that have been used for bonds forecast the drift in the
interest rate, but the volatility should here be modified to reflect the
uncertainty of payments in the floating leg (exercise 5.07).
The volatility reduces the overall curvature and therefore also reduces
the value of the swap: this can be understood financially from the spot
rate fluctuations above and below the swap rate, which tend to cancel
out in time and reduce the value of the swap.
SYLLABUS Previous: 5.2 Credit derivatives
Up: 5.2 Credit derivatives
Next: 5.2.2 Cap/floorlets
