Math 623, W 2007: Homework 6.
For full credit, your solutions must be clearly presented and all code included.
In this homework you will calibrate two fixed income (short rate) models to real (but
not current) data. The following table gives the yields of zero coupon bonds, computed
using
annual
compounding, for various maturities. The table also gives the market implied
(flat) volatilies of atthemoney caps.
Maturity
Yield
Volatility
0
0.0480
—
1
0.0454
0.1520
2
0.0456
0.1620
3
0.0462
0.1640
4
0.0471
0.1630
5
0.0481
0.1605
7
0.0502
0.1555
10
0.0526
0.1475
15
0.0556
0.1350
20
0.0575
0.1260
Each cap can be described as follows. Consider dates 0
< T
0
< T
1
<
· · ·
< T
n
=
T
with
T
i
= 3(
i
+1) months. The cap pays (
T
i

T
i

1
)(
L
(
T
i

1
, T
i
)

R
)
+
at time
T
i
for
i
= 1
, . . . , n
,
where
R
=
R
(
T
) is the swap rate and
L
(
T
i

1
, T
i
) the LIBOR rate. The terminal time
T
is
the maturity listed in the table. The implied cap volatilities come from Black’s formula.
(1)
(a) Compute the price
p
(0
, T
) today of the zero coupon bond maturing at
T
for
0
≤
T
≤
20. Use linear interpolation
on the yields
to do this. Display your
result in a plot of
p
(0
, T
) vs
T
.
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 Fall '08
 CONLON
 Math, Zerocoupon bond, TI, cap volatilities

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