Lecture 19
Interest Rate Models
19.1
Basic elements in a bond market
Consider a market money account
B
=
{
B
t
}
with
B
0
= 1,
B
t
= exp(
R
t
0
r
u
du
),
t
∈
[0
, T
], where
the short rate
{
r
t
}
is assumed to be a stochastic process. The bank account
B
and related term
structures (
{
B
(
t, τ
)
}
, or
{
Y
(
t, τ
)
}
, or
{
f
(
t, τ
)
}
) in a bond market serve as underlying assets similar
to stocks in a stock market. We will introduce several interest rate models and use them to price
various derivatives defined on the underlying assets. Note that each fixedincome asset or derivative
may have its own maturity time.
Let (Ω
,
F
, P
) be a probability space equipped with a filtration
{F
t
}
t
∈
[0
,T
]
. Assume
{
r
t
}
is an
adapted process with
P
‡
R
T
0

r
t

dt <
∞
·
= 1. The filtration
{F
t
}
t
∈
[0
,T
]
is usually generated by a
multidimensional Brownian motion (called shocks or factors) as a source of uncertainty. Since our
main purpose is derivative pricing, we assume the existence of a risk neutral measure
Q
, equivalent
to
P
following the Girsanov transformation (detail skipped), under which the discounted processes
of all underlying assets are martingales.
This will rule out arbitrage opportunities, according to
the (continuoustime) Fundamental Theorem of Asset Pricing, which we have not presented. For
0
≤
t < τ
≤
T
, three equivalent term structures can be defined as follows:
•
Zerocoupon bond
:
Let
B
(
t, τ
) be the time
t
price of a bond with maturity
τ
(called
τ
bond)
and the par value
B
(
τ, τ
) = 1. The riskneutral valuation principle implies
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 Spring '08
 Staff
 Mathematical finance, Short rate model, Rate Models

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