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3.5 Hedging a bond with another bond (Vasicek)
[ SLIDE
Ito 
no random 
no arbitrage 
Vasicek 
price of risk 
VIDEO
modem 
LAN 
DSL]
In contrast to the price of a share (which can never drop below zero
because of the regulations of bankruptcies), it is in principle
possible to live with negative interest rates: these have even been
observed in Switzerland in the 1960s, albeit only for a short period.
This motivates the development of stochastic models for long term
interest rates where the random walk of the spot rate increments
is based on (3.3.1#eq.1) and the distribution chosen somewhere between
(normal walk assumed by Vasicek [23]) and
(assumed by Cox, Ingersoll and Ross [5]).
Here we follow the classical derivation from Vasicek using
and apply Itô's lemma to calculate the stochastic price increment
for a bond
of maturity

(3.5#eq.1) 
The drift and volatility of a bond
depend on the spot rate
and can be parametrized using market data
.
Having no anticorrelated underlying as for the case of stock options, the
trick here is to create here a portfolio that is long one bond
and short a number
of bonds
with a different
maturity
. The portfolio value and its incremental change per
time step become

(3.5#eq.2) 

(3.5#eq.3) 
Choosing
to eliminate the random
component, the portfolio becomes deterministic and, using noarbitrage
arguments, earns exactly the riskfree spot rate

(3.5#eq.4) 
Substitute the value for
, insert (3.5#eq.1) for the
increments and move all the terms with the same maturity on the
same side of the equation to obtain

(3.5#eq.5) 
This shows that the socalled
market price of risk
is independent of the maturity and can therefore
be used to parameterize the market.
Rewriting the bond drift
in (3.5#eq.1) in terms of the
market price of risk (3.5#eq.5), the properly hedged portfolio
(3.5#eq.2) finally yields the bond pricing equation

(3.5#eq.6) 
which can be solved using the normalized terminal payoff at maturity
as the terminal condition.
Boundary conditions depend on the model for the spot rate, e.g.
which can be used with
when
and keeping
finite for small
.
Using (3.5#eq.6) to rewrite the deterministic component of a
single bond (3.5#eq.1), Itô's lemma can be put into another form

(3.5#eq.8) 
or

(3.5#eq.9) 
showing on the left hand side that a higher return can be earned exceeding
the riskfree interest rates, provided that the investor accepts a certain
level of risk
. Indeed, the portfolio grows by an extra
per unit of risk
.
This justifies the interpretation of
as the market price of risk,
with investors that are either risk seeking or risk averse depending whether
is positive or negative.
SYLLABUS Previous: 3.4 Hedging an option
Up: 3 FORECASTING WITH UNCERTAINTY
Next: 3.6 Computer quiz