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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 ) and (assumed by Cox, Ingersoll and Ross ).
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 anti-correlated 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 no-arbitrage arguments, earns exactly the risk-free 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 so-called 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.      (3.5#eq.7)

which can be used with when and keeping finite for small .
Using (3.5#eq.6) to re-write 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 risk-free 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