Fixed Income Securities: Valuation, Risk, and Risk Management

CHAPTER 16

DYNAMIC HEDGING AND RELATIVE VALUE TRADES

The no-arbitrage pricing methodology we discussed in Chapter 15, Section 15.2.1, is also at the basis of dynamic hedging strategies and relative value trades. In the next section we examine the concept of dynamic replication in continuous time. Recall that we looked at the concept of dynamic replication in no arbitrage models in Chapter 10, Section 10.2.2, in the context of binomial trees. The discussion in the next section extends those concepts to the more general continuous time framework.

16.1 THE REPLICATING PORTFOLIO

Consider again the portfolio II(r, t) in Equation 15.11 in Chapter 15, given by a long position in bond Z₁ (r, t) and short Δ units of bond Z₂(r, t):

Recall that according to the hedging rule in Equation 15.14, we choose Δ equal to the ratio of the sensitivities of the two bonds to the interest rate:

The process for II_t, according to Ito’s lemma, is

Let us now rewrite this equation as

Notice an interesting fact. If we know (a) the value of “Δ”; (b) the change in the price of bond 2 between t and t + dt, ”dZ_{2, t ...}