Advances in Risk Management

70 MANAGING INTEREST RATE RISK UNDER NON-PARALLEL CHANGES

These measures do not depart from a continuous-time model for interest
rates but they are arbitrarily specified. Several exceptions are Ingersoll, Skel-
ton and Weil (1978), Cox, Ingersoll and Ross (1979), Chen (1996), Munk
(1999), Jeffrey (2000) and Wu (2000).
Our main goal is to define and apply duration measures based on the
continuous-time model presented in Moreno (2003). Similarly to Chen
(1996), we will obtain several measures of generalized duration to reflect
the changes in the stochastic factors of this model. We can analyse the sensi-
tivity of a bond portfolio to changes in the yield curve and we can compute
hedging ratios to immunize a bond portfolio. Last but not least, we can solve
the limitations of the conventional duration.^4

4.2 THE MODEL

We briefly present the two-factor model for interest rates that has been
proposed and analysed in Moreno (2003).
This model assumes that the price, at a certain time, of a default-free
discount bond depends only on the time to maturity and two state variables:
the long-term interest rate, denoted byL, and the spread (the difference
between the short-term (instantaneous) riskless interest rate,r, and the long-
term interest rate), denoted bys. This selection of state variables allows us
to use the assumption that both variables are orthogonal.^5
After choosing the state variables, we assume that their dynamics over
time are given by the following system of stochastic differential equations:

ds=β 1 (s,L)dt+σ 1 (s,L)dw 1

(4.1)

dL=β 2 (s,L)dt+σ 2 (s,L)dw 2

wheretdenotes calendar time, andw 1 andw 2 are Wiener processes with
E[dw 1 ]=E[dw 1 ]=0,dw^21 =dw^21 =dtand (by the orthogonality assumption)
E[dw 1 dw 2 ]=0.β 1 andβ 2 are the expected instantaneous rates of change in
the state variables andσ 12 andσ^22 are the instantaneous variances of changes
in these two variables.
LetP(s,L,t,T)≡P(s,L,τ) be the price, at timet, of a default-free discount
bond that pays $1 at maturityT=t+τ. Applying Itô’s lemma, setting up a
hedge portfolio, and assuming no-arbitrage conditions, we obtain the partial
differential equation (PDE) that the price of this bond for all maturities must
satisfy:

1

2

[σ^21 (.)Pss+σ^22 (.)PLL]+[β 1 (.)−λ 1 (.)σ 1 (.)]Ps

+[β 2 (.)−λ 2 (.)σ 2 (.)]PL+Pt−rP= 0 (4.2)

where subscripts denote partial derivatives.