RDP 2008-09: A Term Structure Decomposition of the Australian Yield Curve Appendix B: Risk-neutral Bond Pricing

Here we examine why bonds should be priced under the risk-neutral measure. To simplify the analysis we work with a single factor model, that is

where all variables are scalars.

Consider the probability space Inline Equation with associated filtration Inline Equation taken as the augmented filtration of Inline Equation (see, for example, Steele 2001). Xt is an Ito process if

for μx and σx adapted to Inline Equation. Ito's lemma then states that for any function F(x,t) such that F is twice differentiable in x and differentiable in t,

Applying Ito's lemma to Equation (B1) we trivially get

Now let PA(rt,t) and PB(rt,t) denote the time t price of two zero-coupon bonds with different maturity dates. Then by Ito's lemma, Pi (i = A, B) will satisfy

Consider a portfolio that is long one A bond and short h B bonds. At time t this portfolio has value

If held for dt, the portfolio's value changes by

Hence we can make the portfolio instantaneously riskless by choosing h = σA/σB. In this case, the portfolio must earn the risk-free rate rt and so

Substituting Equations (B6) and (B7) into Equation (B8) and setting h = σA/σB leads to



Hence the ratio (μrtP)/σ is independent of the choice of bond, and so there must exist a function λr such that

holds for any bond price P.

Now substituting μ and σ as identified by Equations (B4) and (B5) into Equation (B9) results in a Black-Scholes type partial differential equation

which is solved subject to appropriate boundary conditions (bonds pay 1 unit at maturity) and the radiation condition P → 0 as r → ∞.

The Feynman-Kac formula then says that the solution to Equation (B10) is given

where rt satisfies

and Inline Equation is standard Brownian motion in the risk-neutral measure associated with Inline Equation