Interest Rate Modelling (Finance and Capital Markets Series)

1.3 Risk-neutral valuation

The mean rate of return on a bond can be written as a function of its variance, the risk-free interest rate and market price of risk. From (1.12) we have:

Hence the bond price dynamics (1.9), may be written in terms of the market price of risk as ^[1] :

Let d = ˆ’ qdt + dz and the above equation becomes:

where is the Wiener process in the risk-neutral world being governed by the probability measure. Since dz = qdt + , the equation describing the dynamics of the short- term rate, (1.4) may be written in terms of as:

Using the dynamics of the bond price in (1.15) and the PDE of the bond price (1.13) with boundary condition P ( T , T ) = 1, the Feynman-Kac theorem ^[2] may be applied to yield the valuation:

Here we take the expectation with respect to which corresponds to the risk-neutral world. corresponds to the equivalent probability measure which utilises risk-neutral probabilities. (As opposed to the utility dependent probability measure, Q , which represents investor specific probabilities.) By introducing the market price of risk q we are able to transform the probability measure from a utility-dependent to a risk-neutral one. The Girsanov Theorem ^[3] defines this transformation. First consider the Wiener process:

Let t ‰ t * ‰ T and define

as the Radon-Nikodym derivative used to define the new probability measure, that is:

Also, expectation with respect to is calculated as:

for any random variable Y . Hence the expected bond price (1.17) may be expressed in terms of the utility-dependent measure as:

^[1] To lighten the notation, the functional dependence of r , q , v , s and ƒ on r , t and T is suppressed.

^[2] The discounted Feynman-Kac theorem is applicable in this case. This theorem defines the relationship between a stochastic differential equation (SDE) and the corresponding partial differential equation (PDE). Considering the SDE:

Let 0 ‰ t ‰ T where T > 0 is fixed, and let h ( y ) be some function. Define:

Then the corresponding PDE is:

See [ 45 ] for more details.

^[3] For more details about the application of Girsanov's Theorem and the Radon-Nikodym derivative in the change of measure see [ 45 ] and [ 41 ].