APPENDIX C Introduction to the Mathematics of Fixed-Income Pricing1

Appendix B introduced the techniques for handling the dynamics and calculus of stochastic variables such as interest rates. In this section we introduce the fundamentals of mathematical finance with respect to fixed-income pricing. An extended discussion of the content of this section can be found in Choudhry (2004).

To begin we need to state the following sets of assumptions, generally adopted from Merton’s2 pricing method:

  • There are no transaction cost or taxes.
  • There exists an exchange market for borrowing and lending at the same rate of interest (no bid-offer spread).
  • The term structure is “flat” and known with certainty.
  • There is a rational and competitive market.
  • Market participants prefer to increase wealth.
  • There are no arbitrage opportunities.

The main prerequisite of mathematical finance that is imperative in understanding fixed income are risk-neutral valuation and arbitrage pricing theory. In this introduction we will establish the probabilistic setting in which these concepts are formulated.

As stated in Musiela and Rutkowski (1997), an economy is a family of filtered space {(Ω,I,μ):μ∈P}3, where the filtration satisfies the usual conditions,4 and P is a collection of mutually equivalent probability measures on the measurable space.5 We model the subjective market uncertainty of each investor by associating to each investor a probability measure from P. Investors with more risky tolerance will ...

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