The third [rule is] to direct my thoughts in an orderly way; beginning with the simplest objects, those most apt to be known, and ascending little by little, in steps as it were, to the knowledge of the more complex.

Descartes (1637, 1968), Discourse on Method and The Meditations

A simple and beautiful theory that agrees pretty well with observation is often closer to the truth than a complicated ugly theory that agrees better with observation.

S. Weinberg (2014), To Explain the World: The Discovery of Modern Science

PURPOSE OF THIS CHAPTER

In this chapter I derive, using the simplest and most intuitive route, an expression for bond prices and related quantities (yields, forward rates, their volatilities, etc) in the Vasicek model.

Doing so is in itself character forming. In addition (as the logic is exactly the same), it also shows clearly the steps that one should follow to obtain the same results for the more complex models presented in later chapters.

After obtaining an analytic expression for bond prices in the Vasicek model, we obtain explicit equations for yields, instantaneous forward rates and their volatilities.

We also discuss under what conditions (ie, for which functional forms of the market price of risk) the approach remains affine as one moves from the real-world to the risk-neutral measure.

The chapter does not just present a derivation of analytical results. It also introduces some issues about model calibration that we will encounter again with more complex affine models.

THE REPLICATION APPROACH TO SOLVING FOR BOND PRICES: THE VASICEK MODEL

The PDE Satisfied by Bond Prices

I am told that a Japanese proverb goes something like: ‘He is a fool who never climbs Mount Fuji; but he is a greater fool who climbs Mount Fuji twice.’ I have always liked this proverb, even if I am not one hundred percent sure what it actually means. I take it to suggest that character-forming endeavours are good, but in moderation.