Derivatives

CHAPTER 41Pricing Fixed Income Derivatives: BOPM

Aims

• To derive an interest rate lattice, which precludes risk-free arbitrage profits and is consistent with the current term structure of interest rates.
• To demonstrate how the BOPM is used to price European and American options such as caps, floors, collars, swaptions, callable bonds, and FRAs/FRNs with caps and floors.

In this chapter we price many different fixed income derivatives using the BOPM with an arbitrage-free lattice for the short-rate of interest. This ‘no-arbitrage’ approach ensures that the interest rate lattice is constructed so it is impossible to make risk-free profits by trading (different bonds) along the current yield curve. The lattice is calibrated so that it exactly mimics the current observed term structure – hence, the current term structure is an input to the BOPM and the output is the derivatives price.¹

Pricing interest rate derivatives is more difficult than pricing options on stocks, currencies, commodities and futures contracts because some of the assumptions of the original Black–Scholes model are unlikely to hold for fixed income assets. In particular:

1. the underlying stochastic process for short-term interest rates is more complex than for stock prices – for example, interest rates are mean reverting and therefore do not follow a geometric Brownian motion (GBM).
2. to price some interest rate derivatives we need not only the possible paths taken by a 90-day interest rate but the possible paths ...