Derivatives

CHAPTER 40Pricing Fixed Income Options: Black's Model and MCS

Aims

• To show how Black's model provides closed-form solutions for the price of European options on T-bonds, on T-bond futures, on caps, floors, collars and on European swaptions.
• To price fixed income options using Monte Carlo simulation (MCS).

In previous chapters we discussed hedging/insurance using options on T-bonds and Eurodollar futures and how caps, floors, collars, and swaptions are used to hedge/insure interest sensitive assets and liabilities such as floating rate bank deposits and loans. In this chapter we concentrate on how to price some of these derivatives. To price fixed income derivatives we can use:

• Black's model which gives closed form solutions
• MCS under risk-neutral valuation (RNV)
• BOPM model with an interest rate lattice (tree)
• Equilibrium term structure approach.

Black's model assumes the price of the underlying asset in the options contract has a lognormal distribution, at maturity of the option. MCS generates a path for the short-rate and prices the derivative under RNV. The BOPM uses a lattice for the ‘short-rate’ of interest. The equilibrium yield curve approach assumes a specific stochastic process for the interest rate and solves mathematically for the derivatives price – the BOPM and the equilibrium yield curve approach are dealt with in Chapters 41 and 49, respectively.

40.1 BLACK'S MODEL: EUROPEAN OPTIONS

Black's (1976) model, which was originally used for pricing options on commodity ...