# CHAPTER 21

# FORWARD RISK NEUTRAL PRICING AND THE LIBOR MARKET MODEL

Chapter 20 introduced the Black formula to price caps, floors, and swaptions. In this chapter we review a recent methodology that provides a no arbitrage underpinning of this formula. In addition, this pricing methodology is particularly convenient for the analysis of other fixed income securities as well. The methodology is called forward risk neutral Pricing, and it is at the basis of the LIBOR market model, or BGM model, from Brace, Gatarek, and Musiela (1997), the article that set out the framework. In this chapter, we also review the Heath, Jarrow, and Morton (HJM) framework, whose pathbreaking result in a famous 1990 article led the way to the current approach to fixed income security pricing.

## 21.1 ONE DIFFICULTY WITH RISK NEUTRAL PRICING

To motivate the logic behind this methodology, it is convenient to review the risk neutral pricing methodology, discussed in Chapter 17. Consider an interest rate model

By no arbitrage, the price *V(r, t; T)* of a fixed income security with payoff *g*_{T} = G(r_{T}, T) at *T* must satisfy the Fundamental Pricing Equation

subject to the boundary condition *V(r, T) = g*_{T} = G(r_{T}, T). The Feynman-Kac theorem says that the price of this security is given by

where the expectation *E* ....]*