Interest Rate Modelling (Finance and Capital Markets Series)

12.2 Initial framework

The mathematical framework used by BGM is much the same as that used by HJM. We have a probability space ( , { F ^t ; t ‰ 0}, ) where is the state space and the filtration { F ^t ; t ‰ 0} is the - augmentation of the filtration generated by the n -dimensional Brownian motion . Sincewe make use of the arbitrage-free results of the HJM analysis, is the risk-neutral probability measure with the corresponding Brownian motion. The trading interval is specified as [0, ] where > 0 is fixed. The following processes are defined on this probability space ^[4] :

• f ( t, T ) denotes the instantaneous, continuously compounded forward rate prevailing at time t for maturity T . The process { f ( t, T ); t ‰ T } satisfies:

  where ƒ ( t, T ) is the forward rate volatility and ƒ *( t, T )= ˆ« ^T _t ƒ ( t, v ) dv .

  P ( t, T )=exp ( ˆ’ ˆ« ^T _t f ( t, u ) du ) describes the price evolution of a T -maturity discount bond, and so:

  where ƒ *( t, T ) may be interpreted as bond price volatility and hence ƒ *( t, t ) = 0 for all t ‰ 0.

• Defining the short- term interest rate r ( t )= f ( t, t ) for all t ‰ 0, the money market account is represented as:

  with initial condition B (0) = 1.

We know that if discounted bond prices , t ˆˆ [0, T ], T > 0 are martingales under some probability measure , then we are in an arbitrage-free framework. Within this framework, the bond price may be represented as ^[5] :

^[4] he initial BGM formulation [ 9 ] of these processes uses r ( t, x ) to represent the instantaneous forward rate prevailing at time t for maturity t + x . I feel this formulation obscures any value it adds and hence I maintain consistency with the notation used in Chapter 11 by using f ( t, T ) to denote the time t instantaneous forward rate for maturity T . The obvious relationship between the two representations is r ( t, x )= f ( t, t + x ).

^[5] See equation (11.28).