12.2 Initial framework


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).



12.3 Model of the forward LIBOR rate

Specifying the above model of the instantaneous, continuously compounded forward rate is equivalent to determining the volatility function ƒ ( t, T )(or equivalently ƒ *( t, T )). Fix some > 0, then the LIBOR rate process { L ( t, T ); t ˆˆ [0, T ], T ˆˆ [0, ]} is defined as:

Imposing a lognormal volatility structure on L ( t, T ), the associated stochastic process may be written as:

where ¼ L ( t, T ) is some drift function and ³ : R 2 R n is the deterministic, bounded and piecewise continuous relative volatility function. Letting h ( t, T )= ( ˆ« T + T f ( t, u ) du , we make use of Ito's Lemma to determine the correct functional form of (12.6). Hence:

Here:

Therefore:

and (12.7) becomes:

Hence by (12.6) we require:

and so (12.8) may be written in terms of the ( T + )-maturity bond price volatility as:

Alternatively, solving (12.9) for ƒ *( t, T + ) we may write this LIBOR stochastic process in terms of the T -maturity bond price volatility as:

Now let us assume [6] ƒ *( t, T ) = 0 for all t ˆˆ ( T ˆ’ ) ˆ 0, T and T ˆˆ [0, ], then a recursive relationship may be used to define ƒ *( t, T )for T ˆ’ t as [7] :

Substituting this recursive relationship into (12.10), the stochastic process describing the evolution of LIBOR may be written purely in terms of LIBOR rate volatilities as:

Letting j = k ˆ’ 1 we have:

[6] This assumption implies the volatility factor disappears for all rates where 0 T ˆ’ t < , that is the time between valuation date and maturity date is less than . This allows for the construction of a tractable model. We have ƒ *( t, t ) = 0 for all t ˆˆ [0, T ] since this is the price volatility of an instantly maturing bond. Relationship (12.9) implies:

Hence, for T = t +

since ³ ( t, t ) = 0 is the volatility of the spot LIBOR rate. So, since ƒ *( t, T )=0 for T = t and for T = t + we let ƒ *( t, T ) = 0 for all T ˆˆ ( t, t + ) as well. This is equivalent to ƒ *( t, T )=0 for t ˆˆ ( T ˆ’ , T ).

[7] By (12.9) we have:

Since, by assumption, ƒ *( t, T )=0 for T ˆ’ t < the first term on the RHS vanishes for T ˆ’ t ˆ’ k < i.e. the term vanishes for k > ˆ’ 1 ( T ˆ’ t ) ˆ’ 1, hence the upper bound for the summation index is k = ˆ’ 1 ( T ˆ’ t ).