11.4 Eliminating the market prices of risk


11.4 Eliminating the market prices of risk

Equations (11.26)-(11.28) depend on a specific set of n bond maturities, with maturities { S 1 , , S n }, for which the market prices of risk exist. These are the n bond maturities which define the equivalent martingale measure.

By making an additional assumption, specified in the condition below, we show that there exists a unique equivalent martingale measure simultaneously making relative bond prices of all maturities martingales. This allows the dependence on n specific bond maturities to be eliminated and pricing equations (11.26)-(11.28) become entirely independent of the number and maturity of bonds used to determine the equivalent martingale measure.

Condition 6

start example

Common equivalent martingale measures. Given Conditions 1 - 3, let Conditions 4 and 5 hold for all bond maturities S 1 , , S n ˆˆ [0, ] with 0 < S 1 < < S n . Also, = .

end example
 

Proposition 3

start example

Uniqueness of the equivalent martingale probability measure across all bond maturities. Given:

  • a family of forward rate drifts { ± ( ·, T ): T ˆˆ [0, ]},

  • a family of forward rate volatilities { ƒ i ( ·, T ): T ˆˆ [0, ]}, i = 1, , n

which satisfy Conditions 1-5, then the following are equivalent:

end example
 

Hence the existence of the unique equivalent probability measure which makes all relative bond prices martingales, is equivalent to the condition making market prices of risk independent of the specific bond maturities chosen , which is equivalent to the above restriction on the drift of the forward rate process. All of the above conditions ensure an arbitrage-free framework.

Examine the derivation and resulting implications of each of the above conditions in turn . (11.29a) implies the martingale approach to bond pricing. Z ( t, T ) is a martingale with respect to the equivalent probability measure , hence:

Then, by Girsanov's Theorem:

where is the Radon-Nikodym derivative defining the equivalent martingale probability measure. From (11.22) and the above proposition, showing the independence of the market prices of risk of bond maturity T and equivalently the uniqueness of the equivalent probability measure across all bond maturities, the Radon-Nikodym derivative may be written as:

and so (11.30) becomes:

When pricing under the original market measure Q , the bond price is explicitly dependent on the money market account B ( T ) and the market prices of risk i ( t ), i = 1, , n . This introduces an implicit dependence on:

  • forward rate drifts under the market measure { ± ( ·, T ): T ˆˆ [0, ]},

  • forward rate volatilities { ƒ i ( ·, T ): T ˆˆ [0, ]}, i = 1, , n ,

  • initial forward rate curve { f (0, T ): T ˆˆ [0, ]}.

Condition (11.29b), requiring the independence of the market prices of risk of bond maturity, is a necessary condition for the absence of arbitrage. It is a standard condition, imposed by many earlier models (e.g. Vasicek [ 50 ], Cox, Ingersoll and Ross [ 18 ] and Brennan and Schwartz [ 10 ]) to derive the fundamental partial differential equation for contingent claim valuation.

Condition (11.29c) imposes a restriction on the functional form of the family of drift processes { ± ( ·, T ): T ˆˆ [0, ]}, which is required to ensure the existence of the unique equivalent martingale probability measure. Not all possible forward rate drift processes will comply with this condition.

Examine closely the derivation of (11.29c). By (11.29b) the market prices of risk are independent of the set of bond maturities specified, so (11.20) is equivalent to (11.17) with ³ i ( t ) = ³ i ( t ; S 1 , , S n ) = i ( t ). Making use of the definitions of a i ( t, T ) and b ( t, T ) in (11.11), equation (11.20) becomes:

Differentiating with respect to T yields the required form of the forward rate drift restriction:

To eliminate the market prices of risk from the forward rate process (11.26), make use of this forward rate drift restriction. Integrating the restriction in (11.29c) over [0, t ] yields:

Substituting into (11.26) yields the forward rate process under the equivalent martingale measure in and independent of market prices of risk as:

From (11.3), we have r ( t ) = f ( t, t ), and the short- term interest rate process may be expressed as:

Here, the market prices of risk have been replaced by a series of forward rate volatilities of various maturities. Hence the short-term interest rate for time t is determined using all possible volatility information contained in the term structure over the time interval [0, t ].

The bond price in (11.27) is not an explicit function of the market prices of risk, these enter only via the short-term interest rate process r ( t ). Hence the formulae for the bond and relative bond prices (11.27) and (11.28) remain unchanged once the market prices of risk have been eliminated, except that the Brownian motion no longer depends on the specific n bond maturities chosen, that is . Additionally the formulae may be applied to bonds of all maturities T, T ˆˆ [0, ].

The original formulation of the CIR [ 18 ] model (see Chapter 2) begins with a characterisation of an equilibrium economy. The functional form of the short-term interest rate process and the market price of risk are determined from within this economy. CIR criticise arbitrage pricing theory on the grounds that it exogenously specifies the functional form of the short-term interest rate and market prices of risk, independently of an underlying equilibrium economy. They show that this may lead to inconsistencies and a model admitting arbitrage.

However, in this model the interdependence of the short-term interest rate process, bond price process and market prices of risk is easily seen in equations (11.26)-(11.28). Additionally, HJM make use of information contained in the bond price process to eliminate the market prices of risk from the pricing formulae. This makes their general pricing framework immune to the criticism of CIR.