11.5 The problem with forward rates


11.5 The problem with forward rates

By (11.32) we see that the forward rate process is completely specified by the volatility functions ƒ i ( ·, ·), i =1, , n . Consider a framework with one source of uncertainty and so one volatility parameter. For practical implementation it is desirable to apply a lognormal volatility structure for all forward rates [ 45 ]. This is because market prices of caps and swaptions assume a lognormal structure of forward rates. Hence set ƒ 1 ( t, T )= ƒ f ( t, T ), where ƒ > 0 is a constant. However, under this volatility structure (11.32) becomes:

Here, the drift of the forward rate grows as the square of the forward rate [ 49 ] and causes the forward rate to explode in finite time. Therefore for calibration purposes an upper bound needs to be imposed:

This problem is not particular to the HJM model, but rather a characteristic of all lognormal models of instantaneous forward rates.



11.6 Unifying framework for contingent claim valuation

HJM impose conditions on the forward rate process to ensure that a unique, equivalent martingale probability measure exists and hence the process is consistent with an arbitrage-free market. This implies that the market is complete and contingent claims may be valued using an approach detailed by Harrison and Kreps [ 23 ] and Harrison and Pliska [ 24 ]. Harrison and Pliska examine martingale theory within a continuous trading environment, presenting a general methodology for contingent claim valuation. First consider some important concepts and definitions characterising this methodology:

  1. If is a set of probability measures, equivalent to initial probability measure Q and making discounted prices martingales, then by Harrison and Pliska [ 24 , Corollary 3.36]: if is a singleton [16] then the market is complete.

  2. In a complete market there are enough non-redundant securities being traded [ 22 ], such that every integrable contingent claim is attainable.

    • By attainable, we mean that there exists some trading strategy, requiring an initial investment and thereafter producing the same cash flows as the contingent claim.

    • A trading strategy may be viewed as some portfolio of securities, where the number of units of each security held changes through time.

  3. Define a contingent claim as a random variable X : R, X 0 which is F T 1 measurable [17] . Since contingent claim X must be integrable, we require < + ˆ .

  4. Denote an admissible , self-financing trading strategy by { N ( t ), N 1 ( t ), , N n ( t )} where N i ( t ) is the quantity of asset P i , i = 1, , n in the portfolio at time t .

    • A trading strategy is admissible if it is self-financing and the value of the associated portfolio remains non negative through time. This implies that an investor enters the trading strategy with positive wealth and is never in a position of debt.

    • A self-financing trading strategy is one where changes in value of the portfolio are due to capital gains (changes in value of the instruments held) only, not due to cash inflows and outflows. Let V t ( ) be the time t value of trading strategy , then by [ 22 , Definition 1] a self-financing trading strategy is one where:

  5. An arbitrage opportunity is some trading strategy such that V ( ) = 0 and [ V T 1 ( )] > 0. The existence of arbitrage opportunities is inconsistent with an equilibrium in the economy.

  6. Harrison and Pliska present a theorem showing that a market is free of arbitrage opportunities if and only if is not empty. Hence the absence of arbitrage opportunities is equivalent to the existence of an equivalent martingale probability measure.

  7. By Harrison and Pliska [ 24 , Proposition 2.9], if X is an attainable contingent claim generated by trading strategy , with time t value V t ( ) and ˆˆ then:

Now, consider the economic framework characterised in the previous sections: Let Conditions 1-6 hold. By Proposition 3, there is a unique equivalent measure making all Z ( t, T ), T ˆˆ [0, ], t ˆˆ [0, T ] martingales. Since is unique, the market is complete and there exists an admissible, self-financing trading strategy, as denoted above, such that the portfolio value satisfies:

where

N ( T 1 )

-

amount held in the money market account at time T 1 ,

N T i ( T 1 )

-

amount of bond with maturity time T i in the self-financing strategy at time T 1 ,

P ( T 1 , T i )

-

time T 1 value of a bond maturing at time T i ,

X

-

time T 1 payout of a contingent claim.

Now, since we are in a complete market where all contingent claims are attainable and where a unique equivalent martingale measure exists, we conclude that arbitrage opportunities do not exist and the time t price of a contingent claim paying X at time T 1 is given by:

Hence given (11.35), the trading strategy which generates X , the time t value of the portfolio is:

Therefore, to value the contingent claim, the dynamics of the short- term interest rate r ( t ) and the relative bond price Z ( t, T ) must be known under the equivalent martingale measure. Since the market is complete, every contingent claim may be replicated by means of some admissible, self-financing strategy consisting of only the money market account and some n bonds with maturities T 1 , , T n ˆˆ [0, ]. From this we conclude that all contingent claims may be valued.

[16] A singleton is a set having a single element.

[17] It is a security entitling the holder to a payment at time T 1 . The magnitude of this payment depends on the history of price movements up to time T 1 , hence it is F T 1 measurable.