By way of an illustrative example, consider the continuous time equivalent of the Ho and Lee [ ** 27 ** ] model (studied in Chapter 10) within the above specified framework.

Consider a model with a single Brownian motion and hence single volatility parameter ƒ _{ 1 } ( ‰ , * t, T * ) ‰ ƒ > 0 where ƒ is a constant. Assume Conditions 1-6 are satisfied with { * f * (0, * T * ): * T * ˆˆ [0, ]} the initial forward rate curve and ( ‰ , * t * ), * t * ˆˆ [0, ] themarket price of risk corresponding to the single source of randomness. From (11.32) the forward rate process with respect to the equivalent martingale measure is:

and hence the short- term interest rate process is specified as:

Since there are no restrictions placed on the evolution of the short-term or forward interest rates, there exists a positive probability of these rates being negative. To determine the bond price dynamics, substitute (11.38) into (11.2) to give:

From the definition of the forward rate (11.1), we have:

and so:

Define the following notation:

| - | time |

| - | option exercise price, |

| - | option expiry date, 0 ‰ |

At expiry of the option, its value is:

Making use of (11.36), the time * t * value of the contingent claim is ^{ [18] } :

An analysis, similar to that used in the initial formulation of the Black- Scholes valuation formula, yields the contingent claim price to be ^{ [19] } :

with

and

The above formula is a modification of the Black-Scholes option pricing formula where the required volatility is the standard deviation of instantaneous returns on the forward bond price, that is the standard deviation of returns at time * t * * of a bond maturing at time * T * . This may be determined from (11.13), the bond price process within the current framework:

and so the standard deviation of the forward bond price is ƒ ( * T * ˆ’ * t * ) ˆ’ ƒ ( * t * * ˆ’ * t * ) = ƒ ( * T * ˆ’ * t * *).

^{ [18] } Here [ ·] denotes the expectation taken with respect to the equivalent martingale measure.

^{ [19] } Here * N * ( ·) is the cumulative normal distribution.

Interest Rate Modelling (Finance and Capital Markets Series)

ISBN: 1403934703

EAN: 2147483647

EAN: 2147483647

Year: 2004

Pages: 132

Pages: 132

Authors: Simona Svoboda

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