The market convention for quoting a term structure is in terms of annualised yields of zero coupon bonds of various maturities. Therefore if * y * is the * N * year rate, then the current price of the associated zero coupon bond ^{ [2] } is:

Calibration of the tree involves finding the one-period yield (short-term interest rate) at each node, such that the observed term structure is matched. The following methodology for calibrating the binomial tree to a market-observed interest rate and volatility term structure is outlined by BDT [ ** 6 ** ].

Consider the interest rate and volatility term structures observed at time

* t * = 0, represented by {( * y _{ i } * , ƒ

At time * t * = 0, the short-term interest rate may be taken directly from the observed term structure as the 1-year yield, hence * r * _{ } = * y * _{ 1 } . To determine the short-term interest rates at time * t * = 1, make use of the observed 2-year yield and associated volatility. The current value of a 2-year zero coupon bond, * P * ^{ (2) } _{ } = * P * ^{ (2) } is calculated as:

At time * t * = 1, the 2-year zero coupon bond has one year left to run, so its prices, * P * ^{ (2) } _{ u } and * P * ^{ (2) } _{ d } (where the subscript indicates an up or down move in the short-term interest rate), may be found as:

where * r _{ u } * and

We also need to match the term structure of volatilities. The standard deviation of the short-term interest rate at time * t * = 1 is matched to the volatility of the 2-year yield ƒ _{ 2 } , hence ^{ [3] } :

The equations (8.3)-(8.5) are solved simultaneously for the four unknowns:

The resulting * t * = 1 short-term interest rates * r _{ u } * and

and hence, we need only match two short-term interest rates to two observed values and can find a unique solution.

Figure 8.3: Tree of short-term interest rates out to 2 years

^{ [2] } Assume all zero coupon bonds have a maturity value of 100.

^{ [3] } Consider a random variable * X * . At time * t * = * t * *, * X * may take on two possible values, * x * _{ 1 } and * x * _{ 2 } , each with probability ½. Without loss of generality, let * x * _{ 1 } ‰ * x * _{ 2 } . Hence:

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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