Interest Rate Modelling (Finance and Capital Markets Series)
13.3 Using market data
Having considered the model formulation that allows us to incorporate observed term structure data into the pricing formulae, we examine how actual data is used in the calibration exercise.
For each day, the continuously compounded rate of interest and historical volatility are available for a discrete set of node points corresponding to terms to maturity, i , i = 1, , N where 1 = 1 day.
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The interest rate with term to maturity 1 day and its corresponding historical volatility are taken as proxies for the instantaneous short-term interest rate r (0) and its volatility ƒ r ( r , 0).
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Making use of (13.2) we calculate B (0, T ) at each of the node points.
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Applying (13.3) we determine the time 0 discount bond prices, P ( r , 0, ·), with maturities at the nodes.
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These term structures of B (0, ·) and P ( r, 0, ·) are applied in (13.4) to obtain the values of A (0, ·) at the nodes.
An interpolation technique must be applied to the term structures of A (0, ·) and B (0, ·) so that values for any maturity term maybe extracted. Cubic spline interpolation was selected for the smoothness of curves it produces (see [ 1 ]).