Frequently Asked Questions In Quantitative Finance

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Chapter 2: FAQs 181

in practice, we have to impose some conditions on the
functionr(t).



  • Forward rates should be positive, or there will be
    arbitrage opportunities

  • Forward rates should be continuous (although this is
    commonsense rather than because of any financial
    argument)

  • Perhaps the curve should also be smooth


Even with these desirable characteristics the forward
curve is not uniquely defined.


Finding the forward curve with these properties
amounts to deciding on a way of interpolating ‘between
the points,’ the ‘points’ meaning the constraints on
the integrals of therfunction. There have been many
proposed interpolation techniques such as



  • linear in discount factors

  • linear in spot rates

  • linear in the logarithm of rates

  • piecewise linear continuous forwards

  • cubic splines

  • Bessel cubic spline

  • monotone-preserving cubic spline

  • quartic splines


and others.


Finally, the method should result in a forward rate func-
tion that is not too sensitive to the input data, the bond
prices and swap rates, it must be fast to compute and
must not be too local in the sense that if one input is
changed it should only impact on the function nearby.


And, of course, it should be emphasized that there is no
‘correct’ way to join the dots.

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