The following notation and convention are used.
- The reset dates are at equal intervals \( \delta \). In other words, \( \delta = t_{k} - t_{k-1} \) for \(k=1,2,3,\cdots \).
- \( R(t) \) is the interest rate over \( [0,t]\) as seen at \( t=0\).
- For \( t \leq t_{k} \), \( P(t, t_{k}) \) is the discount factor over \( [t,t_{k}]\) as seen at \( t\).
- For \( t \leq t_{k} \), \( 0 \leq k \), \( F_{k}(t) = F(t, t_{k}, t_{k+1}) \) is the forward rate
over \( [t_{k},t_{k+1}]\) as seen at \( t\) .
- \( m(t) \)
denotes the index of the next reset date on or after \( t\).
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\( \varsigma_{k}(t) \) denotes the volatility of \( F_{k}(t) \) at \( t \).
- It is assumed that \( \varsigma_{k}(t) \) depends only on \(k - m(t) \). Then,
define \( \Lambda_{k -m(t)} = \varsigma_{k}(t) \).
Suppose, furthermore, that \( \Lambda_{0}, \Lambda_{1}, \Lambda_{2}, \ldots, \) are given.
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Define \(\displaystyle \sigma_{B}(k) = \sqrt{ \frac{1} {t_{k}} \cdot \delta \cdot \sum_{j=0}^{k-1} \Lambda_{k}^2} \).
