Interpreting Reset Dates and Bond Volatility in the Libor Market Model
Summary
This short Q&A explains two symbols in the Libor Market Model: the index m(t) for the next reset date and the volatility term v_k(t). Given ordered reset times, m(t) is the smallest index whose reset time is at or after the current time. For example, when t falls after the second reset and no later than the third, the index is three. The answer identifies v_k(t) as the volatility of the zero-coupon bond price P(t,T_k), pointing to the surrounding equation as context.
The document offers definitions and a simple timing example rather than a derivation, calibration procedure, or empirical evidence. It does not reproduce the model equations or discuss how the volatility is estimated, so readers need the cited textbook context to interpret the notation fully. A separate comment recommends another interest-rate modeling reference, but gives no supporting comparison.
Key ideas
- The reset index m(t) identifies the first reset time that is not earlier than t.
- When t lies between consecutive reset dates, m(t) points to the later date.
- The term v_k(t) is identified as volatility of the zero-coupon bond P(t,T_k).
- The explanation clarifies notation but does not derive or calibrate the model.
Tags
Full text
# Libor Market Model definitions in Options, Futures & Other Derivatives, Hull 9th Ed, p744
# Libor Market Model definitions in Options, Futures & Other Derivatives, Hull 9th Ed, p744
Re: Options, Futures & Other Derivatives, Hull 9th Ed, p744.
- What does "m(t)" represent? I am struggling to understand the definition provided of: "Index for the next reset date at time t; this means that m(t) is the smallest integer such that t <= t_m(t)"
- In equation 32.8 (lower half od page 744), what do the parameters "v" represent? I can't seem to find it defined anywhere.
Thank you.
## Answer by rvignolo (score 2, accepted)
https://quant.stackexchange.com/a/61617
Regarding 1:
$m(t)$ returns the index in a tenor structure formed by the "reset times" such that $t \leqslant t_{m(t)}$. So, for example, given a tenor structure formed by the reset times $t_1, t_2, \dots$ and a time $t$ such that $t_1 < t_2 < t \leqslant t_3 \dots$, then $m(t) = 3$. Now, it is clear that $m(t)$ refers to the index of the next reset time.
Regarding 2:
$v_k(t)$ seems to be the volatility of the zero coupon bond $P(t, T_k)$, see the equation between (32.7) and (32.8).
Lastly, I would strongly recommend you to follow a different book for these kind of subjects. For example, Interest Rate Modeling from Andersen and Piterbarg. I think that learning quantitative finance from Option, Futures & Other Derivatives is not a great idea.
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