Interpreting the Drift in the LIBOR Market Model
Summary
The document asks for an intuitive explanation of the drift that appears when a forward rate in the LIBOR market model is expressed under a different forward measure. It presents the adjacent-rate drift adjustment, which depends on the correlation between rates, their volatilities, and the later forward rate.
The accepted response interprets the adjustment through delayed payment and convexity. When the earlier rate is paid with a lag, its value is affected by how that rate covaries with the discounting bond; the drift compensates for this stochastic relationship. The denominator is described as the deterministic discount effect over the accrual interval. This links the formula to the deterministic and stochastic components of a change of measure. The answer is qualitative and does not provide a trading replication, numerical example, or full derivation; it is an intuition to accompany the stated model assumptions.
Key ideas
- Changing the forward measure introduces a drift adjustment for rates other than the numeraire rate.
- The adjacent-rate adjustment depends on correlation, volatility, and the later forward rate.
- Delayed payment creates a convexity effect linked to covariance between the rate and discounting bond.
- The denominator represents a deterministic discount adjustment over the accrual period.
- The explanation is intuitive and does not establish a trading replication or numerical result.
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Full text
# Intuition of drift in Libor market model
# Intuition of drift in Libor market model
I studied the classical Libor market model, where the dynamics of rate $F_k$ from time $T_{k-1}$ to $T_k$ are given by $$ dF_k(t)/F_k(t) = \sigma_k(t) dZ_k(t) $$ under the forward measures $Q^k$ (where we use $P_k(t)$, the bond that matures at $T_k$, as numeraire). Then, it follows by a change of measure approach that for $i=k-1$, the dynamics under this measure $Q^k$ are $$ dF_{k-1}(t)/F_{k-1}(t) = \sigma_{k-1}(t) \left(dZ_{k-1}(t) - \frac{\rho_{k,k-1} \sigma_k(t) F_k(t)}{1+\tau_k F_k(t)} dt \right). $$ The above formula are taken from the book of Brigo and Mercurio and there the general formulation for a rate $F_i$ with $i<k$, resp., $i>k$ can be found.
I was able to technically follow the derivation of the drift, but what is the intuitive understanding of it? Can we use a trading argument to understand this equation?
This might be quite basic but I did not find such an intuitive explanation in the literature. Any comment or reference is highly appreciated.
## Answer by Arshdeep (score 3, accepted)
https://quant.stackexchange.com/a/79360
Paying F(K-1) at a lag (K) is a delayed payment and involves a convexity adjustment that can be understood as a consequence of 2 parts - a "stochastic part" where if rates rise whenever F(K-1) is higher (and vice versa, if rates fall), we systemically lose out. The drift adjusts this valuation. This is why you have covariance between the forward and the bond Z(t,T(K-1),T(K)).
The denominator is the "deterministic" part of the convexity adjustment that just penalizes the delay based on the prevailing discount between k-1 and k.
You will now realize that the Radon-Nikodym derivative is ultimately a deterministic adjustment (A0/B0) times a stochastic adjustment (A(t)/B(t)), this is exactly that! So you can tie this back to the math.
Hope this helps.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.