When a Libor Market Model Volatility Parametrization Is Bounded
Summary
The document considers whether a Libor Market Model volatility function, built from a maturity-dependent linear term multiplied by an exponential and then shifted by a constant, remains bounded. It assumes the parameters and the rate’s maturity are constants and focuses on the product of the linear and exponential terms, since multiplying by a constant and adding another constant do not change boundedness.
For the usual time interval from zero to the rate’s maturity, residual maturity is nonnegative. A positive exponential decay parameter damps the linear factor, keeping the product bounded over that finite interval. The response also notes that if time ranges over the entire real line, boundedness depends on the decay parameter’s sign and behavior at the two extremes. The explanation is a brief qualitative argument rather than a full classification of all parameter combinations; it does not analyze edge cases such as a zero decay parameter or specify bounds.
Key ideas
- The volatility function is a constant scale times a shifted maturity-dependent term.
- For nonnegative residual maturity, a positive decay parameter suppresses linear growth.
- A finite time interval and an unrestricted time domain require different boundedness checks.
- For an unrestricted domain, behavior at both extremes depends on the decay parameter’s sign.
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Full text
# Prove Volatility Parametrization of Libor Market Model is Bounded/Not Bounded
# Prove Volatility Parametrization of Libor Market Model is Bounded/Not Bounded
How can I prove that the function $$\sigma_i\left(t\right) = k_i\left[\left(a+b\left(T_i-t\right)\right)e^{-c\left(T_i-t\right)}+d\right]$$ is bounded/unbounded?
$\sigma_i\left(t\right)$ is the chosen volatility parametrization in the Libor rate dynamics
$$dL_i\left(t\right)=\mu_i\left(t\right)L_i\left(t\right)dt+\sigma_i\left(t\right)L_i\left(t\right)dW_i\left(t\right)$$
I have no clue how to start, any help is appreciated. Thanks in advance.
Edit:
The instantaneous volatility can be decomposed in the following parts $$\sigma_i\left(t\right) = g\left(T_i\right)f\left(T_i-t\right)$$ where $g\left(T_i\right)=k_i$ is the component specific to the individual forward Libor rate and $f\left(T_i-t\right)=\left(a+b\left(T_i-t\right)\right)e^{-c\left(T_i-t\right)}+d$ is the component depending on the residual maturity $T_i-t$.
## Answer by oliversm (score 1, accepted)
https://quant.stackexchange.com/a/27511
If I have read the question correctly then I will assume that $a$, $b$, $c$, $d$, $T_i$, and $k_i$ are constants. If this is the case then the only term which we need to show is bounded is $$ \big(a + b(T_i - t)\big)\exp\big(-c(T_i-t)\big). $$ If we assume that we are only considering the temporal domain $0 \leq t \leq T_i$ such that $T_i - t \geq 0 $ then we can ensure that the the product will remain bounded if we have $c>0$ as this will exponentially dampen the product of the two terms.
If though for whatever reason we have the more general case where $ -\infty \leq t \leq \infty$ then it will all depend again on the sign of $c$, but we would only need to consider these two extreme limits.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.