Calibrating Libor Market Model Volatility to Coterminal Swaptions
Summary
The document describes a Libor Market Model calibration problem. Forward Libor rates are simulated under the spot measure with a maturity-dependent volatility function that combines a decaying exponential term and a constant component. Correlations between rates are specified to decay exponentially with separation between rate indices. The calibration targets a strip of coterminal swaption volatilities.
After calibration, the chosen volatility function, which has four constant parameters, does not fit the market swaption volatilities accurately. The author asks whether this indicates that the model is underspecified and whether a more flexible parametrization would be suitable for the set of market quotes. No calibration results beyond the stated poor fit, alternative parametrizations, or recommended solution are provided. The example raises a practical modeling issue: fit depends on the flexibility of the volatility specification as well as on calibration, but the post does not establish that adding parameters alone would resolve the mismatch.
Key ideas
- The example uses a maturity-dependent parametric volatility function in a Libor Market Model.
- Rate correlations are specified to decay exponentially with index distance.
- Calibration targets coterminal swaption volatilities, but the chosen four-parameter form fits them poorly.
- The document asks whether greater parametrization flexibility is needed but offers no answer or validated alternative.
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# Volatility Parametrization Libor Market Model - Underspecified Model?
# Volatility Parametrization Libor Market Model - Underspecified Model?
Does the volatility parametrization that I have chosen give an underspecified model? Which volatility parametrization in the Libor Market Model would suit the best for the particular case described here under?
I have implemented the Libor Market Model in Matlab and am simulating paths of forward Libor Rates under the spot measure, whose dynamics are given by $$dL_n\left(t\right)=\sigma_n\left(t\right)L_n\left(t\right)\sum_{j=q\left(t\right)}^n \frac{\tau_j \rho_{j,n} \sigma_j\left(t\right)L_j\left(t\right)}{1+\tau_j L_j\left(t\right)}dt + \sigma_n\left(t\right)L_n\left(t\right)dW\left(t\right)$$ where $$L_n\left(t\right):=L\left(t;T_n,T_{n+1}\right),$$ $$\tau_n = T_{n+1}-T_n,$$ For the volatility parametrization I have chosen: $$\sigma_n\left(t\right) = \left(a+b\left(T_n-t\right)\right)e^{-c\left(T_n-t\right)}+d$$ where $a,b,c,d$ are constants and $T_n-t$ represents the time to maturity. For the correlation I have taken $$\rho_{i,j}\left(t\right) = e^{-\beta\left|i-j\right|}$$ with $\beta=0.05$ constant. I have calibrated the Libor Market Model to the co-terminal market swaption volatilities 1Y15Y,2Y14Y,...,14Y1Y,15Y1Y.
After calibration, I see that the market swaption volatilities are not fitted accurately with this volatility parametrization consisting of 4 degrees of freedom. My question is if it is possible that this model is underspecified with this parametrization and if it would be a wise choice to choose another parametrization with more degrees of freedom if we are calibrating to 15 swaption vols. If so, which volatility parametrization would you recommend?
Thanks in advance.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.