Calibrating an LMM to Price Fixed-Strike Bermudan Swaptions
Summary
The document describes a calibration problem for a Libor market model used to value a Bermudan swaption. The proposed volatility structure has a parametric time-to-maturity shape and period-specific scaling factors, while correlations follow an exponential form with a fixed parameter. The initial calibration fits at-the-money swaption volatilities using the Rebonato approximation; a second adjustment fits the diagonal of a fixed-strike volatility matrix. The goal is to price the swaptions on the Bermudan’s underlying swaps consistently with the market at the Bermudan strike.
The central concern is that the Rebonato approximation is most accurate near at-the-money, so matching fixed-strike volatilities through that approximation may not carry over to prices from Monte Carlo simulation. The document raises this mismatch as an unresolved question and supplies no proposed correction, calibration results, or validation evidence. It therefore frames a model-calibration issue rather than establishing a solution; pricing accuracy may depend on the chosen model dynamics and calibration instruments.
Key ideas
- The described LMM uses a parametric volatility term structure and an exponential correlation structure.
- The proposed first calibration fits at-the-money swaption volatilities through the Rebonato approximation.
- Period-specific volatility scalings are then adjusted to fit a fixed-strike volatility matrix diagonal.
- A good approximation near at-the-money may not reproduce Monte Carlo prices at other strikes.
- The document identifies the calibration mismatch but does not provide a tested remedy.
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# Pricing back swaptions corresponding to underlying swaps of Bermudan Swaption in calibrated LMM
# Pricing back swaptions corresponding to underlying swaps of Bermudan Swaption in calibrated LMM
I do not know to which swaption volatility matrix I have to calibrate the LMM in order to price back correctly the swaptions corresponding to the underlying swaps of a Bermudan Swaption.
My problem: For the LMM, I use the simple correlation form $\rho_{i,j}=e^{-\beta|i-j|}$ with beta fixed. This parameter is not taken into account in the calibration. For the volatility I use the form $$\sigma_i\left(t\right) = \phi_i \left(\left(a+b\left(T_i-t\right)\right)e^{-c\left(T_i-t\right)}+d\right)$$ Initially I take $\phi_i=1$ for all $i$ and calibrate the parameters $a,b,c,d$ to the ATM swaption market volatilities matrix by minimizing the MSE between these volatilities and the ones determined by the Rebonato swaption volatility approximation formula. We obtain: $$V_{ATM swaptions}^{Reb} = V_{ATM swaptions}^{market}$$ Afterwards we determine the parameters $\phi_i$ such that the diagonal of a fixed strike $K$ market volatility matrix is fitted exactly: $$V_{K swaptions}^{Reb} = V_{K swaptions}^{market}$$ It is important that the diagonal swaption volatilities (=swaption vols of swaptions corresponding to the underlying swaps of a Bermudan Swaption) of a fixed strike $K$ swaption vol matrix are priced correctly when we want to price a Bermudan Swaption with strike $K$.
However the Rebonato approximation formula is only accurate for ATM strikes. Hence, comparing the Rebonato approximated price with strike $K$ and the price obtained by a Monte Carlo routine with these calibrated parameters, we would in general see that: $$V_{K swaptions}^{Monte Carlo} \neq V_{K swaptions}^{Reb} = V_{K swaptions}^{market}$$ Hence the swaptions corresponding to the underlying swaps of a Bermudan Swaptions are not priced back correctly.
Does anybody know how I can improve my calibration routine in order to price back correctly the swaptions corresponding to a fixed strike?
Any help is appreciated. 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.