Adding a Displacement to a LIBOR Market Model for Negative Rates
Summary
The discussion considers whether a constant displacement can adapt a standard LIBOR market model to negative interest rates. It proposes applying the shift to the forward rates used in the diffusion and drift numerators, while the accepted response clarifies that the cash-bond denominator remains based on the undisplaced forward rate. This distinction matters when modifying the model’s drift structure.
For caplet calibration, the response describes using displaced forwards and strikes in Black pricing for both model and market values. It also points to published work on displaced diffusion LIBOR market models as support for the formulation. The exchange gives a concise modeling correction and calibration approach, but does not present a derivation, numerical calibration, or discussion of how the choice of shift affects other instruments or the full curve model. Those details require further analysis before applying the setup beyond caplets.
Key ideas
- A constant displacement can let a LIBOR market model represent rates below zero.
- Apply the displacement to forward rates in the diffusion and drift numerators.
- The drift denominator is tied to the undisplaced forward rate because the cash bond is not displaced.
- Caplets can be calibrated with displaced forwards and strikes in Black pricing.
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Full text
# Can you shift a standard libor market model with regard to only at-the-money options?
# Can you shift a standard libor market model with regard to only at-the-money options?
Suppose I have an LMM defined using the spot measure as in Brigo and Mercurio:
$dF_k(t) = \sigma_k(t)F_k(t)\sum^k_{j=\beta(t)}\frac{\tau_j\rho_{j,k}\sigma_j(t)F_j{t}}{1+\tau_jF_k(t)}dt + \sigma_k(t)F_k(t)dZ^d_k(t)$
And suppose I want to incorporate negative interest rates with a shift.
Most papers/conversations I have seen consider only a shift adjustment in the context of a stochastic volatility model like SABR-LMM.
Is it acceptable to add a shift to the "basic" LMM such as:
$dF_k(t) = \sigma_k(t)\bar{F}_k(t)\sum^k_{j=\beta(t)}\frac{\tau_j\rho_{j,k}\sigma_j(t)\bar{F}_j{t}}{1+\tau_j\bar{F}_k(t)}dt + \sigma_k(t)\bar{F}_k(t)dZ^d_k(t) \\ \bar{F}_k(t) = F_k(t) + \delta$
where $\delta$ is a shift that indicates a specified lower bound on the interest rates?
I believe to do this, you could use the Rebonato volatility approximate to calibrate, but you would adjust the Black volatilities to also incorporate the shift parameter.
Is it that simple, or am I making a mistake regarding some unintended downstream effect? Are there any resources that come to mind that do this sort of thing?
## Answer by BrownianBread (score 2, accepted)
https://quant.stackexchange.com/a/66151
The cash bond doesn't require a displacement, as such the denominator in your drift term should be $F_k(t)$ not $\bar{F}_k(t)$, i.e.
$dF_k(t) = \sigma_k(t)\bar{F}_k(t)\sum^k_{j=\beta(t)}\frac{\tau_j\rho_{j,k}\sigma_j(t)\bar{F}_j{t}}{1+\tau_jF_k(t)}dt + \sigma_k(t)\bar{F}_k(t)dZ^d_k(t) \\ \bar{F}_k(t) = F_k(t) + \delta$.
For the calibration to caplets it is straightforward and you use the displaced forward and strike in a Black formula for both model and market prices.
A simple search for displaced diffusion libor market model gives the paper by Beveridge and Joshi that agrees with the above: https://fbe.unimelb.edu.au/__data/assets/pdf_file/0004/2591824/195.pdfShown 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.