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?


1 Answer 1


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.pdf


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