# Is it possible to perform a VaR analysis based on the forwards obtained by the LMM?

I am in the process of building a LMM model and I ideally want to use this not only to price LIBOR swaps at the current time but also provide a price distribution in a future time. For example we have that the forecasted IRS and/or forecasted LIBOR rate will follow a LN(5,0.05) distribution. Such that I can then use simulation to obtain the different price paths a portfolio of IRS can take and use this to obtain a VaR of different portfolios. As I understand it, the LMM allows us to model a lognormal volatility structure on the LIBOR rate and thus allows us to obtain a forecast distribution of the LIBOR rates such that we can obtain portfolio prices under these simulated forecasted results. However, I am struggling to see how this is actually achieved. Truthfully, I am a little lost on the derivation of the LMM. I would appreciate it if someone could point me into the right direction and if someone could confirm that what I am trying to do is indeed possible.

• I agree I should have clarrified more. A distribution will allow me to perform a simulation on the possible price paths an IRD will take which will lead to me be able to use value at risk of the portfolio Jan 19, 2023 at 9:02
• My mistake, should be clear enough now hopefully. Jan 19, 2023 at 10:51
• Looks better now, thank you! Jan 19, 2023 at 11:34

To answer your last question first: yes, you can absolutely use a LIBOR-market model to get an estimate of the Value-at-Risk (VaR) of a portfolio of swaps.

To be concrete, let's suppose that you want to know the 1-year 95% VaR of a portfolio of swaps with maturity up to 5 years with pay/receive dates 3 months apart.

The steps would be:

1. Pick a discretization of time. We choose equal length periods of 3 months
2. Collect current curve data. Something like 3m LIBOR, 6m LIBOR, 9M LIBOR, 12 LIBOR, 2y swap, 3y swap, 5y swap
3. Interpolate/bootstrap the swaps curve so that you have 3-month forward rates starting each 3 months for 5 years
4. Specify the dynamics of the first four years worth of those forward rates as geometric Brownian motions (log-normal) and fit each of the relevant parameters (time-dependent drift, volatility and correlation)
5. Simulate each of the forward rates for four time steps (1 year)
6. Using each simulated forward curve, reprice your swaps (the longest of which now has maturity of 4 years)
7. Measure VaR from your simulated swap prices

I've oversimplified a fair bit here. You might need much more precise treatment of time than my coarse "every 3 months" approach. And where the rubber really meets the road is in step 4. That's where you have to specify not only the volatility of each forward rate but the full correlation matrix between them. You might need to reach for some additional external information such as swaption prices to have something to calibrate to. If I recall correctly, the LMM does not admit a closed-form pricing formula for swaptions but pretty good approximations are available.

Also, a year might be a little too long to let the LMM run. Certainly, you wouldn't want to go much longer. The issue is that, even with a strong correlation matrix, the forward rates have nothing pulling them back towards each other and so eventually they drift off and you can get some funky forward curve shapes.

In terms of references, I don't have my copy of Brigo and Mercurio to hand but I believe it contains a fairly complete discussion of the LMM. I also found this paper useful.

The LMM can be a little intimidating at first glance. It might be worth looking first at the Black model. Once you are comfortable with it you can approach the LMM as a multi-variable version of the Black model.

Finally, to quote the preface of Brigo and Mercurio, who are themselves quoting Douglas Adams: Don't Panic.

• Thank you so much for your detailed answer! Jan 26, 2023 at 11:39