Why Lognormal Libor Market Model Rates Can Explode in Simulation

# Exploding Libor Rates in Libor Market Model


# Exploding Libor Rates in Libor Market Model












I have implemented the Libor Market Model in Matlab. When I generate a number of paths, I notice that some of them explode. Does anybody have an idea what could cause this?

I already tried solving the problem by decreasing the timestep (up to dt=0.001) in order to reduce the error and also by simulating with the log-Euler scheme instead of the 'normal' Euler. In both cases it did not resolve the problem, since some of the Libor rates paths are still diverging.

Specifics:

I simulate the 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 $$L_n\left(t\right):=L\left(t;T_n,T_{n+1}\right),$$ $$\tau_n = T_{n+1}-T_n,$$ $$\sigma_n\left(t\right) = k_n \left[\left(a+b\left(T_n-t\right)\right)e^{-c\left(T_n-t\right)}+d\right],$$ index function $q\left(t\right)$ is defined by $$T_{q\left(t\right)-1}\leq t < T_{q\left(t\right)},$$ $W$ is a Brownian Motion under the spot measure.

## Answer by Mark Joshi (score 6)

https://quant.stackexchange.com/a/34965

this is a well-known problem. One solution is to make volatility zero when rates exceed a certain high level.

It's less problematic than it looks because any cash-flows generated will be divided by a rolling money market account which has huge value and so the deflated cash-flows are very small.

## Answer by Bond007 (score 1)

https://quant.stackexchange.com/a/34963

The rates will explode in the current low rates environment my friend where empirically they are at a too low level to use a log-normal model if you want to preserve your log-normality please use a shifted log normal distribution instead to a convenient rate cut off of around 2%. This happens mainly on EUR market. Hope this help

## Answer by danp (score 1)

https://quant.stackexchange.com/a/35883

The explosion of the forward rates in the log-normal LMM simulated in the spot measure seems to be related to the explosion of the Eurodollar futures prices in this model which was studied in this paper

Eurodollar futures pricing in log-normal interest rates models in discrete time

The Eurodollar futures prices are given by the expectation of the Libor in the spot measure, so an explosion in the former quantity is a signal that the Libor distribution becomes heavy tailed. Sampling from such a heavy tailed distribution will produce a path with extremely large Libor values.

The plot in Figure 4.1 shows a lower bound on the ED futures convexity adjustment, which is seen to explode at a certain volatility. This is an exact analytical bound which does not involve any simulation.

A similar explosion of the MC paths happens also in stochastic volatility models such as the log-normal SABR model, simulated by Euler time discretization. The simplest setting where this phenomenon appears is the bank account compounding interest in discrete time, assuming that the interest rate follows a geometric Brownian motion. An introduction to this phenomenon is given in Section 2 of the above paper.

## Answer by danp (score 0)

https://quant.stackexchange.com/a/85429

The Libor Market Model is less used these days after the Libor transition to SOFR - although similar market models have been proposed for SOFR. More popular with practitioners seem to be the Cheyette, or Markovian HJM type models. The same explosive phenomenon appears also in these model, under the log-normal rates specification. For example, the plots below show this explosion in a simple 1-factor Cheyette model. Some control over the explosion can be achieved by adjusting the mean reversion - the explosion can be delayed beyond a reasonably large maturity. More details are given in this paper.