Roulette wheels have been adding value to casino owners in Monte Carlo for hundreds of years, but more recently the mathematical technique named after that city--also used in quantum physics and atomic bomb research--has found a valuable application in finance. This Learning Curve looks at one simple application of Monte Carlo analysis to calculate the risk that LIBOR will exceed a set level in a given period of time. This allows the risk of a floating-rate investment to be quantified and helps to answer questions such as:
* What are the range of funding cost outcomes in a
five-year period?
* If 8% is breached, how long will it remain above that
level?
* What is the probability that floating rate will be cheaper
than fixed-rate debt in the next 10 years?
Projecting volatility cones from today's spot level is a well-known and popular technique for visualizing risk. However, such cones tend to assume a lognormal distribution, where the range of possible moves increases with the square root of time. In other words, the returns on the rate follow a pure random walk. This is fine for foreign exchange rates, where the lognormal distribution is a good model for what we actually observe, but for interest rates it presents problems. The most serious of these is that a lognormal distribution assumes that if an interest rate goes to very high levels, such as would be caused by hyper-inflation, that it is still equally probable that the rate will go up rather than down. This is definitely not the case.
In reality, we would expect that hyper-inflation is not a stable and sustainable situation and that rates would be bound to come down sooner rather than later. It is certainly not reasonable that they continue to spiral into the stratosphere. It is also unreasonable for interest rates to stay down at very low levels, such as we see in Japan, unless it is a symptom of deep economic malaise. This is in contrast to fx rates which are equally happy at any level and may double or halve in normal economic environments as they have no reasonable levels to which they must return.
It is therefore desirable to take mean reversion into account when calculating the probabilities associated with future levels of interest rates. The problem is that it is not possible to do this with a simple, analytical formula such as that for a lognormal distribution. Also, there are questions which it is not possible to answer with such formulae, in particular the probability that a particular level is touched at any time over a given period. This is because this probability is path-dependent: it depends not only on where the rate ends up at the end of the period, but also the path it took to get there.
Monte-Carlo simulation is a technique which allows us to calculate these path-dependent probabilities and also allows us to incorporate features such as mean reversion. It does this by using random numbers to simulate the paths that a rate might take in the future according to a model of how that rate behaves. This process is repeated enough times to build up a picture of the distribution of possible outcomes at the desired point in the future. As the simulation calculates individual paths, this allows us to calculate how many of the paths transgressed a particular level very easily.
For example, take our problem of calculating the risk that an interest rate would go above a given level. One form of the general Vasicek model allows us to incorporate a variable amount of mean reversion into the simulated paths. The amount of mean reversion and the volatility can be calculated from fitting it to historical data or current swaption volatilities. Having derived these parameters simulated paths are calculated by using random normally distributed numbers as input to the stochastic part of the Vasicek equation.
As these paths do not follow a normal or log-normal distribution, building volatility cones based on standard deviations is no longer correct. Instead, we can calculate probability envelopes, which are equivalent to volatility cones, by finding the levels within which a certain number of the paths fall. Similarly, the probability of touch can be calculated by counting all the paths that ever go above the desired threshold. The graph opposite illustrates the probability envelopes generated using this method, with the percentiles chosen to be equivalent to +/- 1 standard deviation.
An interesting feature of these envelopes is that they do not spread out over time as one would expect from normal volatility cones. This is an effect of mean reversion, where paths that stray too far from the mean path shown are pulled back to the center. Mean reversion thus acts to constrain the spread of possible outcomes. If it were not taken into account, the probability of the rate rising to high levels, such as 8% or higher, would be greatly over-estimated.
This week's Learning Curve was written by Chris Attfield, senior analyst in strategic risk management advisory at Bank Onein London.
