Monte Carlo Method of Option-Free Bond Valuation

If you don't want to build a tree, let a computer do it for you, in a way.

The __Monte Carlo__ method takes the same starting interest rate level and variability input and then generates random rate paths through time within these bounds. What is your estimated probability that the resulting paths will be the same as a standard, lognormal tree?

You got it. If you modeled six periods, an interest rate tree would use 64 interest rate paths. $$\displaystyle 2^6 = 64 $$ Someone performing a Monte Carlo simulation would never consider using so few paths. Generating 500 or 1,000 paths is more reasonable. Even if 64 were generated, there's just no way that the same binomial up and down paths would be generated from this method.

No. It's actually a pretty extreme value.

No. Generating random paths wouldn't create the exact same result with certainty. If it did, there wouldn't be any point in one of the two methods.

With each of these hundreds of paths, a potential bond value is calculated. If it's the same distribution of paths with enough random paths generated, how would you expect simple, option-free bond valuations to compare between the interest rate tree and the Monte Carlo method?

Absolutely.

No. They should be the same.

If both are calibrated correctly, the same average present value can be found with either method.

So what's the benefit? Well, the Monte Carlo method is really helpful for bonds where the interest rate affects the cash flows. Mortgage-backed securities are a fine example—higher interest rates, higher prepayments. So this can be built into the Monte Carlo simulation to get some extra insight beyond what you would expect with an interest rate tree. Again, hundreds of paths are generated, and hundreds of potential valuations are calculated. What's the last step?

No. You wouldn't have any idea which of these is the most likely, so just find a simple average.

Excellent.

Two extra elements that may be needed in a Monte Carlo simulation is a "drift term" and a mean reversion. If the rate paths don't allow the benchmark bond to be correctly valued, then the drift term is a slight adjustment added to all interest rates generated in the simulation to make this work. The mean reversion incorporates upper and lower bounds so that interest rates are never predicted to become too high or too low. Even a journey of 1,000 Monte Carlo iterations begins with these first steps.

To summarize: [[summary]]

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