The following article is reprinted from the September, 1997 issue of
On the Edge, the Interactive Data Fixed Income Analytics bimonthly newsletter.
Back-to-Basics: Why Do We Use "Monte Carlo" Simulations?
Teri Geske
Senior Vice President, Product Development
When valuing certain types of derivatives or securities with embedded options, we often talk about a “Monte Carlo” simulation or analysis. The concept of a Monte Carlo simulation was developed around the turn of the century (yes, by someone observing gamblers at the casinos) and has been used since the 1970’s to price financial instruments. In this month’s column, we thought it would be useful to review what we mean by a Monte Carlo simulation and discuss when and why it is used.
In a nutshell, a Monte Carlo simulation is used to estimate an instrument’s current value by determining the (arithmetic) average of the payoff under a large number of simulated environments, discounted back to the present. The Monte Carlo simulation creates each “environment” by generating values (based on some probability distribution) for the underlying variable that determines the payoff. For fixed income securities where interest rates are the critical variable, the simulation considers different interest rate paths that the security could follow; for currency options, a Monte Carlo analysis would generate exchange rates, and so on.
To decide whether or not a Monte Carlo simulation should be used, we determine whether or not the payoff for the security depends upon the path taken by the underlying variable. Consider an investment which offers the following payoff one year from today:
Assuming rates are equally likely to rise or fall, the expected payoff from this investment is (0.50 x $10) + (0.50 x $5) = $7.50. To determine the fair price of this investment today, we would discount the expected payoff at the one year risk-free rate (assuming no credit risk from the counterparty). In this situation, it does not matter what path interest rates follow between now and the payoff date; all that matters is the level of rates at the time the payoff is determined. So, a valuation model need only consider the possible interest rate environments at the payoff (maturity) date to determine the current value of the investment.
Now consider an investment with a more complicated payoff:
In this situation, we cannot simply consider the interest rate environment at the end of the period to determine the payoff; we need to know the path that rates followed over the period. This is the kind of “path-dependent” valuation problem for which a Monte Carlo simulation is used, because it allows us to sample the possible interest rate paths which determine the amount (and possibly the timing) of cashflows to the investor. Monte Carlo simulations are used to determine the price, effective duration, convexity and OAS of mortgage-backed securities, index amortizing swaps, securities with periodic caps, barrier options, etc., because these instruments all have path-dependent payoffs.
It is important to remember the relationship between volatility estimates and a Monte Carlo simulation. Volatility rates are used to determine the range of interest rate environments that the Monte Carlo simulation will sample. A volatility rate of 10% means that, within one standard deviation, interest rates are expected to fluctuate +/- 10% annually. If the volatility estimate is increased to 20% or decreased to 5%, the results of the Monte Carlo simulation can vary dramatically, as different interest rate environments would be sampled with different probabilities, most likely affecting the expected payoff to the security in question.
A Monte Carlo simulation is typically not the valuation technique of choice if other methods are available, because it is cumbersome and time-consuming compared to solving a mathematical equation. In fact, the process of generating enough random paths to adequately sample the possible environments that affect a security’s payoff is referred to as the “brute force” method, as it can take a very large number of paths to produce a sufficiently accurate result. In a random Monte Carlo valuation, accuracy is a function of the square root of the iterations (paths) used in the simulation; therefore, increasing the number of paths from 100 to 1,000 only increases accuracy by approximately 3.16 times (v100 = 10; v1,000 = 31.6). This has serious implications for processing speed, and limits the usefulness of a Monte Carlo simulation where an acceptable level of accuracy cannot be achieved by running a fairly small number of paths.
Because of this limitation, different types of “reduced path” methods of Monte Carlo simulations have been developed, which can produce accurate results using a smaller number of paths than a truly random approach. In BondEdge, we use a proprietary “Representative Path” model when valuing path-dependent securities1 , including mortgage pass-throughs, ARMs, CMOs and FRNs. How can one determine when “acceptable” accuracy has been achieved? If increasing the number of paths only changes the result by a small amount (e.g. less than a bid-ask spread), you have probably reached the point where the cost of running more paths outweighs the benefit. (Caution: for some complicated security types, is difficult to determine whether you have reached this point because the answer does not stabilize until the number of Monte Carlo paths increases dramatically).
Mathematicians and financial experts continue develop ways to improve the speed and accuracy of Monte Carlo simulations. At Interactive Data Fixed Income Analytics, we have done work on a type of Monte Carlo analysis known as a “quasi-random” simulation, which can adjust for the “redundancy” of paths which typically occurs using a truly random approach. We hope this brief review of Monte Carlo simulations has been useful (and has given you something to think about the next time you are rolling the dice at the tables).
1For additional information, please refer to the Interactive Data Fixed Income Analytics Publication, “Term Structure Dynamics and Mortgage Valuation” by O. Cheyette, Ph.D.
