In almost every graduate and intern level training course - and sometimes even in advance training programmes for quants and traders - the question eventually pops up: *why do we use Monte Carlo simulation in estimating the value of a complex derivative? * And at an even more fundamental level: *what after all is this Monte Carlo simulation that we apply to pricing of complex financial derivatives? *

Say, we had any arbitrary function on the variable and we had no idea how to find the value of this function. The function could be very simple or very complex, doesn't matter; what is pertinent is that we do not how this function behaves with respect to the input and output. One measure of the value of this function will be the expected value, or the mean value. Expected value can also be expressed as the probability weighted average value of a function. And why do we take a probability weighted average as the estimate of the value? This is because we have no deterministic way of finding out the value and so we try to assign probability to different outcomes - outputs - from the function and then add those outputs to get an average output.

Those of us in the financial markets will be very familiar with the notion of expected value. The expected value of Dollar-Yen is the one year forward rate for Dollar-Yen currency pair. This is what the market expects Dollar-Yen to be in one year's time given certain parameters of interest rates today.

Let's take another simple example: say, there is an asset whose value is a complex function of interest rate, GDP, inflation, and many other variables and we have not idea how exactly the value of this asset varies with all these parameters, i.e. we do not have an exact mathematical equation which expresses the value of this asset with all these independent variables. This asset could be a stock, an index or the property price. Now assume that this asset is currently trading at $100 and you are reasonably sure (with a high degree of confidence) that it has a 60% chance of going up to $120 and a 40% chance of going down to $80 in one year's time from now. Therefore, rather than bothering with a complex equation or formula which would express the asset's value you simply multiply the probabilities with the values and get an estimate of the asset's value in a year's time from now.

**Expected Value = $120*60% + $80*40% = 104 **

Thus you conclude that the asset's expected value - the average value - in one year's time is $104. This is an estimate of the average; only that we have used the probabilities to calculate that average.

Now, let say that you think that this is taking too much liberty with probabilities and generalizing the process too much. So you go around and start talking to various experts and do more thinking in your head and fine tune your estimate as: the asset has 5% chance (read probability) of moving up to $180 and 5% chance of moving down to $45 - these are the outliers and extreme moves and hence there is low probability of that happening; and you also expect that there is a 35% probability of the asset hitting $115, 20% probability of it hitting $130 and 20% probability of it hitting $75. Therefore, the probability weighted average value of the asset in one year's time would be $92.5.

**Expected Value = $45*5% + $75*20% + $115*35% + 130*20% + $180*5%
**

=> Expected Value = $92.5

Since there are more probability outcomes (based on expert's opinion) this estimate for expected value is better the previous one.

A Monte Carlo simulator - an engine that does Monte Carlo simulation - does something similar to the above exercise. The only difference is that instead of going around talking to experts in the market and the academia the engine simply draws a series of random numbers from a normal distribution - with a mean of one and standard deviation of one - and assigns those random numbers to the arbitrary values of the asset price generated from a generating function - say, there is a function like the one above, which somehow generates some arbitrary value of . These random normal numbers do the job of probability estimates and then the engine simply sums up all those randomized values of the asset price - along a path - and produces the expected value for the asset.

Once an expected value of the asset is generated the derivative payoff can be easily found out from the expected asset value at the end of the path. Continuing on the previous math example let's say that we need evaluate the value of the arbitrary function . The value of the function can be evaluated (an approximation can be made regarding its value) by using the integral:

In the above integral is the probability density function of with support B such that the sum of all probabilities over B is one, i.e. . Essentially, we are trying to find out the area under the curve . In fact, the integral above is the expected value of the function . Here is the expectation operator.

Now we cannot estimate the output of the function using a functional relationship with respect to ; however, is a function of a series of iid random draws of from the probability density function and therefore, using the law of large numbers we can express the expected value of the function as

The notation above indicates there is almost sure convergence*.

This is principle behind the Monte Carlo approach. A call option payoff is given by the maximum of the terminal value of the spot price minus the strike price. The payoff function, with a constant strike price of , looks like:

Actually, this function is a two dimensional function (which may not be very evident from the first look at the function). The payoff of the option is dependent on the underlying price of the asset which in the above is denoted by . But since the value of the asset itself changes with time, that is itself is a function of time, the option payoff is both a function of the asset price and time . More generally, we should write,

Therefore, the expected value of the above payoff function - the probability weighted average - would simply be the area under the curve . Instead of tracing a single curve, we in fact trace a path of the asset - where the asset will be over a certain period of time and how it gets there - over a two dimensional area on a graph. One of the axis is the time axis and the other the spot axis. Therefore, the expected value of the payoff of the option is given by:

The integral is evaluated by first summing the probability weighted path of the asset and then finding the terminal value of the asset from that path. Then that terminal value is input in the above payoff function.

**Reference: ** "Monte Carlo Simulation for Advanced Option Pricing: A Simplifying Tool" by Jason Fink, James Madison University, College of Business; and also see "Options: A Monte Carlo Approach" by Boyle, P in Journal of Financial Economics (May 1977). *

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