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# Monte Carlo Simulations and How They Work

## Monte Carlo Simulations and How They Work

#### 14 March 2023

What is the Monte Carlo Simulation used for?

The Monte Carlo Simulation is a mathematical technique that predicts possible outcomes of an uncertain event and is most commonly used for fair valuing put and call options.

HOW DOES THE MONTE CARLO SIMULATION WORK?

The only way to explain the Monte Carlo Simulation is via an illustrative example:

To calculate the price of a European call option assume where the underlying share price (S) is R195, the strike price (X) is R200, the risk free rate of return (rf), the volatility (s) is 30% and the time to expiry is (t) is 0.25.

STEP 1

Calculate for each of the Monte Carlo Paths the change in S

The role of Monte Carlo simulation is to generate several future values of the share based on which we can calculate the future value of the call option. The changes in the share prices can be calculated using the following formula:

∆S = Srf∆t + Ss∈√t

 A B Price - S 195 2 Strike - X 200 3 Days to Expiry 63 4 Fraction of a Year - t 0,25 5 Risk Free rate - rf 5% 6 Historical Volatility - s 30% 7

In this equation, ∈ represents the random number generated from a standard normal probability distribution. In this example, this number is calculated using the Rand() function in Excel.

For the purpose of this example, 1000 random iterations are generated. In reality, the higher the number of iterations, the more accurate will be the result, as the distribution tends to a normal one.

Assume column 1 is A and Column 2 is B; the Excel formula will be as follows:
Iteration 1:

=\$B\$2*EXP((\$B\$6*\$B\$7^2)*\$B\$5+\$B\$7*SQRT(\$B\$5)*NORM.S.INV(RAND()))

Assume the random number is 0.576548, ∆S is 13,26. This number will change each time the spreadsheet is recalculated.

STEP 2

Once ∆S is calculated the future value of the share price (S + ∆S) is calculated. The share price at expiry date of the option will be: R195 + 13,26 = R208,26

As the R208,26 is greater than R200 the options will be exercised.

If FV of S is less than R200 then the options will not be exercised.

For example:

• Iteration is column C
• Stock price at T is column D.
• Iteration 1 is line 3

 Iteration Stock Price at T Call Expiry Value 1 169,02 - 2 194,04 - 3 185,82 - 4 208,26 8,26 5 249,22 49,22

MAX(0;D6-\$B\$3)

STEP 3

The option value at expiry will be given by the formula: =MAX (0,S-X)
= MAX (208,26-200)
= 8,26

The above 3 steps will be repeated 1 000 times and averaged to get the simulated option value.

STEP 4

The average of the 1 000 option values is 11.10. This number will change each time the spreadsheet is recalculated.

If you require assistance in valuing options or would like to find out more about The Monte Carlo Simulation, chat with one of our valuation experts today.

Contact your local Moore firm HERE.