Think Value

The main advantage of the Monte Carlo simulation method over binomial trees is that it can be used to price options whose payoff depends not only on the underlying asset’s final price at expiration but also on the path it takes over time. For example, the payoff of Asian options is determined by the average price over a period. There are also options with periodically determined payoffs.

Additionally, compared to binomial trees, Monte Carlo is more practical when the number of variables and branches increases. For instance, if an option depends on multiple underlying assets, a binomial tree must account for all possible paths, and the computational resources required grow exponentially. In contrast, Monte Carlo achieves simulation by sampling complete paths, so the computational effort grows only linearly. The same logic applies to the number of branches.

The main limitation of Monte Carlo is that it cannot easily calculate the value of early-exercisable options, so it is less suitable for American options.

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How the Method Works

Monte Carlo simulates many possible scenarios for the development of events, summarizes the resulting random outcomes, and averages them to estimate an unknown variable or outcome.

Example – Estimating π:
Suppose we draw a circle inside a square and randomly fire many bullets at it, with each point equally likely to land inside or outside the circle. The proportion of hits inside the circle can be used to estimate the value of π. Using an Excel program, with 1,000 “shots,” we obtain an estimate of π ≈ 3.11, with a standard error of 0.1.

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Calculating an Asian Option

Suppose we have a 5-month PUT option whose payoff depends on the average stock price at the end of each month. Using the risk-neutral approach, we simulate the stock price paths, calculate the final payoff for each path, and discount it at the risk-free rate.

Intermediate calculations allow us to estimate the risk-neutral probability of an up move. In our example, it is 49%. Using Excel’s RAND() function, we convert the results into 0s and 1s, representing down and up moves. This generates one possible path showing how the stock price evolves from its initial value ($50).

Many paths can be simulated this way (see the Excel file at the end of the text).

Schematic:

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Shortcuts for Simulation

The above simulation includes only 5 steps and 20 paths. In practice, numbers are much larger, increasing computation time and resource usage. Therefore, several techniques are used to achieve accurate results with fewer calculations.

Antithetic Variable Technique:
This balances simulation outcomes to avoid simulating only one direction of paths and wasting resources on unnecessary calculations. In our example, a path generated using RAND() can be combined with a second path without additional simulation: the second path equals 1 minus the first (note: this does not always produce a perfectly symmetric opposite path, depending on the risk-neutral probability).

Control Variate Technique:
This uses the known error of a similar quantity to control the unknown variable. For example, if we have two similar European options and know how the payoff differs under fixed vs. variable volatility for one, we can calculate the second option under fixed volatility and adjust it to estimate the value under variable volatility.

Stratified Sampling:
Instead of selecting random variables freely from the full population, the probability space is divided into intervals, and samples are taken from each interval. Otherwise, important regions might be missed, and unnecessary computations might repeat in the same interval. In our example, instead of using RAND() freely (which can cause clustering), we can divide the range [0,1] into 4 segments and perform 10 iterations per segment, which sufficiently spreads the sampled values.

Moment Matching:
This adjusts selected variables to match known distribution parameters. Distribution moments may include: mean, variance, skewness, and kurtosis. For example, if the sampled variables are supposed to follow a normal distribution with mean 0 but the sample mean is 1, we shift all variables by -1 to match the mean.

The risk-neutral probability of an up move is 49%. Instead of using RAND() for each iteration, we can take 49 ups and 51 downs for each step, shuffle the sequences, and assign them to paths. This ensures the correct proportion of up and down moves and significantly reduces computation time and effort.

Quasi-Random Sequences:
This method deterministically selects variables. It resembles stratified sampling but is more flexible because it does not require dividing the probability space into fixed segments in advance. First, the initial random numbers are selected carefully, and in subsequent steps, new numbers are chosen from the selected range. This method is associated with the Soviet mathematician I.M. Sobol.

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Excel Examples: Monte Carlo π estimation & Asian option calculation

Source:
Options, Futures & Other Derivatives, John C. Hull

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