Monte Carlo simulation

How do you value or measure something no formula covers?

Simulate it. Monte Carlo simulation draws thousands of random scenarios from probability distributions you choose, works out the result in each one, and reads off how often each kind of outcome occurs.

Use it when the answer depends on a whole path of prices, when a security has complex embedded options, or when you want the distribution of a portfolio’s gains and losses. It fits the lognormal price model from the previous lesson: draw normal returns, convert them to prices, and see where the prices end up.

Run it with few trials and then many. With ten paths the answer jumps around, with a thousand it settles.

What are the steps?

Memorize

Six steps: three to specify the simulation, three to run it.

Monte Carlo simulationspecify it, then run it

• Specify the simulation Memorize steps 1 to 3
  • 1. Quantity of interest what to value or measure, and the starting values
  • 2. Time grid K sub-periods of length $\Delta t$
  • 3. Model and distributions a distribution for each risk factor, with parameters
• Run the simulation Memorize steps 4 to 6
  • 4. Draw and convert random numbers into prices or paths
  • 5. Compute the outcome payoff or balance, discounted if valuing
  • 6. Repeat, then summarize N trials; average or read the distribution

The thing varying from path to path is a risk factor , an uncertain input such as a share’s return. Each gets a probability distribution and parameters. Computers produce the draws by random number generation : a program creates numbers that behave as random draws from a chosen distribution, usually starting from a standard normal.

A common model for a stock moves the price by an expected drift plus a random shock, where $Z$ is a standard normal draw and $\sigma$ is the volatility for the step.

$$\Delta S = \mu\, S_{prev}\,\Delta t + \sigma\, S_{prev}\, Z$$

How does it value a contingent claim?

A contingent claim is a security whose value depends on another asset, such as an option. Take a claim that pays the excess of a share’s final price over its average price during the year. No simple formula exists, because the payoff depends on the whole path. So you simulate.

One simulation trial is one complete pass through steps 4 and 5: a full path of prices, then the payoff and its present value.

$$C_T = \max\left(S_T - \bar{S},\, 0\right), \qquad C_0 = \frac{C_T}{(1+r)^{t}}$$

That is one trial. Repeat steps 4 and 5 for thousands of trials, then average the present values. The average is the Monte Carlo estimate of the claim’s value.

$$\hat{V} = \frac{1}{N}\sum_{i=1}^{N} C_{i,0}$$

Many trials pay zero (the final price at or below the average) and a few pay a lot. Only the average of all of them is the value.

The same machinery handles other claims. A lookback claim pays the final price minus the lowest price on the path: record the minimum instead of the average. With a final price of 61.00 and a path minimum of 54.20, the payoff is 6.80.

How many trials are enough?

More trials make the estimate more precise. The standard error of the estimate falls with the square root of the number of trials, so each halving of the error costs four times as many trials.

$$SE = \frac{s}{\sqrt{N}}$$

This is sampling error only. If the assumed distribution is wrong, ten million trials estimate the wrong thing very precisely.

What is it used for, and where does it fall short?

Monte Carlo simulationa complement to analytical methods

• Uses Memorize
  • Valuing complex claims path-dependent, embedded options, no formula
  • Risk and return distribution of portfolio gains and losses, VaR
  • Planning chance a retirement pot or liability plan fails
  • Sensitivity analysis change an assumption, rerun, compare
• Limits Memorize
  • Depends on assumptions wrong distributions, wrong answers
  • Statistical, not exact an estimate with an error, not a proof
  • No cause and effect shows what happens, not why
  • Heavy to run needs computing power

A natural use is sensitivity analysis : because you control the assumptions, you can change one (a volatility, a correlation, a distribution), rerun and see how much the answer moves.

Monte Carlo complements analytical methods rather than replacing them. It gives statistical estimates, not exact results, and a formula, where one exists, shows cause and effect that a simulation does not.

Exam tip Monte Carlo answers “what could happen, and how likely?” It does not answer “why?”. Its single biggest weakness is the assumed distributions.

Go deeper Pseudo-random numbers and seeds

Computers produce pseudo-random numbers with a deterministic recipe started from a seed. The same seed gives the same sequence, which is useful for checking a model. A different seed gives slightly different results, which is why two runs of the same simulation never match exactly. The gap between runs is the standard error at work.

Go deeper Simulate only where you must

If a closed-form formula exists, use it: it is exact, instant and shows how the answer depends on each input. Simulation earns its place when payoffs depend on the path, when several risk factors interact, or when the instrument has embedded options. The cost is that you must make distributional assumptions, so the quality of the result is bounded by the quality of those assumptions. The next lesson shows an alternative that draws from observed data instead.