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.

A closed-form solution gives an exact answer through standard operations, with no iteration. An analytic pricing formula is a closed-form solution for a security’s price; shares and plain bonds usually have one, many complex securities do not.

Use simulation 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 uses the log-normal price model from Learning Module 6: draw normal log returns, turn them into prices, and see where the prices end up. Because you control the assumptions, you can change one and see how the result moves.

On the first tab, price a call: run 10 trials, then 10,000, and watch the estimate settle on the formula value. The second tab turns the same engine on a retirement plan.

What are the four steps, for Monte Carlo?

Memorize

Monte Carlo follows the four steps of every simulation, and nothing is drawn until the setup is done. One pass through steps 2 and 3 is a trial (one path).

Monte Carlo simulationthe same four steps as every simulation

• 1. Setup payoff, horizon, distribution and parameters: price, volatility, risk-free rate
• 2. Scenario generation draw Z from a standard normal; turn it into a log return
• 3. Scenario evaluation build the price path, compute the payoff
• 4. Results average the payoffs, discount at the risk-free rate

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 produces draws that behave as random. A uniform draw is turned into a standard normal with the inverse normal function, NORM.INV(RAND(), 0, 1) in a spreadsheet.

How is one price step simulated?

Use it

To value an option you simulate the share in a risk-neutral world. Each step adds a log return $\tilde r$ to the log price, where $Z$ is a standard normal draw, $\sigma$ is the volatility and $\Delta t$ the step length in years.

$$\tilde r_t = \left(r_f - \tfrac{\sigma^2}{2}\right)\Delta t + \sigma\sqrt{\Delta t}\,Z, \qquad P_t = P_{t-1}\,e^{\tilde r_t}$$

The first term is the risk-neutral drift : the risk-free rate less half the variance, the log-return drift that makes the expected price grow at the risk-free rate. The second is the random shock, with $\sigma$ scaled to the step by $\sqrt{\Delta t}$. This price model is called geometric Brownian motion (Learning Module 6).

Why does the risk-free rate appear twice?

Option values come from a riskless hedge, so the simulation is run in a risk-neutral world. The share is grown at the risk-free rate (drift $r_f - \sigma^2/2$ in log terms, not zero and not the share’s own expected return), and the average payoff is discounted at the same rate, $e^{-r_f T}$. Every simulated path is treated as equally likely.

One trial means little: many trials pay nothing. Averaging the discounted payoffs over many trials (our run, same inputs) gives 6.54 for 10 trials, 8.32 for 100, 7.77 for 1,000 and 7.35 for 10,000. The closed-form value from the Black–Scholes–Merton (BSM) formula is 7.35. As trials increase, the simulation settles on it.

Exam tip Growing the share at its expected return of 10% instead of 4% in the example above gives about 9.60, far above the 7.35 a market would pay.

How does it value a path-dependent option?

A contingent claim is a security whose value depends on another asset, such as an option. A path-dependent option pays according to the prices along the way, not only the final price. An Asian option pays the excess of the final price over the average price during its life, if positive. Its payoff depends on the whole path, so no simple formula exists; this is where Monte Carlo is essential.

$$V_T = \max\left(P_T - \bar P,\, 0\right)$$

A path ending below its average pays 0.

One path proves nothing: this one is worth 7.37 today, the next may pay zero. The value is the discounted average over thousands of paths.

$$\hat V_0 = e^{-r_f T}\,\frac{1}{N}\sum_{i=1}^{N}\text{payoff}_i$$

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, discounted once, is the value.

How many trials are enough?

More trials give a more precise estimate, at the cost of run time. How many is a trade-off: a quick one-step valuation may settle with thousands of trials, a complex path-dependent book can need far more. Precision never fixes a wrong model: if the assumed distribution is wrong, more trials estimate the wrong value more precisely.

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

Monte Carlo simulationa complement to analytical methods

• Strengths Memorize
  • Wide applicability non-linear payoffs, fat tails, changing volatility
  • Correlation handling joint moves of many assets, lesson 5
  • Forward looking scenarios history never produced
  • Precision with enough trials
  • Non-normal distributions skew and kurtosis can be modeled
• Limits Memorize
  • Computationally intensive large runs are slow and costly
  • Model dependency wrong distributions or correlations, wrong results
  • Complexity needs modeling expertise
  • Parameter estimates volatility and correlation must come from somewhere
  • Less insight into cause and effect a formula shows why; a simulation shows what

Typical products are path-dependent options (such as Asian options), American-style options, convertible bonds with embedded options, and mortgage-backed securities, where the borrowers’ right to prepay makes the cash flows depend on the path of interest rates (Fixed Income). Simulating several assets whose returns move together is the subject of lesson 5.

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

The main danger is model risk . Assuming constant volatility and log-normal prices for a share that shows volatility clustering or sudden jumps gives misleading values. Rerunning with changed assumptions is a robustness check.

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 biggest weakness is dependence on the assumed model.

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 assume distributions, so the result is bounded by the quality of those assumptions. Historical simulation and bootstrapping (lessons 2 and 3) avoid that by drawing on observed data, at the price of never going beyond it.