Simulation of Financial Asset Prices and Returns

Many investment questions have no neat formula: how bad could next month's loss be, how uncertain is an estimated Sharpe ratio, how likely is a retirement plan to run out of money, what is a path-dependent option worth? Simulation answers them by generating many possible outcomes. This module covers the three main approaches, historical simulation, bootstrap resampling and Monte Carlo simulation, how each is used in practice, and how to choose between them.

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What you will learn

1. Describe historical simulation, explain its use in investment applications such as value at risk and stress testing, and calculate risk measures from historical scenarios.

Historical simulation applies past risk-factor changes to today's portfolio, one scenario per historical period.
No distributional assumption: fat tails, skew and crisis correlations are captured as they occurred.
VaR is a low percentile of the simulated outcomes; expected shortfall is the average loss beyond it.
Used for VaR, stress tests that replay past crises, and testing withdrawal rules.
Cannot produce losses worse than history and depends heavily on the look-back window.

2. Describe bootstrap resampling and explain how it is used to estimate the precision of statistics and to simulate investment outcomes.

Bootstrap: resample the observed data with replacement, recompute the statistic, repeat many times.
Bootstrap standard error = standard deviation of the resampled statistics (divide by B - 1).
Percentiles of the resampled statistics give a confidence interval without a formula.
Resampling historical returns builds many wealth paths that keep the empirical distribution's shape.
It cannot generate values outside the sample and assumes independence unless blocks are used; the jackknife leaves one observation out at a time.

3. Describe Monte Carlo simulation, explain how it is used to value instruments and assess risk, and generate simulated asset prices from random draws.

Monte Carlo generates scenarios from an assumed model and random draws, not from the historical record.
Steps: specify model and parameters, simulate paths, compute outcomes, repeat many times, summarize.
Lognormal step: S(t + dt) = S(t) exp[(mu - sigma^2 / 2) dt + sigma x square root of dt x z].
For valuation, use the risk-free drift and discount the average payoff at the risk-free rate.
Flexible for path-dependent and multi-factor problems, but subject to model risk and sampling error.

4. Compare historical simulation, bootstrap resampling and Monte Carlo simulation, and select the most appropriate method for a given investment problem.

Historical: actual past sequence; bootstrap: resampled past observations; Monte Carlo: draws from an assumed model.
Only Monte Carlo can generate single-period outcomes beyond the observed data.
Monte Carlo suits new or path-dependent instruments; historical scenarios are the most transparent.
Simulation standard error = standard deviation of outcomes / square root of N; halving it needs four times the trials.
All methods carry model risk: changing the method or window can change reported risk sharply.

Module at a glance

Historical simulation replays actual past risk-factor changes on today's portfolio; no distribution is assumed.
VaR is a percentile of simulated outcomes; expected shortfall averages the losses beyond it.
Historical simulation cannot produce losses worse than its look-back window contains.
The bootstrap resamples the data with replacement; the spread of resampled statistics estimates their standard error.
Monte Carlo draws from an assumed model: S(t + dt) = S(t) exp[(mu - sigma^2 / 2) dt + sigma x square root of dt x z].
Value derivatives by simulating with the risk-free drift and discounting the average payoff.
Simulation error falls with the square root of the number of trials; all methods carry model risk.

This free module has 4 lessons, 12 practice questions with full explanations.

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