Bootstrapping: simulating from the data you have

How can you simulate without assuming a distribution?

Use the data you already have. Bootstrap resampling pretends the sample you hold is the whole population and draws from it at random. You never see the true population, but a random sample should resemble it, so repeated draws from the sample play the part of repeated draws from the population. Each data point has the same chance, 1/n, of being picked on every draw, because it goes back into the pool each time.

Monte Carlo asks you to choose a distribution. Bootstrapping skips that choice: the empirical distribution , the pattern of values actually observed, is the distribution.

Historical simulation also uses past data, but differently: it replays each date once, as it happened (lesson 2). Bootstrapping redraws the data at random, so the same observation can appear twice and others not at all.

Draw months to build a path, then run a thousand paths and compare the bootstrap histogram with the normal curve. The bootstrap keeps whatever skew the data had. On the second tab, run 2,000 resamples to see how uncertain a volatility estimate from 24 months really is.

How does it work?

Memorize

Bootstrap simulation runs the four steps of every simulation. Only step 2 is special: draw one observation at random, record it, put it back, and repeat.

Bootstrap simulationresample the data you have

• 1. Setup collect the sample; define the output and its formula
• 2. Scenario generation
  • Draw a random position from 1 to n Memorize RANDBETWEEN in a spreadsheet
  • Look up that observation
  • Put it back Memorize with replacement, probability 1/n every draw
  • Repeat for every step of the path
• 3. Scenario evaluation sum the log returns, price, payoff or statistic
• 4. Results thousands of paths, then mean, spread, percentiles

The rule that matters is sampling with replacement : each item drawn goes back into the pool before the next draw, so it can be drawn again. Drawing a new sample from the one you hold in this way is resampling : in a resample the same size as the original, some items appear several times and others not at all.

Because every draw faces the same n observations, each one has the same selection probability 1/n on every draw, whatever was drawn before. Take 50 returns, 10 of them above +3%. The chance that a draw is above +3% is 10 ÷ 50 = 20%, even if the last three draws were all above +3%. Taking the earlier draws out of the count (7 ÷ 47 = 14.89%) is the without-replacement trap.

To simulate a path, number the observed returns, draw a position at random, copy that return into the path, put it back, and repeat once per step. A path’s log return is the sum of its draws, and the end price is the start price times e raised to that sum.

$$P_T = P_0\, e^{\sum_{i=1}^{k} \tilde r_i}$$

That average is not a fair price for the option. The draws carry whatever drift the share had in the past, while option pricing grows the share at the risk-free rate (lesson 4). With a strong past uptrend, the bootstrapped value comes out well above the closed-form value.

The simulated end prices spread fairly symmetrically, but the payoffs are heavily skewed: many paths pay zero and a few pay a lot, because the payoff is not linear in the price.

The time grid must match the data. Monthly observations feed monthly steps, so a year of simulated returns needs 12 draws. Everything after the draws is identical to Monte Carlo: compute the outcome on each path, repeat thousands of times, then summarize. Only the source of the random numbers differs.

Bootstrapping is also a tool for statistics. Draw many resamples of the same size as the data, compute the statistic in each (a mean, variance, skewness, kurtosis) and look at how it varies. That gives the sampling distribution of a statistic without an analytical formula.

How does bootstrapping estimate a standard error or a confidence interval?

Draw a resample of the same size as the data, with replacement. Compute the statistic you care about (a mean, a median, a volatility, a correlation). Repeat thousands of times. The standard deviation of those statistics is the bootstrap standard error . The 2.5th and 97.5th percentiles of them give a 95% interval, a bootstrap percentile confidence interval .

In a spreadsheet, =INDEX(B:B, RANDBETWEEN(2, N)) makes one draw, STDEV() gives each resample’s volatility, SMALL() and LARGE() read the percentiles, and CONFIDENCE.NORM() gives a normal-theory margin. Confidence intervals built from a formula are in Learning Module 7.

Strengths and weaknesses

A bootstrap path is built only from what has been seen. With log returns between −8% and +15%, a three-draw path can fall at most 3 × 8% = 24% in log terms (the price ends at e^−0.24 = 78.7% of its start) and rise at most 45% (e^0.45 = 156.8%). No bootstrap path can leave that range, however many are run.

So a calm sample gives calm results. Ten quiet years contain no crash, and a bootstrap from them cannot invent one. Monte Carlo can, if its distribution allows it.

Bootstrappingredraw the observed data

• Strengths Memorize
  • Minimal assumptions no distribution needed
  • Works with small samples reliable variability estimates from limited data
  • Simple no analytical solution needed
  • Standard errors and confidence intervals for almost any statistic
  • Dilutes outliers compared with historical simulation, an extreme point is drawn only now and then
  • No large external data set works from the sample at hand
• Weaknesses Memorize
  • Computationally intensive thousands of resamples
  • Sensitive to sample quality a biased sample gives biased results
  • Breaks time links not ideal for autocorrelated data
  • Limited extrapolation nothing beyond the observed range
  • No analytical insight estimates, not explanations

	Historical	Bootstrap	Monte Carlo
Source of draws	the past data as it happened	the observed sample	a distribution you specify
Replacement	no	yes	not applicable
Assumes	past = future	sample = population	process can be modeled
Unseen extremes	no	no	yes
Typical use	risk management, stress tests	standard errors, confidence intervals	complex securities, what-if analysis
Main risk	window not representative	sample not representative	wrong model

Where the draws come fromsame four steps, different source of scenarios

• Historical each past date once, no replacement
• Bootstrap Memorize observed data, with replacement
• Monte Carlo an assumed distribution

Like Monte Carlo, bootstrapping gives statistical estimates rather than exact results and complements analytical methods.

Exam tip Draws from a specified distribution: Monte Carlo. Draws from observed data, with replacement: bootstrap.

Go deeper Why plain bootstrapping breaks time patterns

Each draw is independent, so any link between one observation and the next (autocorrelation, volatility clustering) is lost. Specialized versions resample whole blocks of consecutive observations; they are not examined at this level.