Learn Monte Carlo simulation of returns

Core Concept

Why this approach?

1. Reality check: Historical backtest = sample size of 1. Monte Carlo = sample size of 10,000.
2. Tail risk: Reveals the worst5% of scenarios that might not appear in your data window.
3. Robustness: If your strategy fails in 40% of simulations, it's fragile—even if the historical backtest looked good.

Mathematical Foundation

Geometric Brownian Motion (GBM)

Most Monte Carlo return simulations use Geometric Brownian Motion, the standard model for stock prices:

Decode each term:

• $S_t$ = stock price at time $t$
• $\mu$ = drift (expected annualized return, e.g., 0.08 for 8%)
• $\sigma$ = volatility (annualized standard deviation of returns, e.g., 0.20 for 20%)
• $dW_t$ = Wiener process increment (random shock from normal distribution)
• $dt$ = tiny time step

Why this form? The $S_t$ multiplier makes returns proportional: a 10% move on a $100 stock is$10, but on a $200 stock it's$20. This matches reality—stocks move in percentages, not fixed dollars.

Discrete-Time Solution

We can't simulate continuous $dt$, so we discretize. The analytical solution over one time step $\Delta t$ is:

where $Z \sim \mathcal{N}(0, 1)$ is a standard normal random variable.

Derivation from scratch:

1. Apply Itô's lema to $\ln S_t$:

   $$d(\ln S_t) = \frac{1}{S_t} dS_t - \frac{1}{2S_t^2} (dS_t)^2$$

2. Substitute the GBM equation:

   $$d(\ln S_t) = \mu \, dt + \sigma \, dW_t - \frac{\sigma^2}{2} dt = \left(\mu - \frac{\sigma^2}{2}\right) dt + \sigma \, dW_t$$

   Why the $-\sigma^2/2$ term? This is the Itô correction. Variance itself contributes to drift because $(\sigma S_t dW_t)^2 = \sigma^2 S_t^2 dt$ (quadratic variation). Without it, we'd overestimate growth.

3. Integrate from $t$ to $t + \Delta t$:

   $$\ln S_{t+\Delta t} - \ln S_t = \left(\mu - \frac{\sigma^2}{2}\right)\Delta t + \sigma (W_{t+\Delta t} - W_t)$$

4. Random walk property: $W_{t+\Delta t} - W_t \sim \mathcal{N}(0, \Delta t)$, so:

   $$W_{t+\Delta t} - W_t = \sqrt{\Delta t} \, Z, \quad Z \sim \mathcal{N}(0, 1)$$

5. Exponentiate both sides:

   $$S_{t+\Delta t} = S_t \exp\left[\left(\mu - \frac{\sigma^2}{2}\right)\Delta t + \sigma \sqrt{\Delta t} \, Z\right]$$

Units check:

• $\mu$ and $\sigma$ are annualized, $\Delta t$ is in years (e.g., 1/252 for daily).
• $\sigma \sqrt{\Delta t}$ has units of $\sqrt{\text{year}} \cdot \text{year}^{1/2} = 1$ (dimensionless).

Estimating Parameters from Data

Given historical daily returns $r_1, r_2, \ldots, r_n$:

1. Sample mean (annualized):

   $$\hat{\mu} = 252 \cdot \frac{1}{n}\sum_{i=1}^n r_i$$

2. Sample volatility (annualized):

   $$\hat{\sigma} = \sqrt{252} \cdot \sqrt{\frac{1}{n-1}\sum_{i=1}^n (r_i - \bar{r})^2}$$

Why multiply by 252? Returns scale linearly with time (adding independent increments), but volatility scales with $\sqrt{\text{time}}$ (variance adds, SD is square root of variance).

Why $n-1$ for volatility? Bessel's correction for unbiased sample variance.

Simulation Algorithm

Setup:

• Historical data: SPY daily returns, 2010–2023
• Estimated: $\hat{\mu} = 0.10$, $\hat{\sigma} = 0.18$
• Simulation horizon: 252 days (1 year)
• Number of paths: 10,000

Algorithm:

For each simulation path i = 1 to 10,000:
    1. Set S[0] = current_price (e.g., $450)
    2. For each day t = 0 to 251:
        a. Draw Z ~ N(0, 1)
        b. S[t+1] = S[t] * exp((μ - σ²/2)/252 + σ/√252 * Z)
    3. Record final value S[252] and path statistics
    
Calculate percentiles of final values:5th, 50th, 95th

Why this step? Each path is an independent "what-if" scenario. The collection shows the range of outcomes your strategy might face.

Given:

• $S_0 = 450$, $\mu = 0.10$, $\sigma = 0.18$
• Random draws: $Z_1 = -0.523$, $Z_2 = 1.246$

Day 1:

Why the negative exponent? The negative $Z_1$ means below-average return this day (bad luck).

Day 2:

Repeat for 252 days × 10,000 paths.

Interpreting Results

1. Expected terminal value: Mean of all final prices

   $$\mathbb{E}[S_T] = S_0 e^{\mu T}$$

2. Value at Risk (95% VaR): The 5th percentile loss

   • If5th percentile is $380, VaR = 450 - 380 =$70

3. Probability of profit: Fraction of paths where $S_T > S_0$

4. Maximum drawdown distribution: Percentiles of worst peak-to-trough decline across paths

Why these matter?

• Mean tells you average outcome (but averages lie—you experience ONE path, not the average).
• VaR quantifies downside: "In 19out of 20 scenarios, I won't lose more than this."
• Profit probability is intuitive for decision-makers.

Scenario: You have a momentum strategy: "Buy when 50-day MA > 200-day MA."

Historical backtest (2015–2023): 12% annual return, max drawdown 18%.

Monte Carlo validation:

1. Estimate $\mu$, $\sigma$ from historical returns.
2. Simulate 10,000 price paths.
3. Apply your strategy rules to each path.
4. Record return and max drawdown for each simulation.

Results:

• Median return: 9% (worse than historical12%—overfitting alert!)
• 5th percentile return: -15% (your strategy can blow up badly)
• In 2,300 paths (23%), max drawdown exceeded 30%

Why this step matters: The historical backtest was lucky. Monte Carlo reveals your strategy is riskier than it appeared. Now you can adjust position sizing or add stop-losses.

Common Mistakes

Why it feels right: Simple multiplication, matches intuition about "average."

Why it's wrong: Returns compound multiplicatively, not additively. The geometric mean accounts for volatility drag. If you gain 50% then lose 50%, you're down 25%, not flat.

The fix:

or use the log-returns formula from GBM directly.

Why it feels right: "Just add drift plus random shock."

Why it's wrong: This systematically overestimates price growth. The missing $-\sigma^2/2$ term accounts for the fact that variance itself acts as a drag (Jensen's inequality for exponentials).

Result: Your simulated returns will be ~$\sigma^2/2$ too high annually. For $\sigma = 0.20$, that's 2% annual overestimation—huge over time!

Why standard GBM fails: The2008 crash was a "20-sigma event" under normal assumptions—should happen once per universe lifetime. But it happened.

The fix: Use Student's t-distribution for $Z$ (heavier tails), or jump-diffusion models that add sudden crash components:

where $J_t$ is a Poisson jump process (rare, large shocks).

For stocks, we don't know the EXACT future, but we know the "game rules" from history: stocks tend to go up about 8% per year on average, but they wigle up and down about 20% along the way (that's called volatility). So we use a computer to play10,000 "pretend futures" that follow those same rules—same average, same wigliness—but each time the random wiggles are different.

Why? Because if you only look at what ACTUALLY happened (one game), you might've gotten lucky. Maybe your stock went up 20% but in8 out of 10 alternate universes it would've crashed. Monte Carlo shows you all those alternate universes so you know if your strategy is REALLY good or just got lucky.

Active Recall Flashcards

#flashcards/stock-market

What is the purpose of Monte Carlo simulation in backtesting? :: To generate thousands of plausible alternate price paths with the same statistical properties (mean, volatility) as historical data, revealing the distribution of outcomes instead of relying on a single historical backtest.

Connections

• 6.2.1-Walk-forward-analysis: Use Monte Carlo to simulate future out-of-sample periods for walk-forward testing.
• 6.2.8-Sharpe-ratio-calculation: Calculate Sharpe ratio distribution across Monte Carlo paths to assess consistency.
• 6.1.4-Maximum-drawdown-analysis: Monte Carlo generates the full drawdown distribution, not just historical max.
• 5.3.5-Value-at-Risk-VaR: VaR is directly computed from Monte Carlo percentiles.
• 4.2.3-Volatility-clusteringGARCH: Advanced models use GARCH for time-varying $\sigma_t$ in simulations.
• 7.1.2-Position-sizing-Kelly-criterion: Monte Carlo tests position sizes across many paths to optimize Kelly fraction.
• 2.1.6-Log-returns-vs-simple-returns: GBM naturally works with log-returns; ensure data preprocessing matches.

"In God we trust. All others must bring Monte Carlo simulations." — Adapted from W. Edwards Deming

Concept Map

Hinglish (regional understanding)