Module 03 · AAPL stochastic paths
Apple as a process, not a point.
Geometric Brownian motion, calibrated on eight Apple annual log-returns, with σ blended against live realized vol. S0 is the last print.
Drift μ
28.3%
Annual log-return mean
Volatility σ
27.0%
blend · annual μ + live realized σ
S0
$336.64
Live tape
Paths
10,000
Stochastic paths
AAPL GBM fan — five years of Wiener noise
GBM statistics
| Metric | Year 1 | Year 3 | Year 5 |
|---|---|---|---|
| Mean | $446.67 | $785.55 | $1,370.63 |
| Median | $430.10 | $707.75 | $1,140.76 |
| P5 | $278.75 | $327.43 | $433.52 |
| P95 | $672.67 | $1,527.48 | $3,069.64 |
| P(> $300) | 91.3% | 96.5% | 98.7% |
| P(> $400) | 60.6% | 88.6% | 96.4% |
| P(< $150) | 0.0% | 0.0% | 0.1% |
GBM finish table
AAPL finish-position probabilities
Each cell is the share of simulated paths that finish a given year inside that price band.
| Horizon | $50–150 | $150–250 | $250–350 | $350–450 | $450–650 | $650–∞ |
|---|---|---|---|---|---|---|
| Year 1 | ||||||
| Year 2 | ||||||
| Year 3 | ||||||
| Year 4 | ||||||
| Year 5 |
Jump-diffusion
AAPL Merton jumps · λ = 0.3 / yr, mean jump −5%
Median comparison
Jump-diffusion versus GBM (median path)
How to read this
GBM assumes log-returns are well-behaved. Jump-diffusion adds Poisson shocks — the model’s way of admitting a regulation case, a demand air-pocket, a China weekend. The medians stay close. The fifth percentile does not. That gap is the point. Drift is the eight-year 10-K mean; volatility is blended with whatever the live session has already printed.