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Monte Carlo Simulation

How Tradient simulates 10,000 stock-price paths to evaluate a strategy's full P&L distribution.

8 min readmonte carlosimulationmath

Closed-form formulas like Black-Scholes give you a single number — fair price, POP, expected move. But a real options position has a distribution of outcomes, not a point estimate. Tradient’s Focus mode uses Monte Carlo simulation to generate that full distribution and show you the percentiles you actually care about.

What Monte Carlo actually does

We simulate 10,000 possible terminal stock prices using geometric Brownian motion (the same process that underlies Black-Scholes), price the strategy’s P&L at each of those terminal prices, and report statistics over the resulting distribution.

The terminal price formula:

  • S_T = S_0 · exp((r − 0.5σ²)·T + σ·√T·Z)

where Z is a standard normal random number. Sample Z 10,000 times, you get 10,000 possible S_T values; each one has a corresponding P&L for the position; average them and you get an expected value, sort them and you get percentiles.

Why bother when we have closed-form POP?

POP only tells you the probability of finishing in the green. Monte Carlo tells you:

  • Expected P&L — the actual mean of the distribution, not a back-of-envelope EV.
  • P&L percentiles— what does the 5th percentile outcome look like? The 95th? Where’s the median?
  • Tail shape — how fat are the losing tails compared to the winning ones? An iron condor has a thin win zone and fat losing tails; a long straddle is the opposite.
  • Distribution histogram — actual visual of where outcomes cluster.

For a bull put spread, POP and Monte Carlo agree. For a complex multi-leg trade or a near-binary outcome, the simulation reveals shape that POP alone hides.

Worked example

SPY at $445. Iron condor: short 430 put, long 425 put, short 460 call, long 465 call. 35 DTE, IV 18%.

  • Net credit: $0.85 → $85 max profit
  • Max loss: $415
  • Closed-form POP: 72%

Monte Carlo (10,000 paths) returns:

  • Expected P&L: −$15
  • P(profit): 71%
  • 5th percentile P&L: −$415
  • 25th percentile: +$85
  • 50th percentile (median): +$85
  • 75th percentile: +$85
  • 95th percentile: +$85

Read the percentiles carefully. 75% of paths end at the max profit — the win zone is wide. But the 5th percentile is the full max loss, and that loss is large enough to drag the overall expected value below zero. This is the income-trader trap visualized: the distribution is “mostly happy with a rare disaster.”

Reading the histogram

Focus mode displays a histogram of the simulated P&L distribution. Look for:

  • A tall spike at max profit — typical of credit strategies where most paths land inside the breakevens.
  • A long left tail — defined-risk credit strategies stop at max loss; undefined-risk strategies (short strangles) keep going.
  • Bimodal shape — long straddles often show two humps, one for the up-move payoff zone and one for the down-move payoff zone, with a dip at the strike.
Same model, same limits
Monte Carlo uses the same log-normal assumption as Black-Scholes. It does not produce fat tails. Real markets do. A 5th-percentile loss in the simulation is not a true worst case — it’s the worst case under the model. Treat it as a lower bound on bad outcomes, not the bad outcome.

Implementation details

A few notes on how Tradient runs the simulation:

  • 10,000 paths by default. Higher counts give marginal precision improvement at much higher compute cost.
  • Seeded RNG for reproducibility. Running Focus mode twice on the same trade gives you the same distribution.
  • Single time step.We jump straight to expiration rather than simulating intermediate days. For European-style payoffs this is correct; for early-exit analysis you’d want a multi-step simulation.
  • Constant IV. Same simplification as Black-Scholes. A path-dependent vol model would be more accurate but much slower and not meaningfully better at this scale.

When Monte Carlo lies

  • Earnings.The implied move on earnings is not log-normal; it’s closer to bimodal. Monte Carlo will smooth it into a single distribution and understate the chance of a big surprise either direction.
  • Vol regime shifts.If IV doubles mid-trade, your actual P&L will diverge from the simulation. Focus mode does not re-run the simulation under stressed vol — yet.
  • Liquidity events.Halts, gaps, hard opens. None of this is in the model. Use position sizing to handle what the model doesn’t.

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