When the future is a distribution, you simulate it.
Every other model in this portfolio is deterministic: same inputs, same answer. But a stock price, a portfolio's loss, an option's payoff — these are random, and you can't integrate your way to them. You simulate thousands of possible futures and average. This program builds that stochastic paradigm — Brownian motion, stochastic differential equations, Monte Carlo — on its canonical application, quantitative finance: pricing options and measuring risk, anchored to the one case where a closed-form answer exists so the simulation can be trusted everywhere it doesn't.
A stock price is a random walk with drift.
Geometric Brownian motion — the foundation of quantitative finance — is a stochastic differential equation: a deterministic drift plus a random kick scaled by volatility, integrated with the Euler–Maruyama scheme. Simulate it many times and the paths fan out into a cone, their endpoints forming the lognormal distribution that underlies option pricing. This is the raw material: not one trajectory, but an ensemble of possible futures.
Accuracy costs paths — unless you're clever.
Monte-Carlo error shrinks only as 1/√N, so each extra digit of accuracy costs a hundred times the paths — the central tax of stochastic simulation. Variance-reduction techniques beat it: antithetic variates and control variates reach the same accuracy with a fraction of the paths (a 2.6× standard-error cut here, worth nearly 7× the work). Knowing these tricks is the difference between a Monte-Carlo pricer that's usable and one that isn't.
The same engine measures how much you could lose.
Turn the simulation on a portfolio instead of a single option and it produces a distribution of profit and loss — and from its tail, the risk numbers regulators and risk desks live by: Value-at-Risk (the loss you won't exceed 95% or 99% of the time) and Expected Shortfall (the average loss when you do). The program computes both, and contrasts a Gaussian Monte-Carlo tail against a fat-tailed historical one — the difference that decides whether a risk model sees the crash coming.
Real markets don't have one volatility.
Black–Scholes assumes a single constant volatility; real option markets don't, and the evidence is the volatility smile — implied volatility that varies with strike. A stochastic-volatility model (Heston), where volatility is itself a random process correlated with price, reproduces it: simulate the coupled processes and the implied-volatility curve skews and the return distribution grows a fat left tail that flat Black–Scholes misses. Stochastic simulation reaching what closed forms can't.
Every number re-derived at sign-off.
The V&V notebook rebuilds each estimator from scratch with fixed seeds and re-derives the requirements against theory, printing a PASS/FAIL board.
| Result | Requirement | |
|---|---|---|
| MC call vs Black–Scholes | 10.50 vs 10.45 | R-01 ≤ 2·SE |
| Convergence rate (log–log) | −0.503 | R-02 ≈ −½ |
| Variance reduction (control variate) | 2.6× | R-03 ≥ 2× |
| Put–call parity / delta | exact / <0.01% | R-04 |
| VaR95 / ES95 ($1M book) | $86k / $108k | R-05 |
A stochastic paradigm, anchored to theory.
The deliverable is ten notebooks and the dossier — stochastic paths and Monte-Carlo estimators, not an acausal network, so there's no custom block or canvas. The SDEs use the Euler–Maruyama scheme (not higher-order or exact schemes), the base process is constant-parameter geometric Brownian motion, pricing is risk-neutral, path counts are modest for notebook speed, and nothing is calibrated to real market data — the risk numbers are illustrative, not production. What the program proves is the paradigm itself: that the platform simulates stochastic systems, not just deterministic ones — validated against Black–Scholes where a closed form exists, and reaching path-dependent options, tail risk, and stochastic volatility where none does.
Simulate your own stochastic system.
Book a walkthrough and we'll set up your process, your payoff or your portfolio, and run the Monte-Carlo pricing and risk studies live.