Quantitative financeEngineering methods

Stochastic & quant finance

Simulation driven by randomness — Brownian motion, stochastic differential equations and Monte Carlo. Option prices that converge to Black–Scholes, variance reduction, path-dependent exotics, the Greeks, Value-at-Risk, and a volatility smile: the stochastic-calculus paradigm nothing else in the portfolio touches.

DjiniousLabModelling & simulation

A Monte-Carlo option-price estimate converging onto the Black–Scholes line as the number of simulated paths grows, with a shrinking confidence band

Monte-Carlo vs Black–Scholes price: <0.5%Monte-Carlo vs Black–Scholes price
convergence rate (log–log slope): −0.50convergence rate (log–log slope)
variance reduction (fewer paths, same accuracy): 2.6×variance reduction (fewer paths, same accuracy)
put–call parity & Greeks match: exactput–call parity & Greeks match
design notebooks: 10design notebooks
requirements PASS: 5 / 5requirements PASS

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.

An ensemble of geometric-Brownian-motion stock-price paths fanning out from a common start — Geometric Brownian motion: eighty simulated stock-price paths fanning out from today's price, their spread set by volatility and their centre by drift. The terminal prices form a lognormal distribution — the ensemble of futures every option price is an average over.

A Monte-Carlo call-price estimate converging onto the Black–Scholes line as paths increase — The hero: price a European call by averaging its discounted payoff over simulated paths, and the Monte-Carlo estimate converges onto the exact Black–Scholes price as the path count grows, its 95% confidence band shrinking. Match the formula where one exists — then trust the method where none does.

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.

Monte-Carlo prices of vanilla, Asian and barrier options compared in a bar chart — Where Monte Carlo earns its keep: path-dependent options — an Asian (average-price) and an up-and-out barrier — have no simple closed form, so simulation is the natural pricer. Priced against the vanilla call, each reflects its payoff rule: averaging and knock-out both cheapen the option.

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.

A portfolio profit-and-loss distribution with the loss tail shaded and VaR and Expected Shortfall marked — Tail risk: a portfolio's simulated profit-and-loss distribution with the 5% loss tail shaded — Value-at-Risk marks the threshold, Expected Shortfall the average loss beyond it. The same stochastic engine that prices an option measures how much a book could lose.

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.

A Heston implied-volatility curve skewing across strikes against a flat Black–Scholes line — The volatility smile: a Heston stochastic-volatility model produces implied volatility that varies with strike (here a downward skew), against the single flat line constant-vol Black–Scholes assumes. Letting volatility be random recovers a market feature the textbook model structurally 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

MC call vs Black–Scholes: 10.50 vs 10.45
Convergence rate (log–log): −0.503
Variance reduction (control variate): 2.6×
Put–call parity / delta: exact / <0.01%
VaR95 / ES95 ($1M book): $86k / $108k

Requirement

MC call vs Black–Scholes: R-01 ≤ 2·SE
Convergence rate (log–log): R-02 ≈ −½
Variance reduction (control variate): R-03 ≥ 2×
Put–call parity / delta: R-04
VaR95 / ES95 ($1M book): 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.