Monte Carlo VaR

An options book has payoffs that bend sharply near the strike; neither a single volatility number nor a fixed historical window captures that curvature well. What you actually want is thousands of "what if tomorrow looked like this" scenarios, each one drawn from a model you trust, with every option in the book actually repriced, not approximated, under each scenario. That is Monte Carlo VaR: simulate the world, then reprice the real portfolio in each simulated world.

The analogy before any symbols

A civil engineer testing a bridge design against wind does not wait for a thousand real storms to happen. She builds a wind model consistent with known physics and historical patterns, generates thousands of plausible gust sequences from it, and runs the actual bridge design through each one, checking whether it holds. The bridge is real, tested exactly as built; only the storms are synthetic. Monte Carlo VaR does the same: your portfolio is priced exactly, with every option's real payoff function; only the market scenarios feeding into it are synthetic, drawn from a model of how prices might move.

The mechanics

Three steps. First, choose a model for how risk factors evolve, typically correlated normal (or fatter-tailed) shocks to each factor, calibrated from historical volatilities and correlations: $\Delta f \sim N(0, \Sigma)$ or a richer process. Second, simulate $M$ independent scenarios, each one a vector of factor moves drawn from that model, for $j = 1, \ldots, M$:

$\Delta f^{(j)} \sim \text{model}, \qquad L_j = V\big(f_{\text{today}} + \Delta f^{(j)}\big) - V(f_{\text{today}}).$

In words: draw a random market move consistent with the assumed model, then fully reprice the actual portfolio (options included, using their real pricing formulas) under that moved market, and record the profit or loss. Third, sort and count, exactly as in Historical Simulation VaR: VaR at confidence $\alpha$ is the $\lceil(1-\alpha)M\rceil$-th worst simulated outcome. The only structural difference from historical simulation is where the scenarios come from, a fitted model instead of the literal historical record, which means you can generate far more scenarios than history provides, and scenarios more extreme than anything that has actually happened yet.

Each path here is one simulated future; a Monte Carlo VaR engine generates thousands like it, reads off tomorrow's endpoint from each, and reprices the portfolio there. More paths means a smoother tail estimate, at the cost of more computation.

Worked example: a single call option

A book holds one call option, strike $100, current spot $100, 30 days to expiry, implied volatility 25%. Simulate daily returns as normal with mean zero and the same 25% annualized volatility (about 1.55% daily). Draw 10,000 one-day scenarios for spot, reprice the call under Black-Scholes at each new spot with 29 days left. Because the option's value is convex in spot (it loses less than proportionally on down moves near the money and gains more than proportionally on up moves), the resulting P&L distribution is skewed, unlike the symmetric spot-return distribution that generated it. Sorting the 10,000 repriced outcomes and taking the 100th worst (99% confidence) might give a VaR of roughly $3.80 per share of $6.50 option premium, a number a linear variance-covariance approximation, which ignores this convexity, would get wrong by 15 to 20%.

Worked example: correlated multi-asset book

A book holds $5m of Asset A and a short $3m of Asset B, correlation 0.6, volatilities 18% and 22% respectively. Simulate 5,000 joint scenarios by drawing correlated normal shocks (via a Cholesky decomposition of the 2×2 covariance matrix), compute portfolio P&L under each, and sort. If the true 99% VaR from a variance-covariance formula is $412,000, the Monte Carlo estimate with 5,000 paths typically lands within a few percent, say $395,000 to $430,000, purely from simulation noise; that spread narrows as more paths are added, roughly proportional to $1/\sqrt{M}$, which is why fixing the number of paths is itself a modelling choice: fewer paths means a noisier VaR estimate even with a perfectly correct model.

What this means in practice

Monte Carlo VaR is the standard for books with meaningful nonlinearity, exotic options, path-dependent structures, or convertible bonds, precisely because it reprices the real payoff rather than approximating it with sensitivities. The cost is computational: a large book with expensive-to-price instruments can take real time per scenario, which is why practitioners often combine full Monte Carlo with faster proxy methods like Delta-Gamma VaR for Option Books for intraday checks and reserve full revaluation for end-of-day.