Monte Carlo From the Implied Density

In the last chapter you did something quietly remarkable. You took a butterfly-arbitrage-free volatility slice, applied Breeden and Litzenberger, and pulled out the full risk-neutral density — the market's own probability distribution over where the asset can land at expiry. You now hold, as a curve, the answer to the question this whole institute keeps circling: how likely does the option market think each future price is? The natural next move is to use that density. And the most powerful way to use a distribution you cannot write down in closed form is to draw from it. That is what this chapter teaches, and it is exactly what sits underneath the fan of forecast paths you see in SVQ.

Let us build the intuition before the mechanics. Suppose you wanted to know the probability that the asset finishes a trade above some level, or the expected payoff of a strange path-dependent structure, or simply what a believable spread of outcomes looks like. If you had a clean formula for the terminal distribution — say, the lognormal that Black and Scholes assume — you could often answer in closed form. But you learned two chapters ago that the real implied density is not lognormal. It has a fatter left tail, it is skewed, sometimes it is visibly bimodal around an earnings date. There is no tidy formula. So instead of solving, you sample: you draw thousands of terminal prices in proportion to how likely the market says each one is, and you let the histogram of your draws become the answer. That is Monte Carlo, and the only thing that makes it honest is whether you are drawing from the right distribution. Draw from a lognormal and you will get a confident, smooth, and wrong picture. Draw from the implied RND and you are letting the market's actual prices speak.

So how do you draw a single number from an arbitrary density you only have as a curve? The workhorse is inverse-transform sampling, and it rests on one beautiful fact. Take your risk-neutral density and integrate it from the bottom up; that gives you the cumulative distribution function, the CDF, which climbs from zero to one as you sweep across price. Now here is the trick. If you generate a uniform random number between zero and one — call it u — and you ask "at what price does the CDF first reach u?", the price you get back is a genuine draw from your density. Feed in a u of one-half and you land on the median. Feed in a u of zero-point-nine-five and you land at the ninety-fifth percentile. Because the CDF spends most of its vertical rise over the price regions where the density is tall, a uniform stream of u-values gets funneled, automatically, into exactly those high-probability regions. You are inverting the CDF: u goes in, a price comes out. Repeat ten thousand times and you have ten thousand terminal prices distributed precisely as the market implies.

Concretely, you take the RND from chapter four, accumulate it into the CDF on a fine price grid, and then for each draw you generate u and read back the price by interpolating where the CDF equals u. That single operation — uniform in, inverse-CDF out — is the whole sampling engine.

Now we must be careful, because a terminal price is not yet a path, and a naive simulation breaks a law it must not break. Under the risk-neutral measure, the discounted asset price is a martingale: the expected terminal price, discounted back at the risk-free rate, must equal today's price. Equivalently, the mean of your draws should sit right on the forward — today's price grown at the risk-free rate to expiry. A correctly extracted RND already integrates to a mean of the forward, but a finite sample of, say, ten thousand draws will miss it slightly by pure luck, and an interpolated tail can introduce a small bias. So SVQ applies a martingale correction: it rescales or recenters the draws so their sample mean lands exactly on the forward. This is not cosmetic. If your simulated mean drifts off the forward, every probability and every expected payoff you compute from the fan inherits a free, fake directional tilt — and a fake tilt is the enemy of treating a number as evidence.

To turn terminal draws into the paths you actually see, SVQ interpolates each draw back toward today through a Brownian bridge whose endpoint is the sampled price, so the intermediate volatility is consistent with the term structure of the surface. Many expiries, many bridges, and you get the smooth, widening cone of trajectories on the screen.