Stress Testing & Scenario Analysis

Everything you have built so far in this course measures risk by looking backward and asking a statistical question. Value-at-Risk hands you a percentile of a distribution you fitted to history. It is a fine summary, but it has a quiet, dangerous assumption baked into it: that tomorrow's distribution looks like the one you estimated. Stress testing exists because that assumption fails precisely when you most need it to hold.

So let us change the question entirely.

A stress test does not ask, "What is the loss I will exceed only one percent of the time?" It asks, "If this specific thing happens, what happens to me?" You name the thing — spot drops twelve percent, the curve steepens, credit spreads double — and you trace that named shock all the way through to your profit and loss. No probability is attached. That is the point. You are not claiming the scenario is likely. You are refusing to let likelihood hide an exposure from you. A one-percent-probability event still wipes out a desk that was unknowingly leveraged into it, and the percentile never warned them because the percentile averaged that event away.

Here is the intuition that ties this back to the throughline of the whole institute. All term we have asked whether a number is genuine evidence that an asset will move. A VaR figure is weak evidence — it is a smoothed, backward-looking summary that washes out structure. A stress test is a conditional statement, and conditional statements are where evidence actually lives. "Given spot falls ten percent, my book loses this much" is a sharp, falsifiable claim about your own exposure. It survives contact with reality in a way that a tail percentile does not.

Now the part most people get wrong, and the part this chapter is really about.

The naive stress test shifts the input and re-prices at the old parameters. You drop spot by ten percent, plug the new spot into your pricer, and read off the new value while leaving volatility, skew, and correlations exactly where they were. This is not a stress test. It is a first-order sensitivity — a delta — wearing a stress test's clothes. The whole reason a ten-percent crash hurts is that the distribution itself changes when spot moves. Volatility spikes. The skew steepens as everyone reaches for downside protection. The correlation between assets that were comfortably independent snaps to one. If you shock the spot but freeze the shape of the distribution, you have stress-tested a world that does not exist.

A real scenario analysis therefore has two moves, in order. First, you apply the shock to the driver. Second — and this is the rigorous half — you re-derive the entire conditional distribution under that shock, and only then do you reprice. The object you reprice against is not a shifted point. It is a shifted distribution.

Let us make that precise. The market already tells you what distribution it is pricing, through the options surface. The risk-neutral density, the RND, is the implied probability distribution of the underlying at expiry that is consistent with every option quote at once. Formally, it is the Breeden–Litzenberger result: the risk-neutral density is the second partial derivative of the call price with respect to strike, discounted forward — the curvature of the call-price-versus-strike curve is the density. Extract that curvature across all strikes and you have recovered the market's full implied distribution, not just its mean and variance.

A stress test done correctly operates on that whole density. You posit the shock — say, spot down ten percent. You then ask what the RND looks like conditional on that shock: re-fit the surface, re-extract the second derivative, and obtain a new density that carries the fattened left tail and steepened skew a real crash brings. Finally you integrate your book's payoff against this new density. The change in that integral, from the base density to the stressed one, is your scenario P&L. Notice what you have done — you have stressed the measure, not merely the point, which is the entire difference between a genuine stress test and a glorified Greek.