Model Risk In Derivatives

A translator can be completely fluent and still mistranslate you, because language is full of idioms that don't survive a literal word-for-word crossing. The translation isn't broken because the translator is bad at grammar — it's broken because the tool itself — word-for-word substitution — is the wrong tool for that particular sentence. Pricing models have the same failure mode. Black-Scholes is not "bad at math." It is a precise, internally consistent machine built on assumptions — constant volatility, no jumps, frictionless trading — and every one of those assumptions is false in some specific, knowable way. Model risk is the money you lose because of exactly that gap between the model's world and the real one.

Where the gap comes from

Model risk has three separate layers, and it helps to keep them apart because they're fixed in different ways.

Model choice. You picked the wrong family of model — say, one that assumes stock returns are lognormal with no jumps, when the real stock jumps on earnings days. No amount of careful calibration fixes this; the model is structurally blind to a thing that actually happens.

Calibration. Even with the right model family, you fit its knobs to a noisy, incomplete set of market quotes (see calibrating a model to market prices). Thin quotes, bad weighting, or an unstable optimizer hand you parameters that fit today's screen and nothing else.

Implementation and use. The model is right, the calibration is right, and someone still applies it outside the range it was built for — pricing a ten-year option with a smile calibrated only out to two years, or feeding it a stale input.

A calibrated price is not a guaranteed price

The core arithmetic fact behind model risk is simple: two models — or the same model with two defensible sets of parameters — can both look "calibrated" and still disagree sharply on a contract neither was directly quoted on.

$$\text{Reserve} = \tfrac{1}{2}\left| P_{model\ A} - P_{model\ B} \right| .$$

In plain English: take the price two reasonable models both produce for the same trade, find the gap between them, and hold back half of that gap as a cushion against being on the wrong side of it. $P_{model\ A}$ and $P_{model\ B}$ are the two candidate model prices; the reserve is the desk's insurance against having picked the wrong one.

Worked example 1 — mispricing a wing strike. A stock trades at $100. The desk's model is calibrated to a flat 20% volatility using only near-the-money options, because that's what's liquid. It then prices a deep out-of-the-money one-year put, strike $70. Using Black-Scholes at 20% vol, that put is worth about $0.22. But the real market — where far-out strikes trade richer because crashes happen more often than a flat lognormal model admits — actually has that strike implying 35% volatility. At 35% vol, the same put is worth about $2.32. The model, calibrated correctly to strikes it could see, is off by more than 10x on a strike it extrapolated to. That roughly $2.10 gap is pure model risk: no market move occurred, no time passed, and yet the "correct" price for that contract was never what the flat-vol model said.

Worked example 2 — a model reserve, from real numbers. A desk uses a local-volatility model to price a barrier option and gets $4.10. A colleague reprices the identical contract with a stochastic-volatility model, calibrated to the same vanilla options, and gets $4.60. Neither model is provably wrong — both reproduce the vanilla smile exactly, and they only disagree on the exotic, where there's no market price to check against. Following the formula above, the reserve is $\tfrac{1}{2}|4.60 - 4.10| = 0.25$ per contract ($0.25 per contract). On a $10 million notional book of similar barriers, that's a reserve in the hundreds of thousands of dollars — money set aside specifically because two "correct" models disagreed, not because anyone made an error.

Widen or fatten the tails on the distribution above and watch how much probability mass appears far from the center. Black-Scholes assumes the thin, bell-shaped version. Real markets — especially around earnings, elections, and crashes — put noticeably more weight in the tails, and every model built on the thin version misprices exactly the contracts that pay off there.

What this means in practice

Every major bank runs an independent model validation function whose entire job is to stress models built by the trading desk — feed them inputs at the edge of their calibrated range, compare them against alternative model families, and set reserves like the one above. Regulators treat model risk as its own risk category, separate from market risk and credit risk, precisely because it doesn't show up in a simple "how much does the market need to move to hurt me" calculation — it shows up when the market does something the model never had a way to represent.

Practice

1. Model A prices an exotic at $6.20, Model B prices it at $5.40, and both are calibrated to the same vanilla quotes. What reserve should the desk hold per contract, and on a 50,000-contract book, what's the total reserve?
2. A model assumes returns are normally distributed with no jumps. Name two specific contract types (think about what happens near a barrier, or the day before an earnings announcement) where that assumption is likely to cost the most money, and explain why in one sentence each.