How Normal and Lognormal Models Produce Different Option Greeks
Summary
Normal and shifted lognormal volatility quotes can represent the same current swaption or cap/floor price. The document explains why this price agreement does not make the models’ Greeks agree: Greeks describe how prices change as market inputs move, and the models can imply different responses outside the calibration point.
It frames a hedge as model-dependent rather than uniquely correct. A useful Greek is one that predicts how the market value will move, but that depends on whether the model captures the underlying dynamics and how option prices co-move with the underlying. Those conditions cannot be known or modeled exactly. The discussion notes that incorrect Greeks have contributed to market-maker gains and losses across asset classes, while discretized hedging losses may outweigh Greek calculation errors. It offers no empirical comparison or quantitative estimate, so it is a conceptual account of model risk and hedging limits.
Key ideas
- Matching today’s option price does not require two models to produce matching Greeks.
- Greeks differ because they measure price changes under each model’s assumptions.
- A hedge is only as reliable as the model’s representation of market dynamics and co-movement.
- No model can capture the true dynamics and calibration perfectly.
- Discrete hedge adjustments can create losses that exceed errors in Greek calculations.
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# Normal vs. Lognormal Greeks for Negative Rates Options # Normal vs. Lognormal Greeks for Negative Rates Options My understanding is that for some of the G10 currencies with negative rates (CHF, EUR), Swaption and Cap / Floor prices are quoted in terms of BOTH, normal and log-normal Vols. That in itself is not controversial, because these vols are self-consistent (you plug the quoted log-normal vol into the (shifted) log-normal formula: you get a specific price. You plug the normal vol into the normal option pricing formula: you must get the same price. Otherwise an identical option would have two different prices and someone would exploit the arbitrage. What about the arising Greeks through and thereby hedging? I have even come across an implementation of the Libor market model which could switch between normal and shifted log-normal diffusion: don't these different models produce different Greeks? Intuitively, they shouldn't, but just looking at the basic Normal (Bachelier) vs. Log-normal (Black-Scholes) option pricing formulas, the Greeks will be different. Doesn't that imply that two different banks using two different models would calculate the risk differently, with one bank inherently miss-hedging? ## Answer by dm63 (score 1, accepted) https://quant.stackexchange.com/a/53820 Yes, different banks using different models will get different Greeks. Some of them will be right and some of them will be wrong. What do we mean by right and wrong? ‘Right’ means that when the market moves, your Greeks closely predict how the market values of the options are moving. There are multiple examples in all asset classes (rates, equities, fx) where models and their associated Greeks have proven to be wrong , resulting in losses (or sometimes gains) at market makers. ## Answer by Arshdeep (score 1) https://quant.stackexchange.com/a/59390 Greeks represent rates of price change. Obviously they vary across models, if they didn't, the models would agree on prices across all market scenarios and thus the models have to be the exact same, which is a contradiction. As far as hedging goes, nobody has the right hedge. For a Greek to be 'true' the model should: - Exactly capture the underlying dynamics, including exact calibration to the true dynamics. This is not possible. - If the option market dynamics are inconsistent with the underlying (but consistent enough to avoid arbitrage), the model should take that into account. This too is not possible. In practice, losses from discretized hedging overpower the errors that creep in during calculation of greeks due to the above.
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