Volatility Surface Arbitrage in a Worst-of Autocallable
An equity derivatives desk is pricing a three-year worst-of autocallable note linked to three European banks. The note pays 8% per annum unless any underlying falls below 70% of its initial level. It autocalls if all three are at or above 100% on any annual observation date. At maturity, if any underlying is below 70%, the investor receives the worst-performing share. The desk calibrates a local volatility model to each stock’s vanilla surface. The model reprices all vanillas within 2bp, but the implied risk-neutral density for one underlying is negative between 80% and 85% of spot. The trader wants to use the model because it gives a fair value 3.5% higher than a stochastic volatility model that shows no arbitrage. The structurer says the difference is due to the local volatility model fitting the skew better. Risk is concerned about hedging the worst-of feature and the note’s gap risk.
1. Explain why a negative implied density is a red flag and how local volatility calibration can produce it.
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A negative implied density means negative risk-neutral probability in that region. That is inconsistent with no-arbitrage: butterfly spreads would have negative prices, and risk-neutral expectations cannot be computed reliably. Any price or hedge ratio from such a surface is unreliable.
Local volatility models can produce this when the calibration is not arbitrage-free. The model may reprice each vanilla within 2bp while the interpolation or extrapolation in strike and maturity violates convexity. For example, a cubic spline through implied volatilities can create a negative second derivative in strike, which maps into a negative density. The finite-difference solver and the local volatility function can also amplify small inconsistencies. Repricing vanillas is necessary but not sufficient: the surface must be free of calendar and butterfly arbitrage. The negative density therefore signals that the model should not be used for pricing without repairing the surface.
2. Advise whether the desk should trade at the local volatility price and outline the model and hedging adjustments you would require.
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The desk should not trade at a price produced by an arbitrageable model. The 3.5% value difference is economically large for a three-year note and is not explained merely by “fitting the skew better”; it is a model-risk signal.
First, repair the vanilla surfaces with an arbitrage-free interpolation and re-check butterfly and calendar conditions. Second, price the note under at least two models that capture joint dynamics, such as a local-stochastic volatility model or a multi-asset stochastic volatility model with calibrated correlation. Local volatility matches each marginal distribution but does not determine the joint law, which is what drives a worst-of payoff. Third, calibrate to worst-of basket options if they are available; if not, set a model reserve at least equal to the difference between models and stress correlation towards 1 and towards 0. Fourth, hedge basket and correlation risk, not only single-name delta, and charge for gap risk. If the desk cannot hedge the worst-of feature, it should widen the spread or decline the trade.