What Is Ultima?
Last reviewed: by Options Analysis Suite Research.
Ultima is the third derivative of option value with respect to volatility (partial3 V / partial sigma3). Equivalently, ultima measures how vomma itself changes when implied volatility moves. Ultima captures the convexity of vega convexity and matters in extreme-vol regimes and for far-OTM tail-risk pricing.
What Is Ultima in Options?
Ultima quantifies the third-order curvature of an option's value as a function of vol. Long-vol positions have positive ultima for ATM strikes; the magnitudes are small relative to vomma but become significant in stress scenarios where IV moves are large. Ultima is the analytical correction to a vega-plus-vomma estimate of P&L when IV moves dramatically.
Two intuitions. First, ultima is the analog of speed in the vol direction - both are third-order convexity measures. Second, ultima becomes operationally relevant during regime transitions when vol moves 10+ vol points: the linear-vega plus quadratic-vomma estimate of P&L is incomplete; ultima adds the cubic correction that captures non-Gaussian tail behavior of long-vol payoffs.
Why Does Ultima Matter?
Ultima is a niche Greek for normal trading but matters in three contexts. First, deep-OTM tail-risk pricing. Far-OTM puts and calls have significant ultima exposure because their value depends on extreme vol regimes; a 10-point IV move on a 5-delta put can produce P&L that the linear-vega estimate underestimates by 30-50% due to vomma plus ultima.
Second, butterfly and condor strategies. These structures are explicit vomma trades; the residual exposure beyond vomma is ultima. For traders running large-size butterfly books, aggregate ultima becomes a non-trivial risk metric during vol-regime transitions.
Third, model calibration validation. The fit quality of stochastic-vol models (Heston, SABR) to deep-OTM market prices is partially a question of whether the model's ultima matches market-implied ultima. Models that nail ATM and partially nail OTM but mismatch deep-OTM are typically failing on the ultima dimension.
How Do Pricing Models Compute Ultima?
- Black-Scholes: closed-form ultima derived by differentiating vomma with respect to sigma. Same ultima for calls and puts.
- Heston (stochastic volatility): Heston naturally generates non-zero ultima through stochastic-vol structure. Calculation by Fourier inversion of third-order vol-derivative.
- SABR: ultima via Hagan formula's vol-of-vol expansion.
- Jump diffusion: ultima includes diffusion and jump components; jump-models tend to produce different ultima profiles than pure diffusion.
- Monte Carlo: ultima via third-order finite difference or pathwise differentiation.
What Are the Special Cases?
- ATM short-tenor: ultima is small.
- OTM long-tenor: ultima is at its largest in absolute terms.
- Deep OTM tail strikes: ultima is a meaningful fraction of total vol-risk.
Related Greeks
Ultima is the third derivative in the vol direction. Its sibling third-order Greeks are speed (third-order in spot), zomma (cross spot-spot-vol), color (cross spot-spot-time), and DcharmDvol (cross spot-time-vol).
Related Concepts
Vega · Vomma · Vol of Vol · Volatility Smile · Tail Risk · Heston · All 17 Greeks
References & Further Reading
- Hull, J. C. (2022). Options, Futures, and Other Derivatives, 11th ed. Pearson.
- Castagna, A. (2010). FX Options and Smile Risk. Wiley.
This page is part of the 17 Greeks reference covering every options Greek with formula, intuition, worked example, and how each pricing model computes it. Browse the full documentation.