Tail Option Pricing Under Power Laws

Brandon Yarckin; Chitpuneet Mann; Damir Delic; Mark Spitznagel; Nassim Nicholas Taleb

REVIEW 4 major objections 4 minor 9 references

Tail Option Pricing Under Power Laws

T0 review · 4 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Under an exact power-law tail, one anchor option price and a single tail index α determine all far out-of-the-money option prices.

desk verdict A mathematically correct practitioner's repackaging of a standard power-law fact, with the real soft spot being the unverified exact-Pareto zone and the weak empirical demonstration. read the letter →

arxiv 1908.02347 v3 pith:3XOYG3C2 submitted 2019-08-06 q-fin.PR

Nassim Nicholas Taleb , Brandon Yarckin , Chitpuneet Mann , Damir Delic , Mark Spitznagel This is my paper · ORCID

classification q-fin.PR MSC 91G2062G3260G70

keywords tailoptionpricingpowerlawKaramataconstantstrongParetoregularvariationindexvolatilitysurfacearbitragebounds

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper argues that in the far option tail, where the survival probability of the underlying follows an exact power law, all relative option prices stop depending on volatility, variance, or any model calibration and are fixed by a single tail index α together with one 'anchor' option price. It proves that for calls with strikes beyond the 'Karamata constant' l, $C(K_2)/C(K_1) = (K_2/K_1)^{1-\alpha}$, and gives the analogous ratio for puts. The result means the entire far wing of the volatility surface can be generated from one observed price and an externally chosen α. The authors position this as a more realistic benchmark than Black-Scholes variants for testing whether tail options are actually overpriced.

What carries the argument

The central object is the 'Karamata constant' $l$, the threshold beyond which the slowly varying function $L$ in the survival function $P(S > x) = L(x)x^{-\alpha}$ is treated as exactly constant, so the strong Pareto law holds exactly. Beyond that threshold the European call price takes the closed form $C(K) = K^{1-\alpha} l^{\alpha}/(\alpha-1)$; when two such prices are divided, $l$ cancels, producing the power-law ratio that is the paper's main pricing machinery. That cancellation is what eliminates every distributional parameter except $\alpha$.

What would settle it

On a liquid underlying, take two different anchors $K_1$ and $K_1'$ above the supposed Karamata constant and compare their implied prices for the same far-OTM strike $K_2$; if the ratio $C(K_2)/C(K_1) = (K_2/K_1)^{1-\alpha}$ yields materially different $\alpha$'s, or if the estimated $\alpha$ changes with the anchor's distance from the money, the exact-Pareto assumption is violated at those strikes.

Watch

Extended reading notes

Core claim

The central discovery is the ratio identity in Result 1: for any two strikes $K_1, K_2$ above the Karamata constant, $C(K_2) = (K_2/K_1)^{1-\alpha} C(K_1)$, where $\alpha$ is the tail index of the underlying's regular-variation survival function. The same argument, applied to geometric returns, replaces strikes by their distance from the current underlying price, and a put-pricing analogue (Result 3, Eq. 7) covers the left tail. Because the normalization constant $l$ cancels in every ratio, the approach requires no estimate of the mean, the volatility, or the scale of the distribution; only $\alpha > 1$ (finite mean) is needed. The paper also shows the implied second derivative of the option price is nonnegative above the anchor, so the pricing rule is locally arbitrage-free, and it derives a lower bound on $\alpha$ from a call-spread inequality.

Load-bearing premise

The entire relative-pricing chain holds only if the underlying's tail is exactly Pareto beyond some strike $l$, and only if the anchor option you start from is itself struck inside that exact-Pareto zone.

Editorial extensions

If this is right

For calls struck beyond the Karamata constant, the whole far-tail price curve is determined by one anchor price and $\alpha$; volatility and variance estimates are unnecessary.
Under geometric returns, the relative-price ratio uses distances from the current underlying price, $(K_i - S_0)$, making the result applicable to real equity index options.
The put-pricing formula gives a comparable one-parameter relative-price rule for downside strikes below $(1-l)S_0$, subject to a normalization correction that is negligible when $\sigma\sqrt{t} \le \tfrac{1}{2}$.
The call-spread arbitrage boundary yields a lower bound on the tail index $\alpha$, tying permissible pricing to the steepness of the Black-Scholes smile at the anchor strike.
Empirically, the method applied to S&P 500 options finds market tail prices that are consistent with a power law but with a thinner tail (larger $\alpha$) than standard calibrated values, contradicting simple 'tail overpricing' claims built on thin-tailed models.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the scaling holds, then any strike-by-strike movement in the far implied-volatility wing is equivalent to a change in $\alpha$; volatility smile analysis and tail-index estimation become the same exercise.
A direct empirical test: compute model-implied $\alpha$ from pairs of far-OTM strikes using several different anchor strikes; stable $\alpha$ supports the exact-Pareto zone, while drift in $\alpha$ marks where $l$ must be moved outward.
The Karamata constant is likely to depend on maturity and on whether prices or returns are used; the paper notes this dependence, which suggests a practical protocol of estimating $l$ on a rolling basis for each tenor.
Because the paper prices only relative to an anchor, claims of absolute option mispricing require an independent estimate of the physical measure's $\alpha$; the methodology itself cannot say whether the anchor is mispriced.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper is a practitioner-facing repackaging of a standard regular-variation fact: under exact Pareto tails, far OTM option prices scale as a power of strike. The central ratio formula, C(K2) = (K2/K1)^(1-alpha) C(K1), is correct and useful in spirit. What is actually new is the packaging: anchoring to a single market option price, eliminating the need for variance or scale estimates, and framing the strong Pareto law via the "Karamata constant." That is a genuinely practical contribution, and the paper deserves credit for stating it clearly.

The math, where it matters, is sound under the stated assumption. The call-side derivation is just integration of a Pareto survival function. The put-side formula is algebraically consistent with a truncated Pareto model, though the presentation with (-1)^(1-alpha) is confusing and the scaling constant λ is introduced with no clean derivation. The arbitrage boundary section is the weakest formal part: Theorem 1's proof is unintelligible (the exponent written in the log-return transformation is wrong as written), and Eq. 10 is a garbled expression that reads like a dropped formula. These are fixable presentation issues, not fatal flaws in the core ratio.

The soft spot that matters is the gap between regular variation and the exact Pareto equality in Eq. 2. The paper says there is a point where the slowly varying function can be "safely" replaced by a constant, but gives no estimator, no diagnostic, and no error bound for where that point is. The stress-test note is right: regular variation alone does not justify Eq. 2, and if the anchor strike sits where L(x) is still drifting, the relative prices carry an uncontrolled bias. The empirical section does not close this gap: alpha is fitted to the same market data, no option chain or source is given, and no error bars or out-of-sample checks appear. So the central claim is conditional on an exact-Pareto zone whose existence and location are asserted, not demonstrated.

Who should read this? Practitioners who want a simple, transparent benchmark for tail option prices and who are comfortable treating the strong Pareto law as an assumed working hypothesis. Researchers in heavy tails will recognize the math immediately and may be frustrated by the sloppy notation, but the framing around relative pricing and the anchor method is worth a look. The paper is not a new theoretical result, but it is a testable heuristic with real-world relevance.

I would accept this for peer review. The core ratio formula deserves scrutiny and the empirical claims need to be either properly validated or toned down, but the paper is coherent, argues a clear position, and would benefit from expert referee feedback. Do not desk-reject it; send it out with a request for major revisions and a demand for cleaner derivations and actual data.

Referee Report

4 major / 4 minor

Summary. The paper proposes a method for extrapolating a single tail option price to all farther out-of-the-money strikes under the assumption that the underlying's survival function is exactly Pareto (the 'strong Pareto law') beyond a point it calls the 'Karamata constant' l. The call-price formula C(K)=l^alpha K^{1-alpha}/(alpha-1), the relative-pricing result C(K2)=(K2/K1)^{1-alpha} C(K1), and the return-based analogue Eq. (6) are derived. A put-side ratio, Eq. (7), is introduced for downside strikes using a truncated Pareto model for negative arithmetic returns. The paper also discusses an arbitrage lower bound on the tail index alpha and illustrates the method on S&P 500 put prices.

Significance. If the strong-Pareto zone exists and the anchor option lies inside it, the paper offers a strikingly parsimonious tool: the far-tail relative volatility surface is determined by one anchor price and an exogenously chosen tail index alpha, with no need to estimate variance or volatility. The algebraic core is simple and, under the stated exact-Pareto assumption, the call-side formulas are correct; the put ratio, although poorly derived, is algebraically consistent with a correctly normalized truncated Pareto model. The paper also makes the useful and correct point that far OTM tails are not a suitable place for Gaussian-based approximations. The practical value of the claims, however, is currently conditional on unverified identification of the exact-Pareto zone and on a more quantitative empirical demonstration than the figures provide.

major comments (4)

[§II, Eq. (2) and Result 1 (Eq. 3)] Equation (2) is not a consequence of regular variation as stated in Eq. (1); it is an equality that holds only under the strong Pareto law P(S>x)=l^alpha x^{-alpha} for all x>=l. The manuscript defines the Karamata constant only informally ('where the slowly moving function is safely replaced by a constant') and provides no estimator, diagnostic, or error bound for l. If the anchor K1 lies in a region where the slowly varying function is not yet constant, the ratios in Eqs. (3), (6), and (7) carry an uncontrolled slowly-varying bias. This is the central load-bearing point of the paper and needs to be addressed by stating the strong Pareto law as an explicit primitive assumption with a validation protocol, or by providing a regular-variation error analysis.
[§III, Eq. (7) and the put density] The derivation of the put formula is internally inconsistent as written: the density f_S(S) begins with a negative sign, the normalization constant lambda = 1/((-1)^{alpha+1}(l^alpha-1)) changes sign with the parity of integer alpha, and for non-integer alpha the term (-1)^{1-alpha} in Eq. (7) is not real. After reconstructing the model as a Pareto distribution for the negative return r=(S0-S)/S0 truncated to [l,1], the ratio in Eq. (7) is algebraically consistent, so the result is salvageable; however, the displayed derivation must be rewritten with a nonnegative density and a clearly defined branch for alpha.
[§V, Figs. 3–4] The empirical claim that market put prices 'tend to fit a power law' is not supported by the evidence shown. The tail index alpha is fitted to the same option chain that is then compared with the model, no option-chain details or error bars are provided, and Eq. (6) is not tested out-of-sample at multiple strikes. Please provide a quantitative evaluation, ideally with log-log slope diagnostics over successive strike intervals, confidence bands, and an out-of-sample comparison.
[§IV, Eq. (10)] The arbitrage lower bound for alpha is presented as a garbled expression with no derivation; as typeset, the inequality is not comprehensible. Since the methodology advertises 'mild arbitrage constraints,' this claim should be derived cleanly and stated as a proper inequality, or removed if it is not central.

minor comments (4)

[§II, Theorem 1] The proof of Theorem 1 is unintelligible: 'phi_rl(rl) = L(s) s^{-log(log alpha(s))/log(s)}' is not a valid transformation and appears to contain a typo. The statement that log returns are not in RV_alpha is standard, since log returns of a Pareto tail have approximately exponential tails, but the proof should be rewritten or replaced by a citation.
[§V] There is a typo 'psycholophastering' (probably 'psychologizing') and 'Finaly' should be 'Finally'.
[Fig. 5 caption] The caption of Fig. 5 does not explain what the labels 'alpha=2' and 'alpha=5' refer to; please clarify the relationship between the plotted curves and the tail index values.
[Remark 1] Remark 1 states that l 'contains all necessary information about the probability distribution below S=l'; this wording is misleading, since for strikes above l the distribution below l is not directly priced, and l acts as a tail scale parameter rather than a summary of the entire lower distribution.

Circularity Check

1 steps flagged · score 2.0 of 10

Core derivation is a direct consequence of the strong-Pareto assumption; only the empirical validation is partly in-sample because alpha is fitted to the same market data.

fitted input called prediction [Fig. 3 caption (Section III, Put Pricing)]
"generating an option prices using a tail index α that matches the market (blue) ("model), and in red prices for α = 2.75. We can see that market prices tend to 1) fit a power law (matches stochastic volatility with fudged parameters), 2) but with an α that thins the tails."

The parameter α is fitted to the same market option prices that are then compared to the model curve, so the visual agreement in Figs. 3-4 is in-sample by construction. This does not provide an out-of-sample test of the relative-pricing formula; it shows only that a Pareto curve with a fitted slope can be drawn through the data. The central derivation itself is independent of this fit, so the circularity is confined to the empirical demonstration rather than to Eqs. 2, 3, 6, or 7.

full rationale

The paper's central derivation is not circular. Equation (2) follows from the stated strong Pareto assumption P(S > x) = l x^{-α} for all x above the Karamata constant, and Result 1 (Eq. 3) is obtained by eliminating the scale l using an anchor option price as a boundary condition. The tail index α is treated as exogenous. Thus the relative pricing formula is a mathematical consequence of the model's assumptions, not a restatement of the data used to fit the model. Similarly, the put formula (Eq. 7) is algebraically derived from the stated transformed Pareto model. The empirical section is the only partly circular element: Fig. 3 explicitly chooses α to match the market data before comparing model prices to market prices, so the agreement is a fitted curve rather than an independent prediction. This lowers the evidentiary value of the empirical demonstration but does not infect the derivation itself. There are minor self-citations ([4], [5]) used for supporting assertions about α stability and variance finiteness, but these are not load-bearing for the central identities. Overall, the paper's core contribution is self-contained conditional on its explicit strong-Pareto premise; the weakness is the unverified location of the exact-Pareto zone, which is a correctness risk rather than a circularity.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The relative pricing rests on one externally supplied tail index α and one anchor market price used to fix scale; the put side further postulates Pareto-distributed negative returns with the same α and a truncation normalization. No formal verification or shipped code accompanies the derivation.

free parameters (2)

Tail index α = 2.75 in Figures 3-4; otherwise exogenous
The relative pricing formulas (Eqs. 3, 6, 7) depend entirely on α, which is not derived but taken from distributional fitting or experience (Section II, Remark 1).
Karamata constant l (scale from anchor price) = derived from anchor option price: l = ((α-1) C_m K1^{α-1})^{1/α}
l is fitted to the anchor option price and cancels in relative pricing ratios, but it sets the absolute price scale in Eq. 2.

assumptions (5)

domain assumption The survival function of S follows an exact Pareto form P(S > x) = l x^{-α} for all x beyond the Karamata constant l.
Eq. 2 assumes equality, not asymptotic equivalence, and Section II defines the Karamata constant as the point where L(x) is safely replaced by a constant. No test establishes that the anchor strike lies in this zone.
domain assumption The tail index α is identical for S and for the arithmetic return (S-S0)/S0, and α > 1.
Remark 2 invokes the scale-free property of the tail index to reuse α from the underlying for return-based pricing. Requires a finite first moment.
ad hoc to paper Negative arithmetic returns (S0-S)/S0 follow a Pareto distribution with the same α, truncated to the unit interval with a normalization λ.
Section III postulates this density to price puts; the truncation and normalization constant λ are introduced specifically for this derivation.
standard math Options are valued as expectations under a probability measure P, and Breeden-Litzenberger positivity holds.
Standard risk-neutral valuation framework invoked in Section II and Section IV.
domain assumption Interest rates are zero in the arbitrage boundary section.
Eq. 10 states 'Assuming 0 rates to simplify,' making the bound approximate for nonzero rates.

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Cite this review

Pith. "Pith review of Tail Option Pricing Under Power Laws." pith.science (2026). https://pith.science/paper/3XOYG3C2

@misc{pith2026190802347,
  author       = {Pith},
  title        = {Pith review of: Tail Option Pricing Under Power Laws},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3XOYG3C2}},
  note         = {Machine review of arXiv:1908.02347}
}

abstract

We build a methodology that takes a given option price in the tails with strike $K$ and extends (for calls, all strikes > $K$, for puts all strikes $< K$) assuming the continuation falls into what we define as "Karamata Constant" over which the strong Pareto law holds. The heuristic produces relative prices for options, with for sole parameter the tail index $\alpha$, under some mild arbitrage constraints. Usual restrictions such as finiteness of variance are not required. The methodology allows us to scrutinize the volatility surface and test various theories of relative tail option overpricing (usually built on thin tailed models and minor modifications/fudging of the Black-Scholes formula).

Figures

Figures reproduced from arXiv: 1908.02347 by the authors.

**Figure 3.** Put Prices in the SP500 using "fix K" as anchor (from Dec 31, 2018 settlement), and generating an option prices using a tail index [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

**Figure 4.** Same results as in Fig 3 but expressed using implied volatility. We match the price to implied volatility for downside strikes (anchor [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

Figures from the paper (1 more)

**Figure 5.** Figure 5: The intuition of the Log log plot for the second calibration [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

[1]

N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular variation . Cambridge university press, 1989, vol. 27

work page 1989
[2]

The pareto-levy law and the distribution of income,

B. Mandelbrot, “The pareto-levy law and the distribution of income,” International Economic Review , vol. 1, no. 2, pp. 79–106, 1960

work page 1960
[3]

Structural probability bounds for the strong pareto law,

D. Dyer, “Structural probability bounds for the strong pareto law,” Canadian Journal of Statistics , vol. 9, no. 1, pp. 71–77, 1981

work page 1981
[4]

N. N. Taleb, The Statistical Consequences of Fat Tails . STEM Publish- ing, 2019

work page 2019
[5]

Finiteness of variance is irrelevant in the practice of quantitative ﬁnance,

——, “Finiteness of variance is irrelevant in the practice of quantitative ﬁnance,” Complexity, vol. 14, no. 3, pp. 66–76, 2009

work page 2009
[6]

Pricing with a smile,

B. Dupire et al. , “Pricing with a smile,” Risk, vol. 7, no. 1, pp. 18–20, 1994

work page 1994
[7]

Gatheral, The Volatility Surface: a Practitioner’s Guide

J. Gatheral, The Volatility Surface: a Practitioner’s Guide . John Wiley & Sons, 2006

work page 2006
[8]

A guide to volatility and variance swaps,

K. Demeterﬁ, E. Derman, M. Kamal, and J. Zou, “A guide to volatility and variance swaps,” The Journal of Derivatives , vol. 6, no. 4, pp. 9–32, 1999

work page 1999

Show all 9 references

[9]

Prices of state-contingent claims implicit in option prices,

D. T. Breeden and R. H. Litzenberger, “Prices of state-contingent claims implicit in option prices,” Journal of business , pp. 621–651, 1978

1978

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular variation . Cambridge university press, 1989, vol. 27

work page 1989

[2] [2]

The pareto-levy law and the distribution of income,

B. Mandelbrot, “The pareto-levy law and the distribution of income,” International Economic Review , vol. 1, no. 2, pp. 79–106, 1960

work page 1960

[3] [3]

Structural probability bounds for the strong pareto law,

D. Dyer, “Structural probability bounds for the strong pareto law,” Canadian Journal of Statistics , vol. 9, no. 1, pp. 71–77, 1981

work page 1981

[4] [4]

N. N. Taleb, The Statistical Consequences of Fat Tails . STEM Publish- ing, 2019

work page 2019

[5] [5]

Finiteness of variance is irrelevant in the practice of quantitative ﬁnance,

——, “Finiteness of variance is irrelevant in the practice of quantitative ﬁnance,” Complexity, vol. 14, no. 3, pp. 66–76, 2009

work page 2009

[6] [6]

Pricing with a smile,

B. Dupire et al. , “Pricing with a smile,” Risk, vol. 7, no. 1, pp. 18–20, 1994

work page 1994

[7] [7]

Gatheral, The Volatility Surface: a Practitioner’s Guide

J. Gatheral, The Volatility Surface: a Practitioner’s Guide . John Wiley & Sons, 2006

work page 2006

[8] [8]

A guide to volatility and variance swaps,

K. Demeterﬁ, E. Derman, M. Kamal, and J. Zou, “A guide to volatility and variance swaps,” The Journal of Derivatives , vol. 6, no. 4, pp. 9–32, 1999

work page 1999