Tail Option Pricing Under Power Laws
Nassim Nicholas Taleb, Brandon Yarckin, Chitpuneet Mann, Damir Delic, Mark Spitznagel
11 Tail Option Pricing Under Power Laws
Nassim Nicholas Taleb ∗†‡ , Brandon Yarckin ∗ , Chitpuneet Mann ∗ , Damir Delic ∗ , and Mark Spitznagel ∗∗ Universa Investments † Tandon School of Engineering, New York University ‡ Corresponding author, [email protected], February 10, 2020 ℓ log SLog Survival Function Fig. 1. The Karamata constant where the slowly moving function is safelyreplaced by a constant L ( S ) = l . The constant varies whether we use the priceS or its geometric return –but not the asymptotic slope which corresponds tothe tail index α . Abstract —We build a methodology that takes a given optionprice in the tails with strike K and extends (for calls,all strikes > K , for puts all strikes < K ) assumingthe continuation falls into what we define as "KaramataConstant" over which the strong Pareto law holds. Theheuristic produces relative prices for options, with for soleparameter the tail index α , under some mild arbitrageconstraints.Usual restrictions such as finiteness of variance are notrequired.The heuristic allows us to scrutinize the volatility surfaceand test various theories of relative tail option overpricing(usually built on thin tailed models and minor modifica-tions/fudging of the Black-Scholes formula). I. I
NTRODUCTION
We start by restating the conventional definition of thepower law class, by the property of the survival function. Let X be a random variable belonging to the class of distributionswith a "power law" (right) tail, that is: P ( X > x ) ∼ L ( x ) x − α (1)where L : [ x min , + ∞ ) → (0 , + ∞ ) is a slowly varyingfunction, defined as lim x → + ∞ L ( kx ) L ( x ) = 1 for any k > , [1].The survival function of X is called to belong to the "regularvariation" class RV α . More specifically, a function f : R + → R + is index varying at infinity with index ρ ( f ∈ RV ρ ) when lim t →∞ f ( tx ) f ( t ) = x ρ
115 120 125 130 K0.20.40.60.81.0Option Price
Black - Scholes Smile Power Law
Fig. 2. We show a straight Black Scholes option price (constant volatility),one with a volatility "smile", i.e. the scale increases in the tails, and powerlaw option prices. Under the simplified case of a power law distribution forthe underlying, option prices are linear to strike. . More practically, there is a point where L ( x ) approachesits limit, l , becoming a constant as in Fig. 1–we call it the"Karamata constant". Beyond such value the tails for powerlaws are calibrated using such standard techniques as the Hillestimator. The distribution in that zone is dubbed the strongPareto law by B. Mandelbrot [2],[3].II. C ALL P RICING BEYOND THE "K ARAMATA CONSTANT "Now define a European call price C ( K ) with a strike K andan underlying price S , K, S ∈ (0 , + ∞ ) , as ( S − K ) + , with itsvaluation performed under some probability measure P , thusallowing us to price the option as E P ( S − K ) + = (cid:82) ∞ K ( S − K ) dP . This allows us to immediately prove the followingresults under the two main approaches. A. First approach; the underlying, S is in the regular variationclass We start with a simplified case, to build the intuition. Let S have a survival function in the regular variation class RV α as per 1. For all K > l and α > , C ( K ) = K − α l α α − (2) a r X i v : . [ q -f i n . P R ] F e b Remark 1
We note that the parameter l , when derived from anexisting option price, contains all necessary informationabout the probability distribution below S = l , whichunder a given α parameter makes it unnecessary toestimate the mean, the "volatility" (that is, scale) andother attributes. Let us assume that α is exogenously set (derived fromfitting distributions, or, simply from experience, in both cases α is supposed to fluctuate minimally [4] ). We note that C ( K ) is invariant to distribution calibrations and the onlyparameters needed l which, being constant, disappears inratios. Now consider as set the market price of an "anchor"tail option in the market is C m with strike K , defined asan option for the strike of which other options are priced inrelative value. We can simply generate all further strikes from l = (cid:0) ( α − C m K α − (cid:1) /α and applying Eq. 2. Result 1: Relative Pricing under Distribution for S For K , K ≥ l , C ( K ) = (cid:18) K K (cid:19) − α C ( K ) . (3)The advantage is that all parameters in the distributions areeliminated: all we need is the price of the tail option and the α to build a unique pricing mechanism. Remark 2: Avoiding confusion about L and α The tail index α and Karamata constant l should cor-respond to the assigned distribution for the specificunderlying. A tail index α for S in the regular variationclass as as per 1 leading to Eq. 2 is different from thatfor r = S − S S ∈ RV α . For consistency, each shouldhave its own Zipf plot and other representations. If P ( X > x ) ∼ L a ( x ) x − α , and P ( X − X X > x − X X ) ∼ L b ( x ) x − α , the α constant will be thesame, but the the various L ( . ) will be reachingtheir constant level at a different rate. If r c = log SS , it is not in the regular variationclass, see theorem next. The reason α stays the same is owing to the scale-free attributeof the tail index. Theorem 1: Log returns
Let S be a random variable with survival function ϕ ( s ) = L ( s ) s − α ∈ RV α , where L ( . ) is a slowly varyingfunction. Let r l be the log return r l = log ss . ϕ r l ( r l ) isnot in the RV α class. Proof. Immediate, thanks to the transformation ϕ r l ( r l ) = L ( s ) s − log ( log α ( s ) ) log( s ) .We note, however, that in practice, although we may needcontinuous compounding to build dynamics [5], our approachassumes such dynamics are contained in the anchor optionprice selected for the analysis (or l ). Furthermore there is notangible difference, outside the far tail, between log SS and S − S S . B. Second approach, S has geometric returns in the regularvariation class Let us now apply to real world cases where the returns S − S S are Paretan. Consider, for r > l , S = (1 + r ) S , where S isthe initial value of the underlying and r ∼ P ( l, α ) (Pareto Idistribution) with survival function (cid:18) K − S lS (cid:19) − α , K > S (1 + l ) (4)and fit to C m using l = ( α − /α C /αm ( K − S ) − α S , which,as before shows that practically all information about thedistribution is embedded in l .Let S − S S be in the regular variation class. For S ≥ S (1 + l ) , C ( K, S ) = ( l S ) α ( K − S ) − α α − (5)We can thus rewrite Eq. 3 to eliminate l : Result 2: Relative Pricing under Distribution for S − S S For K , K ≥ (1 + l ) S , C ( K ) = (cid:18) K − S K − S (cid:19) − α C ( K ) . (6) Remark 3
Unlike the pricing methods in the Black-Scholes modifi-cation class (stochastic and local volatility models, (seethe expositions of Dupire, Derman and Gatheral, [6] [7],[8], finiteness of variance is not required neither for ourmodel nor for option pricing in general, as shown in[5]. The only requirement is α > , that is, finite firstmoment. III. P UT P RICING
We now consider the put strikes (or the correspondingcalls in the left tail, which should be priced via put-callparity arbitrage). Unlike with calls, we can only consider thevariations of S − S S , not the logarithmic returns (nor those of S taken separately).We construct the negative side with a negative return forthe underlying. Let r be the rate of return S = (1 − r ) S , andLet r > l > be Pareto distributed in the positive domain, Fig. 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 α that matches themarket (blue) ("model), and in red prices for α = 2 . . We can see that market prices tend to 1) fit a power law (matches stochastic volatility with fudgedparameters), 2) but with an α that thins the tails. This shows how models claiming overpricing of tails are grossly misspecified. with density f r ( r ) = α l α r − α − . We have by probabilistictransformation and rescaling the PDF of the underlying: f S ( S ) = − α (cid:16) − S − S lS (cid:17) − α − lS λ S ∈ [0 , (1 − l ) S ) where the scaling constant λ = (cid:16) − α +1 ( l α − (cid:17) is set in away to get f s ( S ) to integrate to 1. The parameter λ , however,is close to , making the correction negligible, in applicationswhere σ √ t ≤ ( σ being the Black-Scholes equivalent impliedvolatility and t the time to expiration of the option). Fig. 4. Same results as in Fig 3 but expressed using implied volatility. We match the price to implied volatility for downside strikes (anchor , , and )using our model vs market, in ratios. We assume α = 2 . . Remarkably, both the parameters l and the scaling λ areeliminated. Result 3: Put Pricing
For K , K ≤ (1 − l ) S , P ( K )= P ( K )( − − α S − α (( α − K + S ) − ( K − S ) − α ( − − α S − α (( α − K + S ) − ( K − S ) − α (7) IV. A
RBITRAGE B OUNDARIES
Obviously, there is no arbitrage for strikes higher than thebaseline one K in previous equations. For we can verify the Breeden-Litzenberger result [9], where the density is recoveredfrom the second derivative of the option with respect to thestrike ∂ C ( K ) ∂K | K ≥ K = αK − α − L α ≥ .However there remains the possibility of arbitrage betweenstrikes K +∆ K , K , and K − ∆ K by violating the followingboundary: let BSC ( K, σ ( K )) be the Black-Scholes value ofthe call for strike K with volatility σ ( K ) a function of thestrike and t time to expiration. We have C ( K + ∆ K ) + BSC ( K − ∆ K ) ≥ C ( K ) , (8)where BSC ( K , σ ( K )) = C ( K ) . For inequality 8 to besatisfied, we further need an inequality of call spreads, takento the limit: ∂BSC ( K, σ ( K )) ∂K | K = K ≥ ∂C ( K ) ∂K | K = K (9)
120 140 160 180 log K0.0010.0100.1001log Option Price
Black - Scholes α = α = α = Fig. 5. The intuition of the Log log plot for the second calibration
Such an arbitrage puts a lower bound on the tail index α .Assuming 0 rates to simplify: (10) α ≥ − log ( K − S ) + log( l ) + log ( S )log erfc (cid:18) tσ ( K ) + 2 log( K ) − S )2 √ √ tσ ( K ) (cid:19) − √ S √ tσ (cid:48) ( K ) K log( S tσ ( K )2 + exp (cid:16) − log ( K )+log ( S )2 tσ ( K ) − tσ ( K ) (cid:17) √ π V. C
OMMENTS
As we can see in Fig. 5, stochastic volatility models andsimilar adaptations (say, jump-diffusion or standard Poissonvariations) eventually fail "out in the tails" outside the zonefor which they were calibrated. There has been poor attemptsto extrapolate the option prices using a fudged thin-tailedprobability distribution rather than a Paretan one –hence thenumerous claims in the finance literature on "overpricing" oftail options combined with some psycholophastering on "dreadrisk" are unrigorous on that basis. The proposed methodsallows us to approach such claims with more realism.Finaly, note that our approach isn’t about absolute mispric-ing of tail options, but relative to a given strike closer to themoney. A
CKNOWLEDGMENTS
Bruno Dupire, Peter Carr, students at NYU Tandon Schoolof Engineering, Bert Zwart and the participants at the HeavyTails Workshop (April 6-9 2019) in Eindhoven he helpedorganize. R
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