Election predictions are arbitrage-free: response to Taleb
aa r X i v : . [ q -f i n . M F ] J u l July 4, 2019 Quantitative Finance Election˙predictions˙are˙arbitrage˙free
To appear in
Quantitative Finance , Vol. 00, No. 00, Month 20XX, 1–8
Election predictions are arbitrage-free: response toTaleb
AUBREY CLAYTON ∗ † ()Taleb (2018) claimed a novel approach to evaluating the quality of probabilistic election forecasts viano-arbitrage pricing techniques and argued that popular forecasts of the 2016 U.S. Presidential electionhad violated arbitrage boundaries. We show that under mild assumptions all such political forecasts arearbitrage-free and that the heuristic that Taleb’s argument was based on is false. Keywords : Bayesian Analysis; Arbitrage Pricing; Forecasting Applications; Probability Theory;Martingales
JEL Classification : C11; C53
1. Introduction
Forecasts of a candidate’s probability of winning an election have become a staple of politicaljournalism over the last decade, with the most prominent being those reported on the websiteFiveThirtyEight, for example in the 2016 U.S. Presidential election; see FiveThirtyEight (2016).Taleb (2018) was motivated to critique the quality of election forecasts by the apparent instabilityof these probabilities through time. FiveThirtyEight reported a probability of Clinton winning the2016 election that ranged from 55 to 85 percent over the final five months. According to Taleb, thiswas indicative of ‘stark errors’ and ‘severe violations’ of standard results in quantitative finance,specifically no-arbitrage option pricing. Imposing a no-arbitrage constraint on election forecastprobabilities, as binary options written on the candidate’s vote share, would therefore, accordingto Taleb, eliminate such instabilities.In this paper we show that Taleb’s argument was mistaken. First, one of the ‘standard results’ ofquantitative finance that his election forecast assessments rely on is false, as we demonstrate witha simple counterexample. Next, we show that binary option prices can easily exhibit as much ormore variability through time as the 2016 Presidential election forecasts without violating any no-arbitrage constraints. We argue that, under mild assumptions concerning the information availableto a forecaster, all such election forecasts are arbitrage-free. Finally, we comment on a problemregarding an ill-defined variable in Taleb’s election model and what this implies for the generalapplicability of option pricing methods to election forecasting. ∗ Corresponding author. Email: [email protected] † The views expressed in this publication are wholly those of the author. They do not necessarily represent the views of theauthor’s employer or any of its affiliates; and accordingly, such employer and its affiliates expressly disclaim all responsibilityfor the content and information contained herein. uly 4, 2019 Quantitative Finance Election˙predictions˙are˙arbitrage˙free
2. Summary of Taleb’s argument
We being by summarizing Taleb’s argument, which consists of five main steps. We also take thisopportunity to fix some notation and definitions:(i) He claims that the forecast probability for an event, such as a candidate winning an election,should be considered as the price of a binary option taking the terminal value 1 if the eventhappens and 0 otherwise. This can be thought of a derivative security on an underlying‘asset’ variable Y t , e.g., the candidate’s number of supporters among the population ofvoters at a given time t , with the election outcome determined by whether Y T exceeds somethreshold l at terminal time T . Arbitrage pricing theory will then dictate limits on thebehavior of these prices/probabilities over time.(ii) He gives a model for the stochastic process governing the underlying asset Y t . To produce a Y t that is bounded in some range [ L, H ], Taleb assumes Y t = S ( X t ) for a ‘shadow process’ X t satisfying a tractable stochastic differential equation, where S is a function taking thereal line to a finite interval. He makes the choice: dX t = σ X t dt + σdW t and S ( x ) = 12 + 12 erf ( x )where erf ( x ) = √ π R x exp( − t ) dt .Note that 12 S ′′ ( x ) + xS ′ ( x ) = 0With that choice in place, Itˆo’s formula implies Y t satisfies dY t = (cid:18) σ S ′′ ( X t ) + σ X t S ′ ( X t ) (cid:19) dt + σS ′ ( X t ) dW t = 0 · dt + σS ′ ( X t ) dW t That is, the process Y t is a bounded martingale.(iii) Since the process X t is the familiar Bachelier-style model for an asset price, the price ofa binary option of X t has a known formula. Arguing that this price is equal to the priceone would obtain for the binary option on Y t with the corresponding threshold value, thisgives the election forecast probability. Substituting x = S − ( y ) into the option pricingformula for X gives the price in terms of the currently observed value of Y t at a given time.Since Y t is a martingale, Taleb argues that the stochastic equation for Y t represents thedynamics under the risk-neutral measure, and therefore the price obtained for the binaryoption represents the probability of Y T exceeding the threshold, that is, the probabilityof the given candidate winning the election. Since the forecast probabilities are derived asoption prices, under this measure they are also martingales.2 uly 4, 2019 Quantitative Finance Election˙predictions˙are˙arbitrage˙free (iv) A heuristic from option pricing suggests that the greater the volatility of the underlyingasset, the closer the binary option price should be to 0 . . .
0, in agreement with theheuristic argument. Observing electoral forecasts such as those made by FiveThirtyEightshow forecast probabilities deviating significantly from 0 . n n X i =1 ( i − p i ) where p , . . . , p n are prices of bets laid on the outcomes of n independent events, and i is the indicator function taking the value 1 if the event occurred and 0 otherwise. In orderto minimize losses according to this score, an agent will place bets in agreement with theirprobabilities for the events. He claims the martingale-pricing technique gives a continuoustime analogue of the same idea.
3. Criticism3.1.
The behavior of option prices with respect to volatility
Taleb’s heuristic regarding the behavior of binary option prices as the volatility of the underlyingincreases is only partially true. He claims that ‘[a] standard result in quantitative finance is thatwhen the volatility of the underlying security increases, arbitrage pressures push the correspond-ing binary option to trade closer to 50%, and become less variable over the remaining time toexpiration.’ It is the case that for a given ‘spot-price’ Y t and a given threshold value l , as thevolatility σ in the stochastic model for Y t is increased, under general conditions the binary optionprice/probability of exceeding the threshold should converge to 0 .
5. For example, if the stochasticprocess for Y t were a simple Brownian motion with drift dY t = µdt + σdW t the conditional distribution for Y T given Y t would be N ( Y t , σ ( T − t )) and so the binary optionprice for strike l at time t would be B ( t, T ) = 1 − Φ (cid:18) l − Y t − µ ( T − t ) σ √ T − t (cid:19) where Φ is the standard normal cdf. This converges to 0 . σ goes to ∞ . Other processes dependingon a volatility parameter will show the same behavior.However, the prices B ( t, T ) one actually observed over time would not stabilize around 0 . Y would also result in a wider distribution for Y t atany given t . Taking the same example above with Y = 0, we have an unconditional cdf for B ( t, T )given by 3 uly 4, 2019 Quantitative Finance Election˙predictions˙are˙arbitrage˙free P [ B ( t, T ) < x ] = P (cid:20) Φ (cid:18) l − Y t − µ ( T − t ) σ √ T − t (cid:19) > − x (cid:21) = Φ (cid:18) l − µ ( T − t ) − σ √ T − t Φ − (1 − x ) − µtσ √ t (cid:19) = Φ l − µTσ √ t − r Tt − − (1 − x ) ! So, as σ → ∞ , we end up with a distribution for B ( t, T ) that is stable but certainly not a pointmass at 0 .
5. In particular, at the midway point t = T /
2, if we have set our threshold l to be equalto the projected mean of Y T , l = µT , then we will have P [ B ( t, T ) < x ] = Φ (cid:0) − Φ − (1 − x ) (cid:1) = 1 − Φ(Φ − (1 − x )) = x and so B ( t, T ) has the uniform distribution on [0 ,
1] regardless of σ .This counterexample shows that Taleb’s second claim, that as volatility increases binary optionprices ‘become less variable over the remaining time to expiration,’ is false. Instead, the variabilityof the paths of the option price may be completely independent of the volatility in the underlying.This has profound implications on his criticism of election forecasts. For example, assuming asimple random-walk model for a given candidate’s vote share, if the process stays close to the value50% we may observe the win-probability fluctuate widely through time despite arbitrarily smallfluctuations in the polls. The figures below show one such simulated path over 1000 days where wehave set the size of daily poll movements to 0.1%. We could just have easily chosen 0.000001% orany small number. Figure 1. Simulated candidate vote share for a random-walk process; step = 0.01% uly 4, 2019 Quantitative Finance Election˙predictions˙are˙arbitrage˙free Figure 2. Simulated candidate win-probability for a random-walk process; step = 0.01%
The size of these fluctuations in poll numbers and forecast win-probability are roughly consistentwith what was observed in 2016.In fact, even in Taleb’s model, if the same s parameter, representing annualized volatility in thepolls, is used to project the polls as well as construct the forecasts, the paths do not stabilize at0 . s = 100%: Figure 3. Simulated candidate win-probability for Taleb’s pricing model; s = 100%
Intuitively speaking, the greater we assume the implied volatility for the pricing model Y t tobe, the greater we must also assume the realized volatility to be; the former pulls option prices intoward 0 . uly 4, 2019 Quantitative Finance Election˙predictions˙are˙arbitrage˙free Election predictions are arbitrage-free
Framing election forecasting as a problem of option pricing adds needless complication and intro-duces the possibility of confusion. Taleb’s argument relies on an analogy to the problem: givena model for the dynamics of an asset price under real-world probabilities P, price an option onthe asset. The Fundamental Theorem of Asset Pricing implies that in a complete market with noarbitrage there is an equivalent probability measure Q under which all assets earn the risk-freerate r on average, including the option. Setting r = 0 would make all assets martingales underthis measure. Thus, the value at any given time for any derivative security is simply the averageof its possible values at any future time; for a binary option this implies the price is equal to theprobability of occurrence for the payoff event.However, the task facing a forecaster is not one of pricing an option but of assigning a realprobability. Suppose this is done over the filtration ( F t ) of information available to the forecasterat any given time. Granting Taleb’s assertion that the probabilities should be thought of as prices,and that these prices should be martingales under the real-world measure , we still needn’t botherwith the artifice of defining a stochastic model for Y t under which it is a martingale and pricingan option written on Y t . Instead, we can achieve exactly the dynamic Taleb desires by consideringany stochastic process at all for Y t and then quoting the probability B ( t, T ) = P [ Y T > l | F t ]These probabilities/prices always produce an ( F t ) − martingale by the tower property of condi-tional expectation: E [ B ( t, T ) | F s ] = E [ E [ Y T >l ] | F t ] | F s ] = E [ Y T >l ] | F s ] = B ( s, T )for any times s < t .Thus, Taleb’s construction of a martingale Y t is sufficient but not necessary for the forecastprobabilities to be martingales, which makes his shadow process X t doubly unnecessary. Taleb’sconstruction requires a bounded martingale that can be written as some monotonic function ofa process for which the option prices were possible to compute, but he could have skipped thatentirely.Different forecasters may disagree in their assessments, and the results may be profitable forone or the other. Whether the forecaster’s probabilities allow for profitable investments over timewill depend on the judgments of the forecaster and the investor. However, since the forecaster’sconditional probabilities are automatically martingales with respect to their own filtration, thereis no possibility for arbitrage unless the two assigned probability measures are not equivalent. Inorder to exhibit arbitrage, then, Taleb would need to show an event a forecaster assigned nonzeroprobability to was actually impossible, or conversely, a forecaster claimed an actually possible eventhad probability zero. The definition of the underlying
Finally, Taleb’s analysis suffers from a lack of precision about the meaning of the ‘underlying’ asset Y t , from which the election results are meant to be determined. He desires it both to be the votecount/share Y T at the terminal date (or number of electoral votes for determining a presidentialelection outcome) and yet also to be measurable by some means at times t < T as ‘an intermediaterealization of the process at t.’ This would be possible if the opinion of every potential voter inthe U.S. were known at every given time, in which case the volatility of Y t would correspond tochanges of opinion . 6 uly 4, 2019 Quantitative Finance Election˙predictions˙are˙arbitrage˙free However, the problem for the forecaster is that the intentions of every person are not known andmust be estimated from polls/samples. Thus, the uncertainty facing a political forecaster such asFiveThirtyEight is that these samples themselves have uncertainty, as do the proportions of peoplewho actually vote in the election. Taleb later defines Y t to be ‘the observed estimated proportionof votes’ (emphasis ours). If we treat Y t as the estimated proportions of voters, as given by thesamples, we would find it inadequate to determine the election of the outcome; elections are decidedby the actual vote counts, not a final random sample of voters on election day. This means thatthe process Y t cannot possibly be both an underlying asset for the derivative and adapted to thefiltration of information available to the forecaster.
4. Conclusion
Taleb’s criticism of popular forecast probabilities, specifically the election forecasts of FiveThir-tyEight, was inspired by a judgment that they tend to fluctuate too much given a reasonableamount of uncertainty in the future changes in popular opinion. He attempted to cast this as aviolation of the principles of option pricing and to produce an alternative model by constructing abounded martingale with known binary option prices exhibiting the dynamics he sought. However,the situation facing the forecaster is one of assigning probabilities, not prices, and this can bedone in the real-world measure without reference to an underlying martingale asset. As long asthese probabilites are updated consistenly according to rules of conditional probability, they willautomatically be martingales. If the forecaster’s probability measure is equivalent to an investor’s,then according to the Fundamental Theorem of Asset Pricing arbitrage is impossible even if theforecasts are treated as prices. There may be profitable, but risky, market opportunities if investorshave different probability assessments than the forecasters, but this is simply an argument thatthe investors’ forecasts are better.Taleb’s heuristic of greater volatility/uncertainty giving binary option prices that stabilize around0 . . , Y t represents in Taleb’s model speaksto a misunderstanding of the kinds of uncertainty facing the forecaster. In quantitative finance, anasset with an observable price is given a stochastic model, and fluctuations in that price determinethe uncertainty of the terminal payoffs of various derivatives. In election forecasting, the “price”itself is unknown and unknowable. Thus, the whole framework of derivative pricing is arguablyinapplicable here, which is no great loss since there was no need for it in the first place.
5. References
FiveThirtyEight, 2016 Election Forecast. https://projects.fivethirtyeight.com/2016-election-forecast Ac-cessed: 2019-4-11, 2016. uly 4, 2019 Quantitative Finance Election˙predictions˙are˙arbitrage˙free Taleb, N.N., Election predictions as martingales: an arbitrage approach.
Quantitative Finance , 2018, ,1–5.Williamson, J., Bruno de finetti. philosophical lectures on probability. collected, edited, and annotated byAlberto Mura. translated by Hykel Hosni. Synthese Library; 340 , 2009, Oxford University Press., 2009, Oxford University Press.