Brexit Risk Implied by the SABR Martingale Defect in the EUR-GBP Smile
aa r X i v : . [ q -f i n . P R ] M a r March 30, 2020 SABR˙Brexit
Brexit Risk Implied by the SABR Martingale Defectin the EURGBP Smile
Petteri Piiroinen , Lassi Roininen and Martin Simon University of Helsinki, Finland Lappeenranta-Lahti University of Technology, Finland Deka Investment GmbH, Germany ()We construct a data-driven statistical indicator for quantifying the tail risk perceived by the EURGBPoption market surrounding Brexit-related events. We show that under lognormal SABR dynamics thistail risk is closely related to the so-called martingale defect and provide a closed-form expression forthis defect which can be computed by solving an inverse calibration problem. In order to cope with thethe uncertainty which is inherent to this inverse problem, we adopt a Bayesian statistical parameterestimation perspective. We probe the resulting posterior densities with a combination of optimizationand adaptive Markov chain Monte Carlo methods, thus providing a careful uncertainty estimation for allof the underlying parameters and the martingale defect indicator. Finally, to support the feasibility ofthe proposed method, we provide a Brexit “fever curve” for the year 2019.
Keywords : Brexit, SABR model, martingale defect, uncertainty quantification, Bayesian estimation,adaptive Markov chain Monte Carlo
JEL Classification : C02, D81, G10, G13
1. Introduction
The British exit (Brexit) from the European Union and the uncertainty surrounding its modalitieshave had an impact on financial markets during the last years. Data-driven approaches to infermarket expectations with regard to Brexit-related events which use option market data have beenproposed recently by Clark and Amen (2017) and by Hanke, Poulsen and Weissensteiner (2018).In this work, we propose a novel forward-looking indicator based on foreign exchange (FX) optionmarket data for quantifying the market expectations with regard to tail risks around importantBrexit-related events. The tail risk observed in FX option markets, i.e., the risk of outlier returnstwo or more standard deviations below the mean, is significantly greater than the risk obtainedunder the theoretical assumption that the returns of the underlying currency pair follows a lognor-mal distribution. Our so-called
SABR GBP martingale defect indicator has first been introducedin the context of quantifying the risk of stock price bubbles, cf. Piiroinen et al. (2018). It is derivedfrom the market price of FX tail risk related to a devaluation of the GBP against the EUR whichis in turn calculated from the prices of out-of-the-money EURGBP options with maturity closeto the event of interest. In 2019 this indicator typically ranged from 0% to 20% and as it rises,the corresponding tail of the return distribution acquires more weight such that the probabilities
The opinions expressed in this article are those of the authors and do not necessarily reflect views of Deka Investment GmbH. arch 30, 2020 SABR˙Brexit of extreme outlier returns become more significant. Within the lognormal SABR model, at somepoint the underlying process becomes a strict local martingale, see Theorem 1 below, and it asbeen shown by Jacquier and Keller-Ressel in Jacquier and Keller-Ressel (2018) that this is indeeda model-independent result: An increase in perceived tail risk increases the relative demand forout-of-the-money options leading to a steepening of the slope of the implied volatility smile. Oneof the key results in Jacquier and Keller-Ressel (2018) is that, under the assumption of fully col-lateralized trades, the total implied variance for a fixed time to maturity in log-strike space attainsan asymptotic slope of 2 if and only if the discounted underlying stochastic process is a strict localmartingale. Strict local martingales have in financial applications been mainly employed to modelstock price bubbles in financial markets, see, e.g., (Cox and Hobson 2005, Jarrow 2017, Jarrow et al. et al. et al. et al.
2. Notation and mathematical setting
Let (Ω , F , F t , P i ), i = £ , e , denote a filtered probability space such that F t satisfies the usualassumptions. On this probability space we will define the stochastic process { S t , t ≥ } to model theforeign exchange rate in the usual FOR-DOM convention capturing a DOM investor’s perspectiveand the process { b S t := S − t , t ≥ } for the DOM-FOR exchange rate which corresponds to thepoint of view of a FOR investor. That is, S t denotes the number of units of domestic currency(DOM) required to buy one unit of foreign currency (FOR) at time t and vice versa for b S t . In thiswork we are interested in the currency pair EURGBP, where EUR is the foreign currency and GBPthe domestic one. We assume that for each currency a risk-free money market account exists suchthat dB i ( t ) = r i ( t )d t, B i (0) = 1 , i = £ , e , where r e denotes the time-dependent continuously compounded foreign interest rate and r £ denotesthe time-dependent continuously compounded domestic interest rate (for the sake of readability wewill suppress the time-dependence in our notation). The DOM investor can trade in the domesticmoney market account B £ ( t ) or the foreign money market account B e ( t ) S t , whereas the FORinvestor may trade in the foreign money market account B e ( t ) or the domestic money market2 arch 30, 2020 SABR˙Brexit account B £ ( t ) b S t . We denote by P £ a domestic equivalent martingale measure , i.e., a probabilitymeasure such that EP £ (cid:26) B e ( T ) S T B £ ( T ) (cid:12)(cid:12)(cid:12) F t (cid:27) ≤ B e ( t ) S t B £ ( t ) (1)and by P e a foreign equivalent martingale measure , i.e., a probability measure such that EP e ( B £ ( T ) S − T B e ( T ) (cid:12)(cid:12)(cid:12) F t ) ≤ B £ ( t ) S − t B e ( t ) . (2)For the rest of this work we assume that a domestic equivalent martingale measure P £ whichsatisfies (1) exists or in other words that the process (cid:26) B e ( t ) S t S B £ ( t ) , t ≥ (cid:27) is a local P £ -martingale. This implies that the market model satisfies NFLVR, cf. Delbean andSchachermayer Delbean and Schachermayer (1994). We do not in general assume that P £ is uniqueas we are going to work with the SABR model which is an incomplete market model – in this settingthe market can be completed in the sense that calibration to observed option market data choosesa particular equivalent martingale measure.An outright forward contract trades at time t at zero cost and leads to an exchange of notionalat time T at the pre-specified outright forward rate F t ( T ) = S t · e ( r £ − r e )( T − t ) . At time T , the foreign notional amount N will be exchanged against an amount of N F t ( T ) domesticcurrency units. FX options are usually physically settled, that is the buyer of a EUR European plainvanilla call with strike K and time to maturity T receives a EUR notional amount N and pays N K
GBP. The value of such plain vanilla options is computed via the standard Black-Scholes-Mertonformula V t ( K, T, φ ) = BSM( F t ; K, T, φ ) = φe − r £ ( T − t ) { F t ( T ) N ( φd + ) − K N ( φd − ) } , where N ( · ) denotes the cumulative normal distribution function, φ = ± d ± = log( F t ( T ) K ) ± σ ( T − t ) σ √ T − t with the Black-Scholes-Merton volatility σ . This volatility for every strike and time to maturitycan be implied from plain vanilla option prices σ M ( F t ; K, T, φ ) = BSM − ( V M t ( K, T, φ ) , F t ; K, T, φ ) . In FX markets, vanilla option prices are commonly quoted via an at-the-money straddle volatilitytogether with quotes for 10-delta and 25-delta risk reversals respectively strangles with expirydates corresponding to overnight maturity, 1, 2 and 3 weeks and 1, 2, 3, 4, 5, 6, 9 and 12 months.Quoting conventions vary depending on the underlying currency pair, expiry and broker, see, e.g.,Reiswich and Wystup (2010, 2012), Wystup (2017b). We use in this work preprocessed market3 arch 30, 2020 SABR˙Brexit data obtained from Refinitiv Financial Solutions which provides composite implied volatilitiesderived from different contributing broker sources versus the corresponding deltas. We convertedthis data into strike space in line with the used market conventions with regard to deltas andATM definition, see Reiswich and Wystup (2010), Wystup (2017b) for a detailed description ofdelta-strike conversion. To sum up, we have for each time to maturity T a vector y which consistsof five mid implied volatility data points y := σ M ( F t ; K , − , T ) σ M ( F t ; K , − , T ) σ M ( F t ; K ATM , T ) σ M ( F t ; K , , T ) σ M ( F t ; K , , T ) with strikes K x,φ , x ∈ { , } , chosen such that the corresponding call (for φ = 1) and put(for φ = −
1) option has a delta of x , together with the corresponding vector of bid ask spreads BA ∈ R .
3. The SABR Martingale Defect in FX Smiles
We fit the volatility smile using the stochastic alpha, beta, rho, or brief SABR model. That is, weassume the following dynamics for the forward process under the domestic equivalent martingalemeasure P £ : d F t ( T ) = α t F t ( T ) β d W (1) t (3)d α t = να t d W (2) t , (4)with fixed elasticity parameter β = 1 (which is a common choice for FX smile modeling), volatilityof volatility ν > P £ -Brownian motions W (1) and W (2) with correlationparameter ρ ∈ [ − , b F t ( T ) := F t ( T ) − under the foreign equivalent martingalemeasure P e are given by d b F t ( T ) = α t b F t ( T ) − β d c W (1) t (5)d α t = ρνα t b F t ( T ) − β d t + να t d c W (2) t , (6)with two correlated P e -Brownian motions c W (1) and c W (2) with correlation parameter − ρ .The following Theorem provides the theoretical foundation of our approach. Theorem Assume that the forward process { F t ( T ) , ≤ t ≤ T } follows the SABR dynamics(3), (4), then we have for all t ∈ [0 , T ] : F ( T ) P e { b F t ( T ) > } = EP £ { F t ( T ) | F ( T ) } . (7) Moreover, we have for all t ∈ (0 , T ] : EP £ { F t ( T ) | F ( T ) } < F ( T ) if and only if ρ > . (8)4 arch 30, 2020 SABR˙Brexit Proof.
The forward F and stochastic volatility α given by equations (3) and (4) are both positiveexponential semimartingales F t = F E ( α · W (1) ) t and α t = α E ( νW (2) ) t , (9)respectively. Here we denote the exponential semimartingale of X with X = 0 as E ( X ) t = exp (cid:18) X t − h X i t (cid:19) and the H · X denotes the stochastic integral with respect to a semimartingale( H · X ) t = Z t H ( s )d X s . By Fatou’s Theorem, the positive exponential semimartingale E ( X ) is a martingale if and onlyif EP £ {E ( X ) t } = 1 for every t >
0. As Cox and Hobson Cox and Hobson (2005) point out, themartingale property follows with an argument due to Sin Sin (1998). According to this work, when β = 1, the expectation of the exponential semimartingale F t /F = E ( α · W (1) ) on the interval [0 , T ]is given by EP £ {E ( α · W (1) ) t } = P { b τ ∞ > t } for every t > P e -probability the stopping time b τ ∞ is the time of explosion ofthe auxiliary process that under P e -probability satisfies the following SDEd v t = νv t d W (3) t + νρv t d t, v = α, where W (3) is a standard Brownian motion under P e . We have to open this up bit more, since theargument reveals the stated properties.First of all, assuming (for the sake of simplicity) the stochastic volatility is bounded, then theNovikov condition implies that the forward F t is a uniformly integrable martingale on [0 , T ]. Inthis case, the Girsanov Theorem implies that the foreign equivalent martingale measure P e is givenby P e { A } = EP £ { F T [ A ] } and the boundedness of stochastic volatility implies that both F and b F stay bounded on [0 , T ],i.e., neither the domestic nor the foreign forward reach zero before time T with probability 1 undereither probability.Therefore, in order that either one is a strict local martingale the stochastic volatility shouldattain unbounded values. The analysis of the SDE already implies that this is the case and moreover,there are no explosions on the bounded interval [0 , T ]. Defining the stopping times τ n = inf { t ∈ (0 , T ] ; α t ≥ n } , n ∈ N and stopped foreign equivalent martingale measures P n, e { A } = EP £ { F τ n T [ A ] } arch 30, 2020 SABR˙Brexit we obtain as in Sin (1998) with Dominated Convergence Theorem and Girsanov Theorem EP £ { F t } = lim n →∞ P n, e { τ n > t } . Note that the stochastic volatility under the stopped foreign equivalent martingale measures P n, e ,n ∈ N satisfies the SDE dα t = [ t ≤ τ n ] ρνα t dt + [ t ≤ τ n ] να t dW (2) t . Note moreover, that if τ n > t , then both F t > b F t >
0, solim inf n →∞ P n, e { b F t > } ≥ lim inf n →∞ P n, e { τ n > t } . This estimate implies that if the limit P n, e { τ n > t } is one, then the forward process F t is a P £ -martingale and P e { b F t > } = 1.Since there are clearly no explosions, when ρ = 0 and by Comparison Theorem, when ρ ≤
0, wecan estimate P n, e { τ ( ρ ) n > t } ≥ P n, e { τ (0) n > t } so there are no explosions when ρ < et al. (2018) that when ρ > EP £ { F t } = P e { τ ∞ > t } < . We already deduced that if there are no explosions on the interval [0 , T ], then necessarily b F t > P e { τ ∞ > t } ≤ P e { b F t > } . In order to show the reverse inequality in the case ρ >
0, we will repeat the previous argument butthis time we use stopping times τ (1) n = inf { t ∈ (0 , T ] ; F t ≥ n } , n ∈ N and second the set of stopped foreign equivalent martingale measures P (1) n, e { A } = EP £ { F τ (1) n T [ A ] } . Repeating the same argument we notice that EP £ { F t } = lim n →∞ P (1) n, e { τ (1) n > t } . By the uniqueness of the SDE we can deduce that the previous limit islim n →∞ P (1) n, e { τ (1) n > t } = P e { τ (1) ∞ > t } = P e { b F t > } . This finally implies the equation (7) by relating the explosion probability with the probability of b F not hitting zero before time t . 6 arch 30, 2020 SABR˙Brexit We should emphasize that the fact that Theorem 1 considers the martingale defect only underthe domestic equivalent martingale measure poses no restriction to generality because of the well-known DOM-FOR symmetry which we recall, for the sake of self-containedness, in the followingLemma.
Lemma Let b V denote plain vanilla option prices under the foreign equivalent martingale mea-sure, then V t ( K, T, φ ) = BSM( F t ; K, T, φ ) = S t BSM( b F t ; K , T, − φ ) = S t K b V t ( K , T, − φ ) , where φ = ± for a call and put option, respectively.Proof. The DOM investor values the call with payoff ( S T − K ) + under P £ at GBP V t ( K, T,
1) = EP £ { ( S T − K ) + | F t } and hence EUR V t ( K, T, /S t . Since ( S T − K ) + = KS T ( K − S T K ) + together with the change of measure formula for the martingale measures for every elementaryevent A ∈ Ω: P e { A } = EP £ { F T [ A ] } and S T = F T , we have EP £ { ( S T − K ) + | F t } = K EP e { ( K − b S T ) + | F t } . This implies that we can interpret the original call in EUR as a put in GBP with payoff K ( K − b S T ) + .The price of this GBP put in EUR is K b V t ( K , T, −
1) = K EP e { ( K − b S T ) + | F t } . Therefore, V t ( K, T, /S t = K b V t ( K , T, − φ = 1 follows. The claim for φ = − − Remark
From the FOR investor’s perspective { b F t ( T ) , ≤ t ≤ T } has a positive mass at zerounder P e if and only if from the DOM investor’s perspective { F t ( T ) , ≤ t ≤ T } is a strictlocal martingale under P £ and vice versa. Consider a contingent claim paying out one EUR at T and set for simplicity r £ = r e = 0 . The FOR investor will price this at one EUR at anytime t independent of what happens to the GBP, whereas the DOM investor values the contract at F ( T ) − EP £ { F t ( T ) | F ( T ) } which equals one EUR as long as { F t ( T ) , t ≥ } is a true martingaleunder P £ . However, if { F t ( T ) , t ≥ } is a strict local martingale under P £ , then the value obtainedby the DOM investor is strictly less than one EUR at any time before T which comes from the factthat under P e , the process { b F t ( T ) , t ≥ } can hit zero in finite time. The economic interpretationof this mathematical property is that both investors fear the extreme event of a total devaluationof the GBP which has a non-zero probability from the FOR investor’s perspective and manifests arch 30, 2020 SABR˙Brexit itself in the presence of a strict martingale rather than a true martingale from the DOM investor’sperspective. Note moreover, that the DOM investor prices the contingent claim under the conditionthat the GBP has some strictly positive value. When pricing under the same positivity condition,the FOR investor obtains the same value strictly less than one EUR as the DOM investor. In analogy to the risk indicator defined in the work Piiroinen et al. (2018) let us define thefollowing quantity d £ ( T ; θ ) = 1 − F ( T ) − EP £ { F t ( T ) | F ( T ) } = P e { b F t ( T ) = 0 } (10)for given SABR parameters θ = ( α, ν, ρ ) T which quantifies the perceived tail risk assigned tothe event of a massive devaluation of the GBP against the EUR. We call this the normalizedGBP SABR martingale defect for maturity T . Moreover, we define what we call the GBP SABRmartingale defect indicator via the limit A £ ( θ ) := lim T →∞ d £ ( T ; θ ) = 1 − exp( − ρα/ν ) . (11)We prefer the indicator (11) over the normalized GBP SABR martingale defect (10) for maturity T because it enables risk comparison between different maturities. We think that this is importantbecause when approaching Brexit related events, the market participant’s expectations and fearsare reflected most prominently in those options expiring shortly after the particular event so thatthe relevant time to maturity decreases while approaching the event. Moreover, we justify ourpreference by the monotony properties of the martingale defect which can be seen from Figure 1below. T =[0.47, 0.64, 0.05] T =[0.55, 0.64, 0.05] T =[0.47, 0.64, 0.15] T d £ ( T ; θ ) Figure 1.: Normalized GBP SABR martingale defect (10) for different parameter vectors θ plottedagainst the maturity T in years on the x -axis. 8 arch 30, 2020 SABR˙Brexit
4. Statistical Method
It has been shown in Jacquier and Keller-Ressel (2018) that in option markets where trades arefully collateralized, the martingale defect can, in theory, be inferred equivalently from both observedput and call implied volatility surfaces. However, as the result is asymptotic in nature it is necessaryto extrapolate from the data observed in the market and, as in Piiroinen et al. (2018), we employthe lognormal SABR model for this task. As we have already pointed out, FX implied volatilitysmiles are usually calibrated using merely 5 broker quotes per maturity time slice and the reliabilityof the 10-delta quotes is somewhat questionable. In order to account for this inherent calibrationuncertainty, we adopt a statistical perspective on the problem of calibrating the SABR model (3),(4) in order to obtain the market implied SABR martingale defect indicator (11). To be precise,all quantities are considered as random variables with certain prior distributions that incorporateour prior knowledge about them. In this setting we can define a statistical inverse problem whosesolution is given by the posterior distribution of the SABR parameters θ = ( α, ν, ρ ) T conditioned onthe observed bid and ask market quotes which in turn yields the posterior probability distributionof the quantity of interest, the martingale defect indicator (11), conditioned on the observed marketquotes. Statistical Inverse Problem
Let (Ω ′ , G , P) denote a probability space and let( Θ , E ) : Ω ′ → R , Y : Ω ′ → R (12)denote random vectors on this probability space. We use capital letters for random vectors andlower case letters for their realizations. The vector ( Θ , E ) represents the quantities that cannotbe directly observed, i.e., the unknown SABR parameters θ = ( α, ν, ρ ) T and an unknown errorvector E which accounts for discrepancies between SABR model and quotes observed in the marketwhereas Y represents the vector of mid implied volatilities. To be precise, for a fixed T we definethe random SABR-parameter-to-implied-volatility-map( Θ , E )
7→ L ( Θ , E ) = f ( Θ ) + E = σ SABR ( F t ; K , − , T, Θ ) σ SABR ( F t ; K , − , T, Θ ) σ SABR ( F t ; K ATM , T, Θ ) σ SABR ( F t ; K , , T, Θ ) σ SABR ( F t ; K , , T, Θ ) + E =: Y , where for a given realization θ , the model implied volatility σ SABR ( F t ; K, T, θ ) is computed via thesecond order asymptotic formula obtained by Paulot, cf. Paulot (2015): σ ( F t ; K, T, θ ) (cid:18) σ ( F t ; K, T, θ ) σ ( F t ; K, T, θ ) T + σ ( F t ; K, T, θ ) σ ( F t ; K, T, θ ) T + o ( T ) (cid:19) with σ ( F t ; K, T, θ ) , σ ( F t ; K, T, θ ) and σ ( F t ; K, T, θ ) defined as in Paulot (2015). This formulacan be evaluated very efficiently which is crucial with regard to the computational cost of ourstatistical sampling method. At the same time the formula can be shown to be highly accurate inthe strike and maturity regimens which we are interested in here. The probability distribution ofthe random vector Y conditioned on the vectors θ and e is given by π ( y | θ , e ) = δ ( y − L ( θ , e )) , arch 30, 2020 SABR˙Brexit where δ denotes Dirac’s delta in R k . Let π pr denote the prior probability density of ( Θ , E ), thenwe may write the joint probability density of ( Θ , E ) and Y as π ( θ , e , y ) = π ( y | θ , e ) π pr ( θ , e ) = δ ( y − L ( θ , e )) π pr ( θ , e ) . (13)For simplicity, we assume here that Θ and E are independent random variables. Then we obtainfrom (13) by integration π ( θ , y ) = π pr ( θ ) π noise ( y − L ( θ )) (14)so that we can formulate the following statistical inverse calibration problem: Compute the posterior distribution of Θ conditioned on the observed market implied volatility quotes y which is given by Bayes’ formula π ( θ | y ) = π ( θ , y ) R R π ( θ , y )d θ . (15) Given the (15), compute the posterior density for our quantity of interest, the martingale defectindicator π ( A £ ( θ ) | y ) . (16) Sampling the Posterior Density
The unnormalized posterior density reads π ( θ | y ) ∝ π ( θ ) π ( y | θ ) , where π ( θ ) and π ( y | θ ) are the prior and likelihood probability density, respectively. We factorizethe prior as π ( θ ) = π ( α ) π ( ν ) π ( ρ ) = [ α ∈ R][ ν ≥ | ρ | ≤ , where we have a flat prior for α , flat prior in R + for ν , and a uniform prior for ρ ∈ [ − ,
1] and fornotational convenience, we have used the Iverson bracket [ · ] as an indicator function:[ B ] := ( , if B is true,0 , otherwise.Our prior construction is an improper prior, however, in practical numerical computations in con-nection with the likelihood density, this posterior density becomes a proper probability density.This means that in contrast to sampling from an improper prior density, sampling the correspond-ing posterior is indeed feasible. For a discussion on using improper priors in MCMC samplingschemes, we refer to Hobert and Casella Hobert and Casella (1996).We assume that the observation error is Gaussian such that the likelihood function is given by π ( y | θ ) ∝ exp (cid:18) −
12 ( y − f ( θ )) T Σ − ( y − f ( θ )) (cid:19) k Y i =1 (cid:20) | y i − f i ( θ ) | ≤
12 BA i (cid:21) , arch 30, 2020 SABR˙Brexit σ M ( F ; K , T ) T K
Figure 2.: Raw input implied volatility surface obtained from linear interpolation of the mid marketquotes for August 28. The anticipated event risk with regard to October 31 is clearly visible forthe expiry dates 1M and 2M.where Σ is covariance matrix of the observation error E , and BA i is the bid-ask spread at the i -th strike in terms of the implied volatilities. The corresponding posterior proves to be difficult tostudy analytically, as we have a non-linear parameter estimation problem with somewhat complexpriors and constraints. To cope with this complexity we use an adaptive combined optimizationand MCMC sampling algorithm. With Nelder-Mead optimization, we compute the maximum aposteriori (map) estimate. Given this map estimate as a start value, we use adaptive MCMC in thesense of Haario, Saksman and Tamminen Haario et al. (2001) for both, obtaining the conditionalmean estimator which approximates the conditional expectation EP { A £ ( Θ ) | y } and for providing uncertainty quantification for all the unknown parameters and the GBP martin-gale defect indicator by estimation of the corresponding marginal distributions. For more detailson the algorithm, we refer to the work Piiroinen et al. (2018).
5. Results
For the observation dates listed in Appendix A, we have computed the conditional mean of the GBPSABR martingale defect indicator for the solution (15) of the statistical inverse calibration problemconditioned on the observed market implied volatility quotes for the currency pair EURGBP. Thedata was obtained from Refinitiv at the New York cut at 10 a.m. ET. Brexit was originally meantto happen on March 29 2019, but on March 22 the EU-27 approved a Brexit extension untilApril 12. On April 10 2019, a further half-year extension was agreed between the UK and theEU-27 at the EU summit, until October 31 2019. Taking this into account, we have chosen therelevant expiry dates according to Table 1. As is illustrated by Figure 2, the perceived eventrisk manifests itself in both magnitude and skewness of the quoted nearby volatility time slices.Table 1.: Expiries used for the computation of the martingale defect indicator.15Jan 12Mar 14Mar 21Mar 29Mar 10Apr 24May 24Jul 28Aug 03Sep 09Sep 17Oct 19Oct2M 2W 2W 1W 2W 1W 5M 3M 2M 2M 2M 2W 2W11 arch 30, 2020 SABR˙Brexit J an 12 M a r M a r M a r M a r A p r M a y J u l A ug 03 S ep 09 S ep 17 O c t O c t A £ ( θ ) Figure 3.: The Brexit “fever curve” given by the conditional mean estimates obtained from samplingthe posterior density (16).Our main result is the Brexit “fever curve” depicted in Figure 3 which shows the conditionalmean estimate obtained from sampling the posterior density (16) of the GBP SABR martingaledefect indicator (10) for the observation dates and expiry dates in Table 1. It should be mentionedthat after October 19 and until December 11 (the date of publication of the preprint of this work),the perceived tail risk measured via the martingale defect indicator stayed below 2% for all expirieslonger than two weeks. In particular the perceived tail risk around the general election on December12 was mainly related to short-term moves. The computation is based on 100000 MCMC samplesper observation day and for the observation error we have assumed zero-mean white noise withunit variance. For illustration purposes the implied volatility smile obtained from the conditionalmean estimate of the random SABR parameter Θ conditioned on the 6M observed market data y for the observation day April 10 is depicted in Figure 4. Note that this expiry is different fromthe one we have used for computing the respective point in the “fever curve” which is 1W. Thebackground is that on April 10 a half-year extension was agreed between the UK and the EU-27,until October 31 2019. As a result, during that trading day the pronounced skewness of the impliedvolatility smile shifted from the short term expiry dates to those expiry dates around October 31,i.e., 5M, 6M and 9M. This is visible in the corresponding GBP SABR martingale defect indicators.While the perceived 1W short term risk measured by our MCMC approximation of EP { A £ ( Θ ) | y } is merely 1 . . α , ρ , ν and the martingale defect indicator A £ ( θ ). We also plot the marginal densities obtained by removing the burn-in period (25% of thewhole chain length), and by using a kernel density estimator with Epanechnikov kernels. By visuallyassessing, we note that we have good mixing of the chains, however, without the optimization stepfor choosing the start value, the chains would not converge.12 arch 30, 2020 SABR˙Brexit σ S A B R ( F t ; K , T , θ ) K Figure 4.: April 10 implied volatility smile obtained from the conditional mean of the random SABRparameter vector Θ conditioned on the 6M observed bid and ask volatility quotes given by thered x markers (o markers are the corresponding mids). The corresponding GBP SABR martingaledefect indicator is 8 .
024 0 5 1010 A £ ( θ ) A £ ( θ ) MCMC sample p a r a m e t e r v a l u e d e n s i t y parameter value Figure 5.: October 19 sampled MCMC chains (upper row) and corresponding estimated densities(lower row). 13 arch 30, 2020 SABR˙Brexit
6. Conclusion
We found that the martingale defect which occurs in the lognormal SABR model in the presence ofextremely skewed implied volatility smiles may be used to asses the market expectations of extremeevents such as a no-deal Brexit. For this purpose we introduced a statistical SABR martingale defectindicator which quantifies the market expectation for the GBP to progressively depreciate againstthe EUR based on observed EURGBP option prices. This forward-looking measure of marketexpectations accounts for the inherent uncertainty due to the small number of reliable volatilityquotes observable in the market and it should be of great use for risk management purposes suchas data-driven risk scenario generation or stress testing. Finally, we would like to point out thatthe “fever curve” we have computed for a timeline of Brexit related events in 2019 quantifiesremarkably well the public perception of economic risk related to a no-deal Brexit scenario.
7. Afterword
This paper is the final version of a preprint dated December 11 2019 that originally appeared onarXiv (https://arxiv.org/abs/1912.05773) on December 12 2019, in advance of the general electionon the same day.
Funding
This work has been funded by Academy of Finland (decision numbers 326240 and 326341, andFinnish Centre of Excellence in Inverse Modelling and Imaging, decision number 312119).
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FX options and structured products , 2017, (John Wiley & Sons: Oxford, UK). arch 30, 2020 SABR˙Brexit Appendix A: Timeline of events for the tested dates
Table A1.: 2019 Brexit Timeline part 1/2. Source: Wikipedia
15 Jan • The First meaningful vote is held on the WithdrawalAgreement in the UK House of Commons. The UKGovernment is defeated by 432 votes to 202.
12 Mar • The Second meaningful vote on the WithdrawalAgreement with the UK Government is defeated againby 391 votes to 242.
14 Mar • The UK Government motion passes 412 to 202 toextend the Article 50 period.
21 Mar • The European Council offers to extend the Article 50period until 22 May 2019 if the Withdrawal Agreementis passed by 29 March 2019 but, if it does not, then theUK has until 12 April 2019 to indicate a way forward.The extension is formally agreed the following day.
29 Mar • The original end of the Article 50 period and theoriginal planned date for Brexit. Third vote on theWithdrawal Agreement after being separated from thePolitical Declaration. UK Government defeated again by344 votes to 286.
10 Apr • The European Council grants another extension to theArticle 50 period to 31 October 2019, or the first day ofthe month after that in which the WithdrawalAgreement is passed, whichever comes first.
24 May • Theresa May announces that she will resign asConservative Party leader, effective 7 June, due tobeing unable to get her Brexit plans through parliamentand several votes of no-confidence, continuing as primeminister while a Conservative leadership contest takesplace.
24 Jul • Boris Johnson accepts the Queen’s invitation to form agovernment and becomes Prime Minister of the UnitedKingdom, the third since the referendum. arch 30, 2020 SABR˙Brexit Table A2.: 2019 Brexit Timeline part 2/2. Source: Wikipedia
28 Aug • Boris Johnson announces his intention to prorogueParliament in September. • A motion for an emergency debate to pass a bill thatwould rule out a unilateral no-deal Brexit by forcing theGovernment to get parliamentary approval for either awithdrawal agreement or a no-deal Brexit. This motion,to allow the debate for the following day, passed by 328to 301. 21 Conservative MPs voted for the motion. • The Government again loses an attempt to call anelection under the Fixed-term Parliaments Act. DominicGrieve’s humble address, requiring key Cabinet Officefigures to publicise private messages about theprorogation of parliament, is passed by the House ofCommons. Speaker John Bercow announces hisintention to resign as Speaker of the House ofCommons on or before 31 October. The Benn Billreceives Royal Assent and becomes the European Union(Withdrawal) (No. 2) Act 2019. Parliament is prorogueduntil 14 October 2019. Party conference season begins,with anticipation building around a general election.
17 Oct • The UK and European Commission agree on a revisedwithdrawal agreement containing a new protocol onNorthern Ireland. The European Council endorses thedeal.
19 Oct • A special Saturday sitting of Parliament is held todebate the revised withdrawal agreement. The primeminister moves approval of that agreement. MPs firstpass, by 322 to 306, Sir Oliver Letwin’s amendment tothe motion, delaying consideration of the agreementuntil the legislation to implement it has been passed;the motion is then carried as amended, implementingLetwin’s delay. This delay activates the Benn Act,requiring the prime minister immediately to write to theEuropean Council with a request for an extension ofwithdrawal until 31 January 2020. Department of Mathematics and Statistics, University of Helsinki, FI-00014 Helsinki, Finland . E-mail address : [email protected] School of Engineering Science, Lappeenranta-Lahti University of Technology, FI-53850 Lappeenranta, Finland . E-mail address : [email protected] Deka Investment GmbH, 60325 Frankfurt am Main, Germany .M. Simon
E-mail address : [email protected]@simon-martin.net