How much is your Strangle worth? On the relative value of the δ− Symmetric Strangle under the Black-Scholes model
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On the relative value of the δ − Symmetric Strangle under theBlack-Scholes model
Ben Boukai ∗ Department of Mathematical Sciences, IUPUI, Indianapolis,IN 46202 , USAMay 19, 2020
Abstract
Trading option strangles is a highly popular strategy often used by mar-ket participants to mitigate volatility risks in their portfolios. In this paper wepropose a measure of the relative value of a delta-Symmetric Strangle andcompute it under the standard Black-Scholes option pricing model. Thisnew measure accounts for the price of the strangle, relative to the PresentValue of the spread between the two strikes, all expressed, after a natural re-parameterization, in terms of delta and a volatility parameter. We show thatunder the standard BS option pricing model, this measure of relative value isbounded by a simple function of delta only and is independent of the time toexpiry, the price of the underlying security or the prevailing volatility usedin the pricing model. We demonstrate how this bound can be used as a quick benchmark to assess, regardless the market volatility, the duration of thecontract or the price of the underlying security, the market (relative) value ofthe δ − strangle in comparison to its BS (relative) price. In fact, the explicitand simple expression for this measure and bound allows us to also study indetail the strangle’s exit strategy and the corresponding optimal choice for avalue of delta. ∗ [email protected]; Tel: + + eywords : Call-put parity, option pricing, the Black-Merton-Scholesmodel, European options Options, as asset’s price derivatives, are the primary tools available to the mar-ket participants for hedging their portfolio from directional risk and / or volatilityrisk. The so-called option’s delta, which typically is denoted as δ or ∆ , measuresthe ’sensitivity’ of the option’s price to changes in the price of the underlyingsecurity, is the primary parameter one considers when using an option to miti-gate directional risk. The option’s delta is seen as the hedging ratio and is oftenalso used (near expiration) by market participants as a surrogate to the probabilitythat the option will expire in the money. With standard option pricing model ofBlack and Scholes (1973), (abbreviated here as the BS model, see below), theseprobabilities are readily available for direct calculations under the governing log-normality assumption of the asset’s returns. Roughly speaking, a trader that sells(or buys) a put option at a strike located one standard deviation below the currentasset’s price, ends up with a 16-delta put contract option (i.e. with δ = − . k − . .Similarly, a trader that sells (or buys) a call option at a strike located one standarddeviation above the current asset’s price, ends up with a 16-delta call option (i.e. δ = . k + . . Thus, the corresponding strangle, which is obtained by selling a (negative)16-delta put option and a (positive) 16-delta call option, is a delta-neutral strategythat is associated, very roughly, with a 0.68 probability for the asset’s price toremain between the two strikes, k − . and k + . by expiration, all as resulting fromthe governing normal distribution assumption. We refer to such a strangle as a , only to indicate the common (absolute) delta value( δ = .
16) of its put and call components.In a similar fashion we use the term a δ − Symmetric Strangle to indicate thestrangle obtained, for some fixed δ ∈ (0 , . δ -unitsput and call option contracts at the corresponding strikes k − δ and k + δ , respectively.Such a strangle would be a delta-neutral strategy o ff ering zero directional risk butpotentially useful for mitigating volatility risk. We further denote by Π δ the priceof (or the credit received from) such δ − Symmetric Strangle . In this paper, we2tudy, for a a given δ , the value of this δ − Symmetric Strangle relative to the widthof the corresponding spread ( k + δ − k − δ ), adjusted for its present value (PV). Moreprecisely, for any a δ ∈ (0 , . relative value of the corresponding δ − Symmetric Strangle as R δ : = Π δ PV ( k + δ − k − δ ) . (1)In Section 2, we show that under the standard BS option pricing model, thestrangle’s relative value, R δ , is independent of the price of the underlying securityand is a function only of δ and the prevailing volatility used in the pricing model.In fact, as we will see in Theorem 1 below, for any given δ ∈ (0 , . R δ ≤ R δ where R δ = − φ ( z δ ) z δ − δ, (2)and where φ ( · ) is the standard normal density ( pd f ), φ ( u ) : = √ π e − u , and z δ ≡ Φ − ( δ ), is usual δ th percentile of the standard normal distribution, whose cumu-lative distribution function ( cd f ) is Φ ( z ) : = R z −∞ φ ( u ) du . We point out that since δ < .
5, we have z δ < R δ .As an illustration, one quickly finds by utilizing (2) that the 16 -delta Symmet-ric Strangle has a relative value of R . = . -delta Sym-metric Strangle has a relative value of R . = .
36. That is to say that underthe standard BS option pricing model, one would expect the price of the 30 -deltaSymmetric Strangle to be at most
36% of the width of the spread between thecorresponding strikes, irrespective of the security’s price, or time to expiry, andirrespective of the prevailing volatility. More generally, it follows from Theorem1, that for a any given δ ∈ (0 , . δ − strangle’s price, Π δ , ascalculated under the BS pricing model , satisfies Π δ ≤ R δ × PV ( k + δ − k − δ ) , irrespective of the security’s price, or time to expiry, and irrespective of the pre-vailing volatility. In Section 3, we illustrate how this measure R δ in (2) maybe used as a benchmark to assess the market pricing (or ’worthiness’) of the δ − symmetric strangle compared to its (relative) price, R δ , as suggested by stan-dard BS pricing model. The explicit expression of R δ as is given in (2) allow usto also address, in Section 4, the strangle’s exit strategy and the corresponding optimal choice of δ for it. 3 Pricing the δ -unit option contract One of the most widely celebrated option pricing model for equities (and beyond)is that of Black and Scholes (1973). Their pricing model is derived under somesimple assumptions concerning the distribution of the asset’s returns, coupled withpresumptive continuous hedging, zero dividend, risk-free interest rate, r , and nocost of carry or transactions fees. While the aptness of these assumptions hasoften been criticized (see for example Yalincak (2012)), it has remained as a lead-ing option pricing model for the retail trading practitioner (e.g.: Sinclair (2010)).However, in its standard form, the BS model evaluates, for a risky asset with acurrent market price µ , the price of an European call option contract at a strike k and t days to expiration as: c µ ( k ) = µ × Φ ( d ( k )) − k · e − rt × Φ ( d ( k )) . (3)Here, using the standard notation, d ( k ) : = log( µ k ) + ( r + σ ) t σ √ t and d ( k ) : = d ( k ) − σ √ t , (4)where σ denotes the standard deviation of the daily asset’s returns, and Φ ( · ) is thestandard normal cdf defined above. The model for the corresponding price of a putoption contract, p µ ( k ), may be obtain from expression (3) of c µ ( k ), by exploitingthe so-called put-call parity which is expressed by the equation µ − c µ ( k ) = k · e − rt − p µ ( k ) , (5)see for example Jiang (2005, Theorem 2.3) or Peskir and Shiryaev (2002) fordetails. This parity implies that the price of the corresponding put option contractis, p µ ( k ) = k · e − rt × Φ ( − d ( k )) − µ × [1 − Φ ( d ( k ))] . (6)There is substantial body of literature dealing with the BS option pricingmodel in (3)-(6), its refinements, its extensions and the so-called, its implied’Greeks’ (i.e. the various partial derivatives of di ff erent orders, representing themodel’s ”sensitivities” to changes in its parameters). The interested reader is re-ferred to standard textbooks such as Wilmott, Howison, Dewynne (1995), Hull(2005), Jiang (2005) or Iacus (2011). 4s we already mentioned in the Introduction, we focus our attention here onthe option’s delta , which we denote by ∆ as a function with a corresponding valueof δ ∈ (0 , r , t and σ , we define for the call and the put contracts optionstheir respective ∆ functions as, ∆ c ( k ) : = ∂ c µ ( k ) /∂µ and ∆ p ( k ) : = ∂ p µ ( k ) /∂µ . Itfollows immediately from the put-call parity equation in (5) that ∆ p ( k ) = − (1 − ∆ c ( k )). It is well known (see for Example, Jiang (2005)) that for the BS pricingmodel in (3), ∆ c ( k ) = Φ ( d ( k )), where d ( k ) is given in (4), and hence ∆ p ( k ) = − (1 − Φ ( d ( k )) ≡ − Φ ( − d ( k )).For its supreme importance to portfolio hedging, the investor / trader often needsto buy (or sell) an option at a strike, k , which is associated with a specified anddesired value δ of the option’s ∆ . For any given δ ∈ (0 , k + δ denote the(unique) solution of the equation ∆ c ( k + δ ) = δ , or equivalently the solution of Φ ( d ( k + δ )) = δ. (7)Accordingly, it follows immediately that k + δ satisfies the equation d ( k + δ ) = Φ − ( δ ) ≡ z δ , (8)and hence, by utilizing (4) in (8) leads to the solution as k + δ = µ · e − z δ ν + ν / + rt , (9)where we have substituted ν ≡ σ √ t throughout. It should be clear from (9) that if δ < .
5, one has z δ < k + δ > µ , so that the corresponding call optionis said to be ’out of the money’ (OTM). Also, note that it follows from (4) and (8)that d ( k + δ ) = d ( k + δ ) − σ √ t ≡ z δ − ν , so that Φ ( d ( k + δ )) = Φ ( z δ − ν ) (10)in (3). Indeed, with the re-parameterization by ( δ, ν ) (with ν ≡ σ √ t ), of theBS option pricing model in (3), we may re-express, upon using the matchingexpressions (7)-(10) in equation (3), the current price of a δ − unit call option in amuch simpler form as c µ ( δ, ν ) ≡ c µ ( k + δ ) = µ × δ − k + δ · e − rt × Φ ( z δ − ν ) = µ × h δ − e − z δ ν + ν / × Φ ( z δ − ν ) i , (11)5or any δ ∈ (0 ,
1) and with ν > Remark 1:
Note in passing that in practice, the option’s δ is often used as a crudeapproximation to the probability the option will end in the money, Pr ( IT M ) , whichby (10), (11) is equal to Φ ( d ( k + δ )) ≡ Φ ( z δ − ν ) . However, since ν ≡ σ √ t > , itimmediately follows that Φ ( z δ − ν )) ≤ Φ ( z δ )) ≡ δ . Hence, for any δ ∈ (0 , and ν > , Pr ( IT M ) ≤ δ and only near expiration, as lim t → Pr ( IT M ) = δ , it holds. Similarly to (11), we calculate the current price of the δ − unit put contractoption by using the put-call parity equation in (5), and by noting that by (7) thecorresponding k − δ strike for the put contract is the same as the strike k + − δ of the(1 − δ ) − unit call option contract, so that k − δ ≡ k + − δ . Accordingly, since z δ ≡ − z − δ ,we obtain from (9) that k − δ = µ · e z δ ν + ν / + rt . (12)Hence, it follows immediately from (5) and (12), that under the ( δ, ν ) re-parameterization,the expression for the current price of the δ − unit put option is, p µ ( δ, ν ) ≡ p µ ( k − δ ) = − δµ + k − δ · e − rt × (1 − Φ ( z − δ − ν )) = − µ × h δ − e z δ ν + ν / × Φ ( z δ + ν ) i . (13)It should be clear from (12) that if δ < . r =
0, one has z δ < k − δ < µ only if ν < − · z δ , in which case, the corresponding put optionis said to be ’out of the money’ (OTM). Hence, we will restrict our attentionto the practical case of the above parametrization in which ( δ, ν ) are such that k − δ < µ < k + δ , or alternatively, ( δ, ν ) ∈ B , where B = { ( δ, ν ); δ > , & ν > , s.t. δ < Φ ( − ν/ } . We further point out that the two strikes, k + δ and k − δ , ( ≡ k + − δ ), need not be symmet-rical with respect of the current price µ of the underlying security (i.e.: µ − k − δ , k + δ − µ ). It is well-known that the occasional observed asymmetry of these equal δ − units strikes is a fixture of the skew in the volatility surface that is a ff ectingthe option pricing model, see for example Gatheral (2006), or Doran and Krieger(2010). δ -Symmetric Strangle Consider now a trader that desires to simultaneously sells (say), at some givenlevel of δ < .
5, the δ − unit put and the δ − unit call contracts so as to form the6OTM’ δ − Symmetric Strangle strategy. The total selling price of this strangle ascalculated under the BS pricing model, is therefore Π δ = c µ ( k + δ ) + p µ ( k − δ ). As ameasure for assessing the ’worthiness’ of this strangle, we consider the ’value’ ofthe selling price, Π δ , relative to the present value of the spread between the strikes,namely, PV ( k + δ − k − δ ) = ( k + δ − k − δ ) × e − rt . We express this relative value measure in(1) as R ( δ, ν ) : = Π δ PV ( k + δ − k − δ ) = c µ ( δ, ν ) + p µ ( δ, ν )( k + δ − k − δ ) × e − rt . (14)Note that by its definition, R ( δ, ν ) ≥ δ ∈ (0 , .
5) and ν >
0, in par-ticular over B . Further, since an European option price is (linearly) homogeneousin µ , and in the strike, k , (see Theorem 6 of Merton (1973)), the ratio R ( δ, ν )in (14), is independent of the current price, µ , of the underlying security. Alsonote that since we account in (14) for the present value of the spread between thestrikes, this quotient is, by construction, also independent of the risk-free interestrate, r . This can fully realized by substituting expressions (9), (11), (12) and (13)in R ( δ, ν ) and simplifying the resulting terms, to obtain, for each δ < . ν ≡ σ √ t >
0, the expression, R ( δ, ν ) = e z δ ν · Φ ( z δ + ν ) − e − z δ ν · Φ ( z δ − ν ) e − z δ ν − e z δ ν , (15)for the relative value of the δ − Symmetric Strangle under the BS option pricingmodel. We point that the values of R ( δ, ν ) in (15) are straightforward to calculatefor any ( δ, ν ). Figure 1 below provides the graph of R ( δ, ν ) for various values of( δ, ν ), with 0 < δ < . < ν <
1, where ν = σ √ t representing realistic valuesfor t (the time in days to expiry) and the model’s daily (implied or historical)volatility, σ . In any case, the properties of R ( δ, ν ), as a function of δ and ν (in B )are of interest. In Appendix A below we show that for a fixed δ < . R ( δ, ν ) ismonotonically non-increasing function of ν (with ∂ R /∂ν ≤
0) and that for a fixed ν > R ( δ, ν ) is monotonically increasing function of δ (with ∂ R /∂δ > Theorem 1
Under the BS model and irrespective of the current price, µ , of theunderlying security, the current risk-free interest rate, r, and irrespective of thetime to expiry, t, and the presumed volatility (either implied or historical), theupper bound to the relative value R ( δ, ν ) , of the OTM δ − Symmetric Strangle with e l t a nu R d Figure 1:
The Relative Value function R ( δ, ν ) of the δ − Symmetric Strangle for δ < . and ν ∈ (0 , . δ ∈ (0 , . , depends only on δ and is given by, < R ( δ, ν ) ≤ R δ , where R δ : = lim ν → R ( δ, ν ) = − φ ( z δ ) z δ − δ. (16) Moreover, lim δ → R δ = , and for all δ ∈ (0 , . , R ′ δ : = dd δ R δ = z δ > . (17) Proof.
That R ( δ, ν ) is a monotonically decreasing function of ν for each fixed δ ∈ (0 , .
5) is seen by direct calculation, ∂ R ( δ, ν ) /∂ν ≤ R ( δ, ν ) and noting that it trivially also independent of µ and r by construction.By another direct application of L’Hopital’s rule to the quotient φ ( z δ ) / z δ along withthe facts that dd δ φ ( z δ ) = − z δ φ ( z δ ) z ′ δ and z ′ δ = /φ ( z δ ) leads to the second assertionas well as to the result stated in (17).The results of Theorem 1 and the bound R δ in (16) provide a benchmark forassessing the value, in relative terms, of a δ − Symmetric Strangle under the BS8ption pricing model in (3)-(6), as applicable to any security (i.e. independentof the current underlying security price µ ), to any expiry (independent of t ), andunder any presumed volatility (independent of σ ). In fact, if ˆ R δ denotes the market(relative) value of a δ − Symmetric Strangle (i.e. the market version of (1)), then,this strangle would be deemed ’well-priced’ compared to its (relative) price underthe BS option pricing model, as long as ˆ R δ ≥ R δ . In Figure 2 below, we graph thevalues of this function, R δ (in (16) or (2)) for all 0 < δ < . . . . . . . d R _de l t a Figure 2:
The relative value R δ as a function of δ . Marked in red is the market relativevalue (current, as of EOD, May, 13th, 2020), ˆ R . = . , of a 34-delta symmetricstrangle with strikes $112 and $120 in IBM (see Example 1 and Table 1 for more details.) Remark 2:
The results stated in Theorem 1 and their derivations are valid inthe BS ’world’, in which the distribution of the asset’s returns assumed to have aconstant variability throughout and do not take into account the volatility ’skew’or ’smile’ that is often being observed by the traders across the discretized op-tions’ grid equipped with bid-ask price spreads. It surely implies that the BSpricing model (with all it inputs) undervalues the δ -Symmetric Strangle, when-ever R δ < ˆ R δ , where ˆ R δ is its market (relative) value (i.e. the market version of(1)). Example 1:
As an illustration of it’s usage, consider the market EOD (end ofday) market pricing of IBM (International Business Machine Corp.) as of May13th, 2020. We find that the 34 − delta symmetric strangle for the June 5th, 2020expiration with the strikes of k = $112 and k = $120 for the sold put and call,9espectively, has a market mid-price of ˆ Π . = $4 .
60 (along with current tickerprice of µ = $115.73, with t =
23 days to expiration, and IV = .
32% (average)implied volatility, so that σ = IV / √ = . R . : = ˆ Π . ( k − k ) = . − = . , for this 34-delta Strangle in IBM, whereas, by using (2), we calculate under theBS pricing model a relative value of R . = .
548 for this 34-delta strangle. Thus,the BS pricing model (with its constant variance assumption, etc.) under-valuesthis strangle (in relative terms) as compared to its actual market value. Similarresult is obtained with the relative value of a 21-delta strangle with 100 days toexpiration in BA (Boeing Co.), which yields ˆ R . = .
156 as compared to R . = . δ − symmetric strangles for additional securities, with di ff erent δ , underlying prices, IV and days to expiration. In all cases listed in the Table, the market (relative)value ˆ R δ exceeded that of the corresponding BS (relative) value R δ . Thus, in thesenoted cases, the BS pricing model (with all its inputs) appears to undervalue thestrangles (in relative terms) as compared to their market (relative) value. Thereader is invited to check the validity of Theorem 1 results and the applicability ofthe bound R δ in (16) as a benchmark for the market pricing (in relative value) of a δ -Symmetric Strangle with any other traded security options at any expiration.Table 1: Oserved market relative values ˆ R δ of the δ − Symmetric Strangle for vari-ous tickers and durations as were priced on EOD ∗ , May 13, 2020, as compared tothe bound R δ (16) calculated under the BS option pricing model. T icker µ IV Days δ k k ˆ Π δ -Price ˆ R δ R δ SPY 281.60 0.3529 37 0.170 250 302 5.19 0.107 0.095LLY 157.93 0.3597 156 0.200 130 185 8.22 0.150 0.133BA 121.50 0.7685 100 0.210 95 175 12.45 0.156 0.147TLT 168.50 0.2029 16 0.255 162 170 1.97 0.246 0.232C 40.60 0.6851 219 0.295 32 52.5 7.05 0.362 0.345IBM 115.73 0.3832 23 0.340 112 120 4.60 0.575 0.548GOOG 1349.33 0.3356 65 0.405 1320 1400 102.65 1.283 1.207 ∗ EOD market pricing were obtained using the TOS platform of TDAmeritrade benchmark for assessing and comparing the market (relative) pric-ing of a major trading strategy (namely the δ − Symmetric Strangle) across varioussecurities and assets, across various durations and irrespective of the underlyingsecurity-specific volatility (implied or historical).
One of the appealing aspects of a δ − Symmetric Strangle is that from the outset,it is a delta-neutral strategy with zero directional risk, initially. Moreover a traderthat sells such a strangle, for some fixed δ < .
5, at the matching two strikes k − δ and k + δ , benefit from a well defined probability of success, that may be calculatedunder the current distribution of the asset’s returns implied by BS option pricingmodel in (3) and (6). Specifically, for a given value δ < . ν >
0, the initial probability that the underlying security price would remain, at expiration, between k − δ , ( ≡ k + − δ ) and k + δ is simply (see Remark 1), α ≡ Φ ( − z δ − ν ) − Φ ( z δ − ν ) . (18)Hence, the expected reward for a trader that sells the strangle for Π δ = c µ ( k + δ ) + p µ ( k − δ ) (as credit) and plans to exit and buy it back for a fraction λ ∈ (0 ,
1] of thecredit received is E λ ( δ ) : = α Π δ − (1 − α ) λ Π δ . In relative terms, this expected reward, relative to the present value of the spreadbetween the strikes, becomes¯ E λ ( δ ) : = E λ ( δ ) PV ( k + δ − k − δ ) ≤ α R δ − (1 − α ) λ R δ , (19)where R δ is given in (2). As was mentioned in the Introduction and pointed outin Remark 1, for small values of ν (i.e. near expiration) we may approximate the ’success’ probability in (18) as α ≈ (1 − δ ). Accordingly, for any given fractional11oss λ ∈ (0 ,
1] the expected relative reward in (19), under this approximationwould be, E λ ( δ ) = (1 − δ (1 + λ )) × R δ . (20) . . . . . . d r dd Figure 3:
The expected relative reward function, E λ ( δ ) as a function of δ with λ = . .The maximal value is achieved at δ ∗ = . at which point, E λ ( δ ∗ ) = . . Observe that E λ ( δ ) ≥ δ ≤ / + λ ) and that, upon using (16) and(17), the equation E ′ λ ( δ ) ≡ − + λ ) R δ + (1 − δ (1 + λ )) R ′ δ = , is seen to have a unique root, δ ∗ , at which point E λ ( δ ) attains its maximal value.That is, for a given fractional lose λ ∈ (0 , δ ∗ = h ( λ ), must solves theequation δ (1 − z δ ) − z δ φ ( z δ ) = + λ ) , (21)at which point E ∗ λ : = E λ ( δ ∗ ) ≥ E λ ( δ ).In Figure 3 above we present the graph of the relative reward function, E λ ( δ ),for a trader who sells a δ − Symmetric Strangle, and wishes, as a matter of strategy,to exit it upon a loss of 50% of the credit received. This case corresponds to12able 2: The optimal choice for δ for the δ − Symmetric Strangle strategy, calcu-lated for ’exits’ with the various fractional loss λ . λ δ ∗ E ∗ α ( δ ∗ )0.25 0.300 0.091 0.4000.40 0.256 0.067 0.4890.50 0.234 0.056 0.5330.60 0.216 0.048 0.5670.75 0.194 0.040 0.6111.00 0.164 0.031 0.670 λ = . δ of δ ∗ = . E ∗ . = . δ as were calculated (as anumerical solution of (21)) for various choices of λ , along with the correspondingvalues of the maximal expected reward E ∗ λ , and the matching initial probabilityof ”success” of this δ ∗ − Symmetric Strangle strategy. As can be seen from Table1, the selling of a ’standard’ 16-delta symmetric strangle with its 0 .
68 ’success’probability should be coupled with an exit strategy that limits losses at 100% ofthe credit received may yield a maximal expected relative reward of E ∗ = . E ∗ . = . In this appendix we study the coordinate-wise behavior of R ( δ, ν ) as given in (15)over the practical domain B . To begin with, note first that since e z δ ν + ν / ≡ φ ( z δ ) φ ( z δ + ν ) and e − z δ ν + ν / ≡ φ ( z δ ) φ ( z δ − ν ) , we may express R ( δ, ν ) in (15), entirely in terms of the standard normal pd f and cd f , as R ( δ, ν ) = φ ( z δ − ν ) · Φ ( z δ + ν ) − φ ( z δ + ν ) · Φ ( z δ − ν ) φ ( z δ + ν ) − φ ( z δ − ν ) . (22)13pon di ff erentiating expression (22) of R ( δ, ν ), with respect to δ and with respectto ν along with the fact that φ ′ ( u ) : = ddu φ ( u ) = − u φ ( u ) we obtain the followingresults. Lemma 2
With R ( δ, ν ) as defined in (15) above we have,a) For each fixed δ < . (so that z δ < ), ∂ R ∂ν = AB ( B + z δ · D ) ≤ , b) For each fixed ν > , ∂ R ∂δ = − ( z δ − ν ) · A · D > , where, D : = ( Φ ( z δ + ν ) − Φ ( z δ − ν )) > , B : = ( φ ( z δ + ν ) − φ ( z δ − ν )) > , andA : = ( φ ( z δ + ν ) − φ ( z δ − ν )) > . Proof.
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