Black to Negative: Embedded optionalities in commodities markets
aa r X i v : . [ q -f i n . P R ] J un Black to Negative ∗ Embedded optionalities in commodities markets
Richard J. Martin and Aldous BirchallJune 12, 2020
Abstract
We address the modelling of commodities that are supposed to have positive price but, onaccount of a possible failure in the physical delivery mechanism, may turn out not to. This isdone by explicitly incorporating a ‘delivery liability’ option into the contract. As such it is asimple generalisation of the established Black model.
Introduction
To the consternation of oil traders, WTI crude prices went below zero on 20th April this year,with the May futures contract trading at − .
63 USD/bbl just before expiration. This prompteda rethink of the fundamentals of option pricing and risk management.The standard pricing model is, of course, Black (also known as Black’76 or Black-Scholes),wherein the underlying asset price follows a geometric Brownian motion. As such a model cannotaccommodate negative prices, there has predictably been a rush for the use of arithmetic (Bache-lier) models instead, but this seems symptomatic of desperately clutching at the nearest simpleprobabilistic tool to hand, rather than of carefully appreciating the fundamental problem. Further,Black works well most of the time, albeit with the usual difficulties of pricing short-dated out-of-the-money options. It seems unwise simply to discard years of accumulated experience of Black inthe following areas: hedging; pricing using volatility surfaces; the concept of convenience yield.It is obvious from Table 1 that the negative price was caused by a failure in the deliverymechanism. With storage tankers at Cushing full, and transportation tankers also, it was verydifficult to store the commodity, and so traders had to pay people to take it off their hands. Thededuction: commodities are not simply assets. What do we mean by this?May Jun JulCLK0 CLM0 CLN017-Apr-20 18.27 25.03 29.4220-Apr-20 − Closing prices of recent WTI oil contracts just before and after the negative price date. Source:Bloomberg ∗ The title comes from the convention (at least in the UK) that on a battery, red is the positive terminal and blackthe negative. In accounting it is the other way round! Rich Dad, Poor Dad , a series of ‘popular finance’ books published over the last couple ofdecades, Robert Kiyosaki defines an asset as something that puts money in your pocket, whereasa liability takes it out. He goes on to point out that many things that people consider to be assetsare in fact liabilities. A better summary would be: they are pure asset and a pure liability takentogether. For example, a house has an intrinsic value, but on the liability side there is the cost ofupkeep, and property taxes .It follows that we should represent a commodity as a hypothetical pure asset A say, whosevalue is always positive, coupled with a liability whose price is likely to be linked to A . Failure ofthe delivery mechanism can give rise to a large liability. It is attractive, from the perspective ofcapital markets vernacular, to regard this as a form of intrinsic optionality that generally expiresworthless—until one day it doesn’t.Before making too strong a strong connection between commodities and securities markets, weshould point out a fundamental difference between them. In the latter, the spot price of the assethas primacy, and the forward or futures price is linked to it by a simple parity argument based oneither buying the asset and delivering it into the forward/futures, or selling it and buying it in theforward/futures market. In oil markets, however, there is no spot contract to speak of; inasmuchas it exists it has different delivery standards and necessarily trades at a basis to each futurescontract. There is no notion of arbitrage between spot and futures or between one futures andanother, because for such a trade to function one must be able to physically deliver into or receivephysical delivery from the contract—the failure of which is the whole crux of the problem, given thecomplexity and expense of receiving barrels of crude oil rather than a conveniently dematerialisedbond or share. See Bj¨ork [1, Ch.20] for information about martingale pricing of forward and futures contracts.Throughout E means expectation under the risk-neutral measure Q (wherein with the rolled-up money market account as numeraire, discounted expectations are martingales), and W t is aBrownian motion under Q .We write A t for the intrinsic asset price at time t . This is the value of the asset if we could storeand deliver it at no cost, and is hypothetical. Doing the obvious thing, we make the A -dynamicsunder Q a geometric Brownian motion, at least in the first instance: dA t /A t = ( r − y ) dt + σ dW t (1)where r is the riskfree rate and y is as usual the convenience yield [4, Ch.2,3].Usually one has for the futures, F t ( T ) = E t [ A T ] , with a similar result holding for forwards , but we are proposing F t ( T ) = E t [ A T − ψ − ( A T )] , (2) We recall the apocryphal tale of the Russian hotel used as collateral against a bank loan. It was so dilapidatedthat the local authorities forced the new owner, i.e. the bank, to bring it to state of good repair—which cost morethan the final value. Thereby the loan had negative recovery. The expectation would be taken under the forward risk-neutral measure. As we are not worrying about stochasticinterest rates in this paper, this difference will not detain us. ψ − is a convex decreasing function. An obvious idea is a vanilla putoption but a better one is a call option on a negative power of A T : ψ − ( A T ) = ℓ ˆ A (cid:0) ( ˆ A/A T ) λ − (cid:1) + . (3)There are three parameters, ˆ A > ℓ ≥ λ ≥
0; the first has the same units as A T and isinterpreted as the price at which ‘things start to go wrong’, and the other two are dimensionlessand control the size of the resulting liability. Not only can the futures price go negative: it is alsounbounded below as A →
0. This is why we have chosen the above formulation rather than asimple put ( ˆ A − A T ) + . Had we made the instrinsic option a constant, we would have just endedup with a shifted lognormal model, which is a well-known idea in other contexts; its disadvantageis that one never knows how much shifting to do.We briefly return to a point we made earlier. It is superficially attractive to regard the followingconstruct as a form of spot price: S t = A t − ψ − ( A t )so that F t ( T ) = E t [ S T ]. But there is a conceptual difficulty with that, as we may well need theparameters of ψ − to dependent of the futures maturity T . This appears to generate a multiplicityof spot prices—but it is legitimate, because in reality the spot contract does not trade.The prescription (3) has good analytical tractability in the sense that the expectation of ψ − ( A T )can be evaluated using Black-type formulae. Therefore the futures price is easy to calculate. Indeed, E t [ ψ − ( A T )] = ℓ ˆ A λ E t [ A − λT ] P − λt ( A T < ˆ A ) − ℓ ˆ A P t ( A T < ˆ A )= ℓ ˆ A ˆ Ae ( λ +1) σ ( T − t ) / F ∗ ! λ Φ( − ˆ d ) − Φ( − ˆ d ) (4)with ˆ d = ln( F ∗ / ˆ A ) − σ ( T − t ) σ √ T − t ˆ d = ˆ d − λσ √ T − tF ∗ = A t e ( r − y )( T − t ) = E t [ A T ]and Φ the standard Normal integral. The quantity F ∗ = F ∗ t ( T ) is interpreted as the futures price onthe intrinsic asset, were such a contract to trade; it is always positive even if F is not. The notation P ν corresponds to the expectation E ν [ Z ] ≡ E [ A νT Z ] / E [ A νT ], and we use standard arguments aboutchange of numeraire. This formula allows the intrinsic asset price (equivalently F ∗ ) to be imputedfrom the futures price; this also requires the specification of ψ − and the volatility of A .Valuation of European options on F t ( T ) can then be done using compound-option formulae.Let T o ≤ T be the option maturity. The main thing to note is that for any option strike K o thereis a unique A ♯ = A ♯ ( K o ) such that E (cid:2) A T − ψ − ( A T ) | A T o = A ♯ (cid:3) = K o and so the exercise decision at time T o reduces to A T o ≷ A ♯ .As regards application we note that the exchange-traded options are vanilla options on eachfutures contract, with the same expiry as the futures ( T o = T ). We will only deal with this case This is not quite correct: the options expire around five days before the futures. For this work we are setting T equal to T o . C K o = E t [( A T − ψ − ( A T ) − K o ) + ]= E t [( A T − ψ − ( A T ) − K o ) ( A T > A ♯ )]= E t [ A T ] P t ( A T > A ♯ ) − K o P t ( A T > A ♯ ) − ℓ ˆ A h E t [( ˆ A/A T ) λ ] P − λt ( A ♯ < A T < ˆ A ) − P t ( A ♯ < A T < ˆ A ) i = F ∗ Φ( d ) − K o Φ( d ) (5) − ℓ ˆ A × K o e ( λ +1) σ ( T − t ) / F ∗ ! λ (cid:0) Φ( − ˆ d ) − Φ( − d ) (cid:1) − (cid:0) Φ( − ˆ d ) − Φ( − d ) (cid:1) , A ♯ ≤ ˆ A , A ♯ ≥ ˆ A and via the same route P K o = E t [( K o − A T + ψ − ( A T )) + ]= E t [( K o − A T + ψ − ( A T )) ( A T < A ♯ )]= K o Φ( − d ) − F ∗ Φ( − d ) (6)+ ℓ ˆ A × K o e ( λ +1) σ ( T − t ) / F ∗ ! λ Φ( − d ) − Φ( − d ) , A ♯ ≤ ˆ A K o e ( λ +1) σ ( T − t ) / F ∗ ! λ Φ( − ˆ d ) − Φ( − ˆ d ) , A ♯ ≥ ˆ A where d = ln( F ∗ /A ♯ ) − σ ( T − t ) σ √ T − td = d + σ √ T − td = d − λσ √ T − t The prices are these discounted by e − rT . It is easy to see that the put-call parity formula C K o − P K o = F − K o is obeyed.If T o < T then the conditions A T o < A ♯ and A T < K o refer to the intrinsic asset price atdifferent times, and the expectations require the bivariate Normal integral. We will deal with thisin forthcoming work. Figure 1 shows results for the June contract CLM0 as of 21-Apr-20, and the same but for theJuly contract CLN0. The left-hand plot shows prices, and the right-hand plot the equivalentBlack volatility (only for positive strikes, of course). It is apparent that the fit for low strikes isexcellent—and for the first time ever we can show a Black model with negative strikes!In both cases the market quotes for the lowest-strike put options indicate positive probabilityof negative futures price at expiry, though obviously this is greater for the front (June, CLM0)contract. Indeed, attempting to impose that the zero-strike put P be worthless would lead to aconvexity arbitrage in the puts (sell 2 × P and buy 1 × P and 1 × P ). The new model shows that4a) CLM0 p u t p r i c e strikemarketmodel 200 300 400 500 600 700 800 900 1000 1100 1200 1300 0 5 10 15 20 25 30 35 40 p u t i m p li e d v o l ( % ) strike marketmodel (b) CLN0 p u t p r i c e strikemarketmodel 100 150 200 250 300 350 400 0 5 10 15 20 25 30 35 40 p u t i m p li e d v o l ( % ) strike marketmodel Figure 1:
Price and implied vol for CLM0 and CLN0 on 21-Apr-20: market and model compared. Marketdata source: Bloomberg.
F σ ˆ A λ ℓ F ∗ CLM0 11.57 140% 19.5 0.793 2.16 19.13CLN0 18.69 130% 24.7 0.310 1.33 22.54Table 2:
Fitted parameters for front two CL contracts on 21-Apr-20. , and indeed some negative-strike puts, should have traded at a positive price, which is entirelyreasonable.Another important point is that the option prices are captured with parameters (Table 2) thathave a reasonable physical intuition. While the volatility σ is high (around 130%), it retains somesort of plausibility: whereas an implied Black volatility of over 1000%, as is needed to mark theCLM0 5-strike put, is essentially useless. We should point out that different parameter sets givevery similar fit, so the model is probably overparametrised: this could be alleviated by fitting tomultiple futures at the same time and requiring, for example, that σ not very too much betweenadjacent contracts. This is a matter for future research.Finally, we are able to make a deduction about the convenience yield, not from the ratio ofadjacent futures prices F (which would fail immediately when prices went negative), but insteadfrom the ‘intrinsic futures’ F ∗ , as F ∗ ( T ) /F ∗ ( T ) = e ( r − y )( T − T ) . (7)As r is context ignorable, and T − T = , we have y ≈ − One thing we specifically set out not to do in this paper is to rewrite from scratch all commodityoption pricing models. Quite the reverse: what we propose integrates perfectly well with allbranches of the subject, as we now briefly justify.
As is well known, no single Black volatility prices all options consistently the market—in any assetclass. This has led to a convenient visualisation tool: the (Black) volatility surface.The primary objective of any extension to the Black model is to flatten the volatility surface;ideally the whole surface would be explained by a small mumber of parameters. For example, L´evymodels deal with the problems of short-dated OTM options by using jumps instead of an artificiallyhigh volatility.We do not suggest that the incorporation of an ‘intrinsic put’ of the type herein described willresult in perfect pricing—but it does permit negative strikes and captures the ‘smile effect’ for lowstrikes. For perfect matching of prices we will need to make one parameter vary, and this had betterbe the volatility σ . With the Black model the implied volatility is unique (and exists provided theoption price does not violate simple arbitrage constraints), because both call and put option pricesare increasing functions of volatility. That the same property carries over to the model here isalmost too obvious to be worth asking about, but there is a subtlety. When we alter σ , the futuresprice will change by (2,4), and we will no longer match the market unless we alter the intrinsicasset price via a bump δF ∗ . Therefore in calibration when we talk about a move δσ in volatility,we need also to apply a bump δF ∗ so that the futures price is held fixed. The resulting sensitivityto σ is not ∂/∂σ but rather (cid:18) ∂∂σ (cid:19) F = ∂∂σ − ∂F/∂σ∂F/∂F ∗ ∂∂F ∗ . (8)It is clear that for a call option, one has ( ∂C/∂σ ) F > ∂C/∂σ > ∂C/∂F ∗ > ∂F/∂σ < ∂F/∂F ∗ > This subject has received some attention over the years and an excellent account is given in [2],into which our work integrates completely. A noticeable feature of commodity options is that thelonger-dated futures are typically less volatile than the shorter-dated ones, and this suggests thatthe price is mean-reverting under Q . With a nod to interest rate theory and in particular the Hull–White model [6, § § x t = ln A t and theconvenience yield y t follow a bivariate Gaussian process of the form (cid:20) dx t dy t (cid:21) = Λ (cid:20) x t y t (cid:21) dt + M ( t ) dt + (cid:20) σ A dW At σ y dW yt (cid:21) (9)where the matrix Λ is constant, and the drift term M ( t ) and the volatilities σ are not allowed todepend on x or y but may be time-dependent. By (1), dx t = ( r − y t − σ A / dt + σ A dW At . (10)The convenience yield follows essentially the Hull–White model, as also suggested by [5], see also[4, Eq.(3.15)], with an important difference in that it can be coupled to the asset price dynamicsvia a parameter β ≥ dy t = κ ( α ( t ) + βx t − y t ) dt + σ y dW yt . (11)When A t is low, y t is more likely to be low/negative (contango) and when A t is high, y t is morelikely to be positive (backwardation). TherebyΛ = (cid:20) − κβ − κ (cid:21) , M ( t ) = (cid:20) r − σ A / κα ( t ) (cid:21) . The effect of the coupling parameter β is to introduce an implicit mean reversion into the assetdynamics, because when the intrinsic asset price is low the futures curve is likely to be in contangoand when high, in backwardation. The long-term variance is reduced, so that long-dated futurescontracts are less volatile than short-dated ones.The obvious attraction of GL2 is that many quantities associated with it, principally the meanand variance of A T , can be calculated in closed form. Indeed, (cid:20) x t y t (cid:21) = (cid:20) x y (cid:21) + Z t e Λ( t − s ) (cid:20) r − σ A / κα ( s ) (cid:21) ds + Z t e Λ( t − s ) (cid:20) σ A dW As σ y dW ys (cid:21) and so the joint distribution of x t and y t is bivariate Normal with mean (cid:20) x y (cid:21) + Z t e Λ( t − s ) (cid:20) r − σ A / κα ( s ) (cid:21) ds Z t e Λ( t − s ) (cid:20) σ A ρσ A σ y ρσ A σ y σ y (cid:21) e Λ ′ ( t − s ) ds, with ρ the correlation between dW At and dW yt . These expressions require matrix exponentiation,achieved by the following lemma :exp (cid:18)(cid:20) a bc d (cid:21) t (cid:19) = e ( a + d ) t/ (cid:20) cosh δt + a − dδ sinh δt bδ sinh δt cδ sinh δt cosh δt + d − aδ sinh δt (cid:21) (12)with δ = ( a − d ) + 4 bc .Contrary to what is implied in [2] there is no requirement that the eigenvalues of Λ be real: allthat is necessary is that both have real part ≤
0, which is automatic provided that β, κ ≥
0. Infact, when δ <
0, equivalent to β > κ/
4, we will have oscillatory behaviour with period 2 π i /δ , asis observed in autoregressive processes with complex-conjugate poles. It is worth noting that wecould delete β and make the mean reversion explicit, by adding a term − κ x x t dt (where κ x ≥ − κ x in the top left-hand elementof Λ. This has been suggested in e.g. [4, Eq.(3.8)], but has a fundamentally different effect in thatthe eigenvalues of λ must be real, so no cyclical behaviour can be generated.The addition of stochastic interest rates via the Hull–White model gives us the GL3 model, andthis presents no further difficulties in analysis provided the interest rate dynamics are not in anyway driven by A or y .In this modelling framework we would fit all futures expiries (and hence their options) at once,as it is a term structure model. This makes the problem higher-dimensional but also imposesrigidity on the structure in the sense that the calibration parameters should not vary too stronglyfrom one maturity to the next. Two restrictions are:(i) One can no longer choose a different volatility for each maturity. The variation of volatilitywith T is determined by σ A in (10) and the parameters κ , β .(ii) As seen in (7), F ∗ t ( T i ) /F ∗ t ( T i +1 ) gives an estimate of the convenience yield and so there cannotbe too wild a variation in adjacent values of F ∗ .Work on this area is continuing. Regarding (ii), it is likely that this year’s events will show a wideexcursion of the convenience yield from its equilibrium level, adding to the catalogue of real-worldexamples in which the Ornstein–Uhlenbeck process fails to capture large deviations . As mentioned earlier, these are the ideal tool for dealing with short-dated OTM options (for anintroduction see e.g. [9]). We make the elementary point that the work in § A T , and of indicator functions ( A T > K ).For nearly all L´evy processes this is straightforward because E t [ A θT ] is given directly by the momentgenerating function, while E t [ A T > K ] is sometimes known in closed form and otherwise has to bedone using inverse Fourier integrals. Either way, the implementation of what has been describedhere does not require new machinery. Proved by diagonalising the matrix. A discussion of this and possible extensions to the OU model is given in [8]. .4 Merton model It might seem strange to drag in structural credit modelling at this point, but there is an obviousparallel. In the Merton framework the equity of the firm is modelled as a call option on its assetsand the debt as a riskfee bond minus a put option. The strike relates to the face value of the firm’sdebt. In the first instance let the firm value A t follow a geometric Brownian motion. The SDEfollowed by the equity E t is, by It¯o’s lemma, dE t = rE t dt + ∆ σA t dW t = rE t dt + σ E ( E t ) E t dW t where ∆ is the call option’s delta, and we find that E t has acquired local volatility given by σ E = σ ∆ A t /E t . Now E t is a convex function of A t , and so ∆ > E t /A t , and we conclude that σ E > σ , the effectbeing less pronounced when the call option is in-the-money. Thus even if the firm value follows ageometric Brownian motion (which, from the perspective of calibrating to term structure of creditspread, is not a workable model) the equity price still acquires a smile by virtue of the embeddedoption. A fuller discussion is in [7]. So, in modelling equities one is likely to benefit from modellingthe assets and liabilities as separate positive processes rather than trying to model the equitydirectly. Were the equity not of a limited-liability firm, it could go negative, and then such anapproach would be essential. The connection with local volatility [3] is that rather than attempting to infer the local volatilitysurface from traded options, or imposing an arbitrary parametric form upon it, we may do better toinstead postulate the existence of certain embedded optionalities, and identify those. The advantageof so doing is that there is a clear financial intuition, and the pricing of options on the asset inquestion is likely to be straightforward using Black-type formulae, as has been done here. To returnto the numerical example here, it is easy to see that to fix up the Black model using local volatilitywould require an exorbitant level of volatility (even before worrying about the problems of negativeprice). On the other hand, incorporating an option into the traded asset seems to remove most ofthe difficulties.
It is not necessarily true that all optionalities have a negative impact on the price. Upward spikesin commodity prices can stem from the opposite kind of difficulty discussed here: low inventories,causing difficulty obtaining the asset. An obvious prescription in the light of this paper is anoptionality of the form ψ + ( A ) = ℓA (cid:0) ( A/ ˆ A ) λ − (cid:1) + where again ℓ, λ ≥
0. This would in principle explain why in commodity markets implied volatilitiesoften increase for high strikes, sometimes known as the ‘inverse leverage effect’.9
Conclusions
We have presented an extension of the Black model that incorporates an intrinsic optionalityinto the commodity price. It avoids the need to switch to the Bachelier model, as some marketparticipants have been forced to do, doubtless causing interpretative difficulties, expense, and lossof historical context. The new model captures a liability caused by failure of the physical deliveryprocess, and can cause negative futures prices. If the intrinsic asset price is well above the price atwhich this option kicks in then the deformation to the original Black model is negligible and so theyears of accumulated experience in handling Black models is preserved. The events of this Aprilare, however, captured with precision and insight.
The opinions expressed in this paper are those of their authors rather than their institutions.Email [email protected] , [email protected] References [1] T. Bj¨ork.
Arbitrage Theory in Continuous Time . Oxford University Press, 1998.[2] J. Casassus and P. Collin-Dufresne. Stochastic convenience yields implied from commodityfutures and interest rates.
J. Finance , 60(5):2283–2331, 2005.[3] B. Dupire. Pricing with a smile.
RISK , 7(1):18–20, 1994.[4] H. Geman.
Commodities and Commodity Derivatives . Wiley, 2005.[5] R. Gibson and E. Schwartz. Stochastic convenience yields and the pricing of oil contingentclaims.
J. Finance , 45:959–976, 1990.[6] J. C. Hull.
Options, Futures, and Other Derivatives . Prentice Hall, 1997. 3rd ed.[7] R. J. Martin. CUSP 2007: An overview of our new structural model. Technical report, CreditSuisse, 2007. .[8] R. J. Martin, R. V. Craster, and M. J. Kearney. Infinite product expansion of the Fokker–Planckequation with steady-state solution.
Proc. R. Soc. A , 471(2179):20150084, 2015.[9] W. Schoutens.