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Table 1: Closing prices of recent WTI oil contracts just before and after the negative price date. Source: Bloomberg

It follows that we should represent a commodity as a hypothetical pure asset A say, whose value is always positive, coupled with a liability whose price is likely to be linked to A. Failure of the delivery mechanism can give rise to a large liability. It is attractive, in capital markets vernacular, to regard this as a form of intrinsic optionality that generally expires worthless—until one day it doesn’t. The option concept relates to previous research linking storage cost and/or price volatility to inventory, which has traditionally been taken into account using the convenience yield [4, §2.2,2.3]. However, it goes further by identifying a problem that occurs when inventory becomes critical: the marginal storage cost suddenly explodes and this can push the front-end futures negative. In extremis, these costs may not be limited to storage but could include unquantifiable legal liabilities arising from a breach of contract. In other words it is not simply storage cost, but cost arising from storage limitation, that is being identified and captured. Before making too strong a connection between commodities and securities markets, we should point out a fundamental difference between them. In the latter, the spot price of the underlying asset has primacy, and the forward or futures price is linked to it by a simple parity argument based on either buying the asset and delivering it into the forward/futures, or selling it and buying it in the forward/futures market. In oil markets, however, there is no associated spot price to speak of: rather there are ‘benchmark prices’ set by observations of the physical OTC spot market by agencies (known as Trade Journals) or by the futures market itself. Consequently there is no notion of arbitrage2 between spot and futures or between one futures and another, because for such a trade to function one must be able to physically deliver into or receive physical delivery from the contract—the failure of which is the whole crux of the problem, given the complexity and expense of receiving barrels of crude oil rather than a conveniently dematerialised bond or share.

1 Modifying the Black framework

1.1 Pricing model 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 P (wherein with the rolled- up money market account as numeraire, discounted expectations are martingales), and Wt is a Brownian motion under P. We are not using any subjective measure in this paper. We write At for the intrinsic asset price at time t. This is the value of the asset if we could store and deliver it at no cost, and is hypothetical. Doing the obvious thing, we make the A-dynamics under P a geometric Brownian motion, at least in the first instance:

dA /A = (r y) dt + σ dW (1) t t − t 2In its strict sense of a trade strategy providing a riskfree profit. The term is more loosely used by traders to describe what are more correctly called relative value trades.

2 where r is the riskfree rate and y is as usual the convenience yield [4, Ch.2,3]. Usually one has for the futures, Ft(T )= Et[AT ], with a similar result holding for forwards3, but we are proposing

F (T )= E [A ψ−(A )] , (2) t t T − T where the intrinsic option ψ− is a convex decreasing function. An obvious idea is a vanilla put option but as previously intimated we want the option price to rise without limit (explode) as A 0. We also want to retain analytical tractability, as will be presently demonstrated. This T → points to a better idea, a call option on a negative power of AT :

λ + ψ−(A )= ℓAˆ (A/Aˆ ) 1 . (3) T T − There are three parameters, Aˆ > 0, ℓ 0, λ 0: the first
has the same units as A and is ≥ ≥ T interpreted as the price at which ‘things start to go wrong’, and the other two are dimensionless and control the size of the resulting liability. The quantities ℓ, A,ˆ λ are allowed to depend on maturity, and therefore be different for different futures contracts. Delivery into the May contract is different from the June, etc, and so the optionalities relating to failure of the physical delivery mechanism may therefore vary substantially from one contract to another. Notice that there is no unique definition of the spot contract any longer, and this may seem odd. But as we have already noted, the spot oil contract in this context is a benchmark price based on observations of the spot OTC market which accommodates a multiplicity of contract specifications. As the futures market is far bigger than the spot market, we are modelling the futures directly rather than regarding it as the derivative of a spot contract and modelling the latter (as is common in capital markets). The prescription (3) has good analytical tractability in the sense that the expectation of ψ−(AT ) can be evaluated using Black-type formulae:

1+λ −λ −λ E [ψ−(A )] = ℓAˆ E [A ]P (A < Aˆ) ℓAˆP (A < Aˆ) t T t T t T − t T = ℓAˆ mΦ( dˆ ) Φ( dˆ ) − 3 − − 2 and so the futures price is
∗ F = F ℓAmˆ Φ( dˆ )+ ℓAˆΦ( dˆ ) (4) − − 3 − 2 with

2 λ Aeˆ (λ+1)σ (T −t)/2 m = ∗ F ! ∗ ˆ 1 2 ln(F /A) 2 σ (T t) dˆ2 = − − σ√T t − dˆ3 = dˆ2 λσ√T t ∗ − −(r−y)(T −t) F = Et[AT ]= Ate

∗ ∗ and Φ the standard Normal integral. The quantity F = Ft (T ) is interpreted as the futures price on the intrinsic asset, were such a contract to trade; it is always positive even if F is not. The notation

3The expectation would be taken under the forward risk-neutral measure. As we are not worrying about stochastic interest rates in this paper, this difference will not detain us.

3 ν ν ν ν P corresponds to the expectation E [Z] E[AT Z]/E[AT ], and we use standard arguments about ≡ ∗ change of num´eraire. This formula allows F to be imputed from the futures price, but bear in mind that there is an implicit dependence on the specification of ψ− and also on the volatility σ. Valuation of European options on Ft(T ) can then be done using compound-option formulae. Let To T be the option maturity. The main thing to note is that for any option strike Ko there ≤ ♯ ♯ is a unique A = A (Ko) such that

♯ E A ψ−(A ) A = A = K T − T | To o ♯ and so the exercise decision at time To reduces to ATo ≷ A . As regards application we note that the exchange-traded options are vanilla options on each 4 futures contract, with the same expiry as the futures (To = T ). We deal only with this case here. Also the exchange-traded options are American-style but we ignore the value of the early exercise option which, by standard arguments, is technically worthless. The expected payouts of the call and put are

+ CKo = Et[(AT ψ−(AT ) Ko) ] − − ♯ = Et[(AT ψ−(AT ) Ko)1(AT > A )] −1 −♯ ♯ = Et[AT ]Pt (AT > A ) KoPt(AT > A ) − ℓAˆ E [(A/Aˆ )λ]P λ(A♯ < A < Aˆ) P (A♯ < A < Aˆ) − t T t T − t T ˆ ˆ ♯ ˆ ∗  m Φ(d3) Φ(d3) Φ(d2) Φ(d2) , A A = F Φ(d1) KoΦ(d2) ℓAˆ − − − ≤ (5) − − · 0, A♯ Aˆ 

≥  and via the same route

+ PKo = Et[(Ko AT + ψ−(AT )) ] − ♯ = E [(K A + ψ−(A ))1(A < A )] t o − T T T = K P(A < A♯) E [A ]P1(A < A♯) o T − t T t T ℓAˆ E (A/Aˆ )λ 1 1(A < Aˆ)1(A < A♯) − t T − T T ♯ ∗ mΦ( d ) Φ( d ), A Aˆ = K Φ( d) F Φ( d )+
ℓAˆ − 3 − − 2 ≤ (6) o − 2 − − 1 · mΦ( dˆ ) Φ( dˆ ), A♯ Aˆ  − 3 − − 2 ≥  where ∗ ♯ 1 2 ln(F /A ) 2 σ (T t) d2 = − − σ√T t − d = d + σ√T t 1 2 − d = d λσ√T t 3 2 − − and the others as before. The prices are these discounted by e−r(T −t). It is easy to see that the put-call parity formula C P = F K Ko − Ko − o is obeyed. ♯ If To < T then the conditions ATo < A and AT < Ko refer to the intrinsic asset price at different times, and the expectations require the bivariate Normal integral. We will deal with this in forthcoming work.

4This is not quite correct: the options expire around five days before the futures. For this work we are setting T equal to To.

4 We note for reference the elementary Black formula for the undiscounted call and put prices:

1 2 CKo = F Φ(d+) KoΦ(d−) ln(F/Ko) 2 σB(To t) − , d± = ± − . (7) PKo = KoΦ( d−) F Φ( d+) σ √T t − − − B o − 1.2 Numerical results An obvious approach to fitting is to minimise the root-mean-square relative pricing error with respect to the parameters σ, Ko, λ, ℓ:

1/2 n ⋆ 2 1 PK PKj E = j − n P ⋆  Kj ! Xj=1   (P ⋆ denotes the market price, P the model). Clearly at least four option prices will be needed, but this is not a difficulty as for oil they are quoted with strikes $1 apart. We have tried this for different contracts and on different dates and the results are consistently very good. However, we should point out that different parameter sets can give similar fit, suggesting that four parameters is too many (we suspect that three would be ideal). To obviate this problem we can control the variation in intrinsic futures price F ∗ between adjacent contracts. The reason for so doing is that we can make an inference about convenience yield, as we shall explain presently. The revised objective function is

1/2 n ⋆ 2 ∗ 2 1 PK PKj F (T ) E = j − + w2 ln i+1 (8) n P ⋆ F ∗(T )  Kj ! i Xj=1 Xi     with w denoting a weighting coefficient (we use w = 5). As a specific example, Figure 1 shows results for the June, July, August contracts CLM0, CLN0, CLQ0 as of 21-Apr-20. The left-hand plot shows prices, and the right-hand plot the equivalent Black volatility (from eq.(7): obviously this can only be done for positive strikes). It is apparent that the fit for low strikes is excellent. And, for the first time ever, we can show a new Black model with negative strikes! The r.m.s. relative pricing errors were 0.014, 0.017, 0.007 for CLM0,CLN0,CLQ0 ≈ respectively, which are lower than the prevailing bid-offers.

F σ Aˆ λ ℓ F ∗ CLM0 11.57 109% 21.7 0.921 2.20 20.42 CLN0 18.69 125% 29.6 0.532 0.52 23.17 CLQ0 21.61 88% 27.4 1.754 0.11 24.64

Table 2: Fitted parameters for front three CL contracts on 21-Apr-20.

In all three cases the market quotes for the lowest-strike put options indicate positive probability of negative futures price at expiry, though obviously this is greatest for the front contract. Indeed, attempting to impose that the zero-strike put P0 be worthless would lead to a convexity arbitrage in the puts (sell 2 P and buy 1 P and 1 P ). The new model shows that P , and indeed × 5 × 0 × 10 0 some negative-strike puts, should have traded at a positive price, which is entirely reasonable. Another important point is that the option prices are captured with parameters, shown in Table 2, that have a reasonable physical intuition. While the volatility σ is high (around 100%), it retains some sort of plausibility: whereas an implied Black volatility σB of over 1000%, as is needed to mark the CLM0 5-strike put using (7), is essentially useless.

5 (a) CLM0 30 1400 market market model model 25 1200

20 1000

15 800 put price 10 600 put implied vol (%)

5 400

0 200 −10 −5 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 strike strike

(b) CLN0 30 400 market market model model 25 350

20 300

15 250 put price 10 200 put implied vol (%)

5 150

0 100 −10 −5 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 strike strike

(c) CLQ0 30 300 market market model model 25 250 20

15 200 put price 10 put implied vol (%) 150 5

0 100 −10 −5 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 strike strike

Figure 1: Price and Black implied vol for front three CL contracts on 21-Apr-20: market and model compared. Market data source: Bloomberg.

6 We return to the matter of convenience yield. Usually this is estimated from the ratio of adjacent futures prices F , but that method fails spectacularly when prices go negative. In view of the modelling herein, the correct approach is to use the ‘intrinsic futures’ F ∗ instead (this is necessarily positive), giving

∗ ∗ (r−y)(Ti−Ti+1) F (Ti)/F (Ti+1)= e . (9)

To obtain convenience yields with reasonable economic intuition we should not have extreme varia- tion between adjacent F ∗’s, and this is why (8) penalises that. Applying this to the example shown5 gives y = 136% as estimated from CLM0–CLN0 and 88% as estimated from CLN0–CLQ0. This − − indicates, as might be expected, extreme contango at the front end but less extreme further out. A variation on eq.(8) is to require that σ not vary too much between adjacent contracts—this idea naturally leads to term structure modelling, which we discuss in the next section. The model can be extended in a number of different directions, as the second half of the paper suggests. However, the main purpose of this paper is to show that the basic ideas hold good, even if the finer details of calibration are a matter for further research and development. Obviously given the closed-form nature of the option pricing, computation and therefore optimisation are very fast, which is an important advantage.

1.3 Connection with the Merton model There are obvious parallels with the Merton structural credit model (i) in the sense of modelling assets and liabilities separately, with April 2020 being a sort of ‘credit event’ in the oil market, and (ii) in that the introduction of an embedded option creates a volatility skew and may in other environments be an explanator of volatility skew/smile. In the Merton framework the equity Et of the firm is modelled as a call option on its assets At and the debt as a riskfee bond minus a put option, struck at the face value of the firm’s debt (K). By put-call parity, the equity is

E = E [(A K)+]= A E [min(A , K)] . t t T − t − t T embedded option

In the first instance let the firm value At follow a geometric| Brownian{z } motion. The SDE followed by the equity Et is, by It¯o’s lemma,

dEt = rEt dt +∆σAt dWt

= rEt dt + σE(Et)Et dWt where ∆ is the call option’s delta, and we find that Et has acquired local volatility given by

σE = σ∆At/Et.

Now Et is a convex function of At, andso∆ > Et/At, and we conclude that σE >σ, the effect being less pronounced when the call option is in-the-money, corresponding to a firm with low leverage. Thus even if the firm value follows a geometric Brownian motion (which, from the perspective of calibrating to term structure of credit spread, is not a workable model) the equity price still acquires a volatility skew by virtue of the embedded option. A fuller discussion is in [8]. More generally this connects with the idea of local volatility [3]: rather than attempting to infer the local volatility surface from traded options, or imposing an arbitrary parametric form upon it, we may do better

5 34 29 Note T2 T1 = and T3 T2 = . We ignore r as it is too small to be worth worrying about. − 365 − 365

7 to instead postulate the existence of certain embedded optionalities, and identify those. As seen here, fixing up the Black model using local volatility would require an exorbitant level of volatility (even before worrying about the problems of negative price). On the other hand, incorporating an option into the traded asset seems to remove most of the difficulties, and the practicality is certainly enhanced by choosing a simple functional form, as here.

2 Advanced skew/smile modelling

An embedded option generates a volatility skew/smile and the results of Figure 1 show that in some cases it can explain most of the skew/smile. However, it is not necessarily true that all the volatility skew/smile is explainable that way. There are more obvious explanations, for example jumps. With that in mind, we extend the work in §1 to exponential L´evy processes. But first we give a brief account of a simple and common construction and show what happens when we introduce an embedded option.

2.1 Volatility surfaces As is well known, no single Black volatility prices all options consistently the market—in any asset class. This has led to a convenient visualisation tool: the (Black) volatility surface. We do not suggest that the incorporation of an ‘intrinsic put’ of the type herein described will result in perfect calibration—but it does permit negative strikes and captures the ‘smile effect’ for low strikes. For perfect matching we will need to make one parameter vary, and this had better be the volatility σ. With the Black model the implied volatility is unique (and exists provided the option price does not violate simple arbitrage constraints), because both call and put option prices are increasing functions of volatility. That the same property carries over to the model here is almost too obvious to be worth asking about, but there is a subtlety. When we alter σ, the calibrated futures price will change by (4) because the embedded option’s value changes, and we will no longer match the market unless we alter the intrinsic asset 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 sensitivity to σ is not ∂/∂σ but rather

∂ ∂ ∂F/∂σ ∂ = . (10) ∂σ ∂σ − ∂F/∂F ∗ ∂F ∗  F ∗ It is clear that for a call option, one has (∂C/∂σ)F > 0 (as ∂C/∂σ > 0, ∂C/∂F > 0, ∂F/∂σ < 0, ∂F/∂F ∗ > 0). The same simple argument cannot be used for the put, but we can just use put-call parity instead. So finding implied volatility in the new model is straightforward.

2.2 L´evy models As mentioned earlier, these are the ideal tool for dealing with short-dated OTM options (for an introduction see e.g. [10]), but exponential L´evy models cannot deal with negative prices and/or strikes. Fortunately the work in §1 is easily generalised, as we now explain, first in a general setting and then with regard to a special case. A L´evy process is specified through its generator L, which is essentially a cumulant-generating function. We can deal either with the intrinsic asset price A or with its corresponding futures F ∗, 6 ∗ ∗ iu L(u)(t−s) and the latter is slightly neater . Define Es[(Ft /Fs ) ] = e , with u a complex variable. 6Because we do not need to invoke the convenience yield and risk-free rate at the outset, only to see them cancel ∗ out later. Put differently, F must be a P-martingale and this imposes a condition on L.

8 The martingale condition is L( i) = 0. For a Brownian motion, L(u) = µiu 1 σ2u2, and the − − 2 martingale condition is µ = 1 σ2. − 2 When jumps are included, L acquires additional terms through jumps of size y and intensity κ(y) 0, via ≥ ∞ 1 2 2 iux L(u)= µiu 2 σ u + (e 1)κ(x) dx; − −∞ − Z and in any L´evy model of finite variation, L may be cast in this form (L´evy-Khinchin theorem). Let us start with the basic Black model without the embedded optionality. Clearly the main 0 1 challenge is calculating the expectation of 1(K1 < AT < K2) under P and P . The probability ∗ ∗ 0 density of x = ln(FT /Ft ), under P , is obtained by Fourier transformation: ∞ 1 L(u)(T −t)−iux fT (x)= e du. 2π −∞ Z This integral can be approximated by the discrete Fourier transform, followed by integration over x [ln(K /F ∗), ln(K /F ∗)] to get the desired probability. We also need the probability under P1, ∈ 1 t 2 t which corresponds to the exponentially-tilted density eνxf (x) const, with ν = 1 in this instance. T × To this end define Lν(u)= L(u iν) L( iν); − − − then the density of x under Pν is obtained using the same integral as above, replacing L with Lν. When the embedded optionality is brought into play we need expectations of powers of AT , ν which are given directly by L, and expressions of the form P (K1 < AT < K2), which have already been dealt with above. Therefore no extra machinery is required to calculate option prices in the new model. Many choices of L´evy model belong to so-called exponential families, by which is meant that if the generator L(u, p) is in the family (p denoting a list of parameters) then so is Lν(u, p). Equivalently, exponential tilting simply alters the parameters from p to pν , so that we may write Lν(u, p)= L(u, pν ). Examples include the Variance Gamma and Normal Inverse Gaussian models, and the double-exponential and Merton jump-diffusions. Now for some families, the density fT and/or probability P(K1 < AT < K2) are known in closed form. Write

∗ P (A > K)=Υ ln(F /K), T t; p and Υ = 1 Υ t T − − ∗ ∗
∗ where we recall that F is short for Ft (T ) = Et[AT ] and FT (T ) = AT . Then the (undiscounted) call and put prices are given by

∗ 1 CK = FtΥ ln(Ft /K), T t; p KΥ ln(At/K), T t; p − − − (11) ∗ 1 PK = KΥln(F /K), T t; p
FtΥ ln(At/K), T t; p
, t − − − an obvious generalisation of (7)7.

The introduction of embedded optionalities creates no problem, and (4,5,6) generalise via the

7And in the Brownian case we note that if p = (µ, σ)=( 1 σ2,σ) then p1 = (µ + σ2,σ)=( 1 σ2,σ), and − 2 2 Υ(x,τ, p) = Φ(x + µτ)/σ√τ
, explaining the relationship between d+ and d−.

9 following substitutions: ∗ − m (A/Fˆ )λeL(iλ)(T t) ∗ Φ(d ) Υ ln(F /A♯), T t; p1 1 t − ∗ ♯ Φ(d2) Υ ln(Ft /A ), T t; p
∗ − − Φ(d ) Υ ln(F /A♯), T t; p λ 3 t −
ˆ ∗ ˆ Φ(d2) Υ ln(Ft /A), T t; p
∗ − − Φ(dˆ ) Υ ln(F /Aˆ), T t; p λ . 3 t −
One concrete example is the double-exponential jump-diffusi
on [7], in which exponentially- distributed positive and negative jumps of mean size ξ and ξ− arrive as independent Poisson + − processes of rate κ+ and κ−. For this model

κ ξ iu κ−ξ−iu L(u)= µiu 1 σ2u2 + + + − 2 1 ξ iu − 1+ ξ−iu − + with µ chosen so that L( i) = 0, and if p = (µ, σ, κ ,ξ , κ−,ξ−) then − + + κ ξ κ− ξ− pν = µ + νσ2, σ, + , + , , . 1 νξ 1 νξ 1+ νξ− 1+ νξ−  − + − +  Kou gives a closed-form expression for Pt(AT > K) as an infinite series of special functions (and in fact uses the symbol Υ for this purpose). Another example is the Merton jump-diffusion (MJD) in which jumps are instead Normally distributed with mean c and variance δ2, and arrive at rate κ. The L´evy generator is − 2 2 L(u) = (µ 1 σ2)iu 1 σ2u2 + κ eciu δ u /2 1 − 2 − 2 − 2 in which the martingale condition is µ = κ(ec+δ /2 1), and if p = (µ,
σ, κ, c, δ) then − − 2 2 pν = µ + νσ2, σ, κeνc+ν δ /2, c + νδ2, δ .

This can be handled using the general formulae above, but an alternative
route to the same answer (the algebra is lengthy but not difficult) is to realise that it is simply a mixture of Black models, as is seen by conditioning on the number of jumps. The MJD only provides a useful advance over a simple diffusion when the jumps are rare and large: very frequent small jumps are indistinguishable from a diffusion. Consequently this method of calculation is computationally effective as the sum can be truncated after only a few terms. For the details, the probability of j jumps in [0, T t] is π = e−θθj/j!, with θ = κ(T t). The − j − futures price is now ∞ ∗ F = F π m Φ( dˆ ) Φ( dˆ ) (12) − j j − 3,j − − 2,j j=0 X
in which λ ˆ (λ+1)σ2(T −t)/2 Ae −λµ(T −t)+λj(−c+λδ2/2) mj = ∗ e F ! ∗ ˆ 1 2 ln(F /A) + (µ 2 σ )(T t)+ jc dˆ2,j = − − σ2(T t)+ jδ2 − ˆ ˆ 2 2 d3,j = d2,j λ pσ (T t)+ jδ . − − p 10 Moving on to the options, we have much as before ∞ ∗ C = π F Φ(d ) K Φ(d ) Ko j j 1,j − o 2,j Xj=0  m Φ(d ) Φ(dˆ ) Φ(d ) Φ(dˆ ) , A♯ Aˆ ℓAˆ j 3,j − 3,j − 2,j − 2,j ≤ (13) − · 0, A♯ Aˆ ∞ 

≥  ∗ P = π K Φ( d ) F Φ( d ) Ko j o − 2,j − j − 1,j Xj=0  m Φ( d ) Φ( d ), A♯ Aˆ + ℓAˆ j − 3,j − − 2,j ≤ (14) · m Φ( dˆ ) Φ( dˆ ), A♯ Aˆ  j − 3,j − − 2,j ≥  where

∗ ∗ µ(T −t)+jc+jδ2/2 Fj = F e ∗ ♯ 1 2 ln(F /A ) + (µ 2 σ )(T t)+ jc d2,j = − − σ2(T t)+ jδ2 − 2 2 d1,j = d2,j + σp(T t)+ jδ − 2 2 d3,j = d2,j pλ σ (T t)+ jδ − − ∞ ∞ 2 (note that π F ∗ = F ∗ and π m p= m, recalling µ = κ(ec+δ /2 1)). j=0 j j j=0 j j − − P P 2.3 Numerical results As an example, we take the market a couple of months later, on 09-Jun-20. The oil price had now rallied to around 40. The results are shown in Fig. 2) with and without the embedded option, and the parameters are shown in Table 3. We have fixed δ = 0.5 throughout: this is to reduce the number of extra paremeters to two (κ, c) and to obviate the aforementioned degeneracy in the limit of very frequent tiny jumps. The MJD alone fits well except for the lowest strikes, which the embedded option takes care of: the fitting error E 0.01 in each case. The value of the embedded option, which is F ∗ F , ≈ − is much lower than in April. In ‘more normal’ market conditions the skew is thereby explained, as one would expect: jumps capture the near-at-the-money volatility and the embedded option the deep-out-of-the-money volatility. The parameters do not vary wildly between contracts. We can also redo the results from 21-Apr-20 using the MJD rather than a diffusion. In fact the incorporation of jumps makes no measurable difference to the fit—the green trace in Fig. 3 is indistinguishable from the blue one in Fig. 1—showing that the entire smile/skew structure can be explained by the embedded option. The red trace in Figure 3 shows that attempting to fit using only jumps, with no embedded option, fails completely. We repeat our previous comment that no exponential L´evy model can deal with negative prices/strikes.

F σκ cδ Aˆ λ ℓ F ∗ CLN0 38.94 56% 0.66 0.30 0.50 11.73 2.00 1.00 38.94 − CLQ0 39.16 44% 0.72 0.32 0.50 15.80 1.25 0.93 39.23 − CLU0 39.38 39% 0.61 0.32 0.50 15.82 1.17 0.99 39.50 − Table 3: Fitted parameters for front three CL contracts on 09-Jun-20.

11 (a) CLN0 12 market market MJD 500 MJD 10 MJD+EO MJD+Emb.Opt.

8 400

6 300

4 put price 200

2 put implied vol (%)

100 0

−2 0 0 10 20 30 40 50 0 10 20 30 40 50 strike strike

(b) CLQ0 12 300 market market MJD MJD MJD+EO MJD+Emb.Opt. 10 250

8 200 6 150 4 put price 100 2 put implied vol (%)

0 50

−2 0 0 10 20 30 40 50 0 10 20 30 40 50 strike strike

(c) CLU0 12 250 market market MJD MJD MJD+EO MJD+Emb.Opt. 10 200

8 150

6

put price 100 4 put implied vol (%)

50 2

0 0 0 10 20 30 40 50 0 10 20 30 40 50 strike strike

Figure 2: Price and Black implied vol for front three CL contracts on 09-Jun-20: market and Merton jump-diffusion with and without embedded option compared. Market data source: Bloomberg.

12 (a) CLM0 30 1400 market market MJD MJD MJD+Emb.Opt. MJD+Emb.Opt. 25 1200

20 1000

15 800 put price 10 600 put implied vol (%)

5 400

0 200 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 strike strike

(b) CLN0 25 400 market market MJD MJD MJD+Emb.Opt. MJD+Emb.Opt. 350 20

300 15

250

put price 10 200 put implied vol (%)

5 150

0 100 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 strike strike

(c) CLQ0 20 300 market market 18 MJD MJD MJD+Emb.Opt. MJD+Emb.Opt. 16 250 14

12

10 200

put price 8

6 put implied vol (%) 150 4

2

0 100 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 strike strike

Figure 3: Refit of 21-Apr-20 using MJD rather than Black (cf. Fig. 1). A jump-diffusion on its own cannot generate enough skew (red trace), but an embedded option can.

13 3 Convenience yield dynamics; Term structure

Another direction in which the modelling can be generalised is in the modelling of convenience yield dynamics. 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 the longer-dated futures are typically less volatile than the shorter-dated ones, and this suggests that the price is mean-reverting under P. With a nod to interest rate theory and in particular the Hull–White model [6, §17.11], it is obvious to consider the following general linear model, combining lognormal intrinsic asset prices with convenience yield dynamics and, optionally, interest rates (for the present purposes we neglect the last of these effects). Option prices can then be calculated in closed form using the methods of §1. By GL2, we mean a model in which the logarithm of the intrinsic asset price xt = ln At and the convenience yield yt follow a bivariate Gaussian process of the form dx x σ dW A t = Λ t dt + M(t) dt + A t (15) dy y σ dW y  t  t  y t  where the matrix Λ is constant, and the drift term M(t) and the volatilities σ are not allowed to depend on x or y but may be time-dependent. By (1),

dx = (r y σ2 /2) dt + σ dW A. (16) t − t − A A t 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 dynamics via a parameter β 0 thus: ≥ dy = κ(α(t)+ βx y ) dt + σ dW y. (17) t t − t y t

When At is low, yt is more likely to be low/negative (contango) and when At is high, yt is more likely to be positive (backwardation). Thereby

0 1 r σ2 /2 Λ= − , M(t)= − A . κβ κ κα(t)  −    The effect of the coupling parameter β is to introduce an implicit mean reversion into the asset dynamics, because when the intrinsic asset price is low the futures curve is likely to be in contango and when high in backwardation. The long-term variance is reduced, so that long-dated futures contracts are less volatile than short-dated ones. Models of this form therefore impose a term structure upon the futures price and also on the volatility, even if the parameters themselves are not maturity-dependent. Backwardation and contango curves can be obtained easily enough in this model, the former by making y0 and α(t) positive, the latter by making them negative. A humped curve is obtained by making α(t) > 0 but y0 < 0. (We assume that the effect of r is negligible.) At a simple level, we can argue that embedded optionalities just accentuate this. As an example, suppose that y0 and α(t) are both negative, giving an upward-sloping curve, and suppose also that the parameters associated with ψ− are not maturity-dependent. Then for the front contracts the embedded puts will be more in-the-money than the back ones, and so the modelled futures prices will be more strongly upward-sloping (as in April 2020). However, we repeat an important point we made earlier. The optionalities may well have different parameters for different maturity dates, and so in times of extreme stress the futures curve can in principle become contorted into weird shapes, as happens with the electricity market because storage in that market is next to impossible.

14 The obvious attraction of GL2 is that many quantities associated with it, principally the mean and variance of AT , can be calculated in closed form. Indeed,

t 2 t A xt x0 Λ(t−s) r σA/2 Λ(t−s) σA dWs = + e − ds + e y yt y0 κα(s) σy dWs     Z0   Z0   and so the joint distribution of xt and yt is bivariate Normal with mean

t 2 x − r σ /2 0 + eΛ(t s) − A ds y κα(s)  0 Z0   and covariance matrix t 2 ′ Λ(t−s) σA ρσAσy Λ (t−s) e 2 e ds, ρσAσy σ Z0  y  A y with ρ the correlation between dWt and dWt . These expressions require matrix exponentiation, achieved by the following lemma8:

δt a−d δt 2b δt a b cosh + sinh sinh exp t = e(a+d)t/2 2 δ 2 δ 2 (18) c d 2c δt δt d−a δt    " δ sinh 2 cosh 2 + δ sinh 2 # with δ2 = (a d)2 + 4bc. − Contrary to what is implied in [2] there is no requirement that the eigenvalues of Λ be real: all that is necessary is that both have real part 0, which is automatic provided that β, κ 0. In ≤ ≥ fact, when δ2 < 0, equivalent to β > κ/4, we will have oscillatory behaviour with period 2πi/δ, as is observed in autoregressive processes with complex-conjugate poles. It is worth noting that we could delete β and make the mean reversion explicit, by adding a term κ x dt (where κ 0 is − x t x ≥ another parameter) into the drift term of (16), replacing 0 with κ in the top left-hand element − x of Λ. This has been suggested in e.g. [4, Eq.(3.8)], but has a fundamentally different effect in that the 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, and this presents no further difficulties in analysis provided the interest rate dynamics are not in any way 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 imposes rigidity on the structure in the sense that the calibration parameters should not vary too strongly from one maturity to the next. Two restrictions are:

(i) One can no longer choose a different volatility for each maturity. The variation of volatility with T is determined by σA in (16) and the parameters κ, β. ∗ ∗ (ii) As seen in (9), Ft (Ti)/Ft (Ti+1) gives an estimate of the convenience yield and so there cannot be 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 wide excursion of the convenience yield from its equilibrium level, adding to the catalogue of real-world examples in which the Ornstein–Uhlenbeck process fails to capture large deviations9.

8Proved by diagonalising the matrix. 9A discussion of this and possible extensions to the OU model is given in [9].

15 4 Conclusions and Final Remarks

We have presented an extension of the Black model that incorporates an intrinsic optionality into the commodity price. It captures a liability caused by failure of the physical delivery process, and can cause negative futures prices, allowing the events of this April in the oil market to be captured with precision and insight. On the other hand in normal market conditions it constitutes only a small deformation from standard model frameworks. It extends to exponential L´evy models in a natural way and gives closed-form solutions for option pricing in the Merton jump-diffusion. As an aside and opportunity for further work, it is not necessarily true that all optionalities have a negative impact on the price. Upward spikes in commodity prices can stem from the opposite kind of difficulty discussed here: low inventories, causing difficulty obtaining the physical asset. An obvious prescription in the light of this paper is an optionality of the form

ψ (A)= ℓA (A/Aˆ)λ 1 + + − where again ℓ, λ 0. This would in principle explain why in commodity
markets implied volatilities ≥ often increase for high strikes, sometimes known as the ‘inverse leverage effect’.

The opinions expressed in this paper are those of their authors rather than their institutions. Email [email protected], [email protected]

References

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[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 contingent claims. J. Finance, 45:959–976, 1990.

[6] J. C. Hull. Options, Futures, and Other Derivatives. Prentice Hall, 1997. 3rd ed.

[7] S. G. Kou. A jump-diffusion model for option pricing. Mgt. Sci., 48(8):1086–1101, 2002.

[8] R. J. Martin. CUSP 2007: An overview of our new structural model. Technical report, Credit Suisse, 2007. www.researchgate.net/profile/Richard Martin16.

[9] R. J. Martin, R. V. Craster, and M. J. Kearney. Infinite product expansion of the Fokker– Planck equation with steady-state solution. Proc. R. Soc. A, 471(2179):20150084, 2015.

[10] W. Schoutens. L´evy Processes in Finance. Wiley, 2003.

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