Pricing a Class of Exotic Commodity Options in a Multi

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Pricing a class of exotic commodity options in a multi-factor jump-diffusion model Crosby, John

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Empfohlene Zitierung / Suggested Citation: Crosby, J. (2008). Pricing a class of exotic commodity options in a multi-factor jump-diffusion model. Quantitative Finance, 8(5), 471-483. https://doi.org/10.1080/14697680701545707

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Diese Version ist zitierbar unter / This version is citable under: https://nbn-resolving.org/urn:nbn:de:0168-ssoar-221107 Quantitative Finance

Pricing a class of exotic commodity options in a multi-factor jump-diffusion model For Peer Review Only

Journal: Quantitative Finance

Manuscript ID: RQUF-2006-0145.R1

Manuscript Category: Research Paper

Date Submitted by the 19-Jan-2007 Author:

Complete List of Authors: Crosby, John; Lloyds TSB Financial Markets, Financial Markets Division

Commodity Prices, Energy Derivatives, Computational Finance, Keywords: Derivatives Pricing, Option Pricing, Financial Engineering, Mathematical Finance, Methdology of Pricing Derivatives

G13 - Contingent Pricing|Futures Pricing < G1 - General Financial JEL Code: Markets < G - Financial Economics

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1 2 3 4 Pricing a class of exotic commodity options in a 5 multi-factor jump-diffusion model 6 7 8 JOHN CROSBY 9 10 Lloyds TSB Financial Markets, Faryners House, 25 Monument Street, London EC3R 8BQ 11 Email address: [email protected]

12 th th 13 28 March 2006, revised 17 January 2007

15 Acknowledgements: The author wishes to thank Simon Babbs, Peter Carr, Andrew Johnson and 16 For Peer Review Only conference participants at the Global Derivatives conference in Paris in May 2006 and at the 17 Derivatives and Risk Management conference in Monte Carlo in June 2006 as well as two anonymous 18 referees. 19

41 Our aim, in this paper, is to price a class of simple European-style exotic commodity options within 42 an extension of the Crosby (2005) model. One of the features of the commodities markets is that 43 options which are considered “exotic” for other asset classes are very common in the commodities 44 markets. Consider an option which pays the greater of zero and the difference between the prices of 45 two commodities minus a fixed strike (which might in practice, be zero). These options are very 46 actively traded. When the commodities are crude oil and a refined oil product (such as heating oil or jet 47 fuel), an option on the price difference is called a crack spread option. These crack spread options are 48 actively traded, not only in the OTC market but also, on NYMEX, the New York futures exchange. 49 When one of the commodities is coal, spread options are called dark spread options and when one of 50 the commodities is electricity, spread options are called spark spread options. Phraseology apart, all 51 these options are options on the difference between the prices of two commodities. The prices in 52 question might be the futures prices to some given tenors or the spot prices of two different 53 commodities. In this paper, we will focus on the case when the prices in question are futures 54 commodity prices because, we can easily include the case of spot prices as a special case of the former 55 (ie as a futures contract which matures at the same time as the option maturity). 56 Another phraseology that is also used for spread options is that of “primary” commodity and 57 “daughter” commodity. A “primary” commodity might be, for example, a very actively traded blend of 58 crude oil (in practice, either Brent or WTI) and a “daughter” commodity would then be either a much 59 less actively traded blend (eg Bonny Light from Nigeria or Dubai) of crude oil or a refined petroleum 60 product such as heating oil, jet fuel or gasoline. The price movements of the “daughter” commodity would closely, but not perfectly, follow those of the “primary” commodity. In practice, many spread

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1 2 3 options involve a “primary” commodity and a “daughter” commodity although, clearly, spread options 4 on two seemingly unrelated commodities, such as natural gas and a base metal, are possible. 5 It would be possible to approximate the prices of spread options by making ad-hoc assumptions such 6 as assuming the price spread is normally or log-normally distributed. However, such assumptions are 7 ad-hoc and are inconsistent with the assumptions typically made about the dynamics of the individual 8 commodities. The disadvantages of these approaches are discussed in, for example, Dempster and 9 Hong (2000) and Garman (1992). We will briefly mention one disadvantage of modelling price spreads 10 ie (arithmetic) price differences as log-normal. Because, of the way that crude oil is refined, through 11 fractional distillation, into a basket of refined products, one would expect the basket of refined products 12 to be always worth more than the same quantity of crude oil and that the difference is the (positive) 13 cost of refining. One might therefore be tempted to expect that the price of a particular refined 14 petroleum product (for example, heating oil or aviation fuel) is always higher (when measured in the 15 same units) than the price of crude oil. In fact, whilst a positive price differential is the more common 16 situation, it isFor empirically Peerobserved (see, forReview example, Geman (2005)) Only that sometimes an imbalance of 17 supply and demand in the international markets results in a negative price differential, albeit usually for 18 just short periods of time. It is also observed that, over a period of time, the spot price of a benchmark 19 grade of crude oil (such as Brent or WTI) can trade both more expensively and, at different times, more 20 cheaply than a given, less actively traded grade of crude oil (such as Bonny Light or Dubai). Clearly, it 21 would not be appropriate, therefore, to model (arithmetic) price differences as log-normal. So therefore, 22 in this paper, we will look at pricing spread options without ad-hoc assumptions about the price spread 23 and consistent with each of the two commodities following the dynamics of the model of Crosby 24 (2005). 25 Quite often the fixed strike of the spread option is, in fact, zero and we will call these “zero strike” 26 spread options. These are the type we will focus on, in this paper. The “zero strike” type of spread 27 option (an option to exchange one asset for another) was first considered by Margrabe (1978) for the 28 case of log-normally distributed asset prices. See also Rubinstein (1991a), (1991b) and Geman (2005) 29 and the references therein. Duffie et al. (2000) consider the pricing of some simple types of exotic 30 options for assets (bonds (both risk-free and defaultable), foreign exchange rates and equities) 31 following affine jump-diffusion processes. Deng (1998) considers the pricing of spread options on spot 32 commodity prices where the underlying spot commodity prices follow affine jump-diffusion processes. 33 In addition, Dempster and Hong (2000) have considered spread options (including the more difficult 34 case of “non-zero-strike”) on options where the underlying assets can follow more general stochastic 35 processes, including processes with stochastic volatility. Duffie et al. (2000), Deng (1998) and 36 Dempster and Hong (2000) all use Fourier Transform methods. 37 There are other actively traded variants on spread options, including options on the price ratio (rather 38 than the price difference). Another variant is that the underlying is actually a single physical 39 commodity but the spread involves the price difference (or ratio) between two futures contracts on that 40 same commodity but with two different tenors. These could be viewed as options on the slope of the 41 futures commodity curve. A somewhat different variant again is that a single commodity futures 42 contract is observed at two different calendar times. This gives rise to forward start and ratio forward 43 start (single leg cliquet) options. Using Fourier Transform methods, we will derive a single unifying 44 form for all these exotic options and some of their generalisations. 45 It is well-known (see Geman (2005) and Crosby (2005) and the references therein) that jumps are an 46 important feature of the commodities and commodity options markets, being both more frequent and 47 larger in magnitude than in, for example, the equity and foreign exchange markets. 48 In Crosby (2005), we introduced a multi-factor jump-diffusion model for commodities and 49 commodity options. It is an arbitrage-free model consistent with any initial term structure of futures 50 commodity prices. The model incorporates multiple jump processes into the dynamics of futures 51 commodity prices. It also allows for a specific empirically observed feature, common in the 52 commodities markets (especially for energy related commodities such as crude oil, natural gas and 53 electricity), that when there are jumps in futures commodity prices, the short end of the futures 54 commodity price curve jumps by a larger magnitude than the long end of the futures commodity price curve. This is a feature that did not seem to have appeared in the literature before. In fact, Deng (1998), 55 and several other papers, such as Hilliard and Reis (1998) and Clewlow and Strickland (2000), include 56 jumps in models for spot commodity prices. None of these models are consistent with any initial term 57 structure of futures commodity prices but even if time-dependent drift terms were introduced to allow 58 for this, they are only able to produce jumps which cause parallel shifts in the term structure of (log) 59 futures commodity prices. We also allow for these latter types of jumps (see Assumption 2.2 in section 60 2) but, in addition, through an exponential dampening feature, we also allow for jumps (see Assumption 2.1 in section 2) which cause long-dated futures commodity prices to jump by smaller

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1 2 3 magnitudes than short-dated futures commodity prices. In Crosby (2005), we explain how jumps which 4 cause parallel shifts in the term structure of (log) futures commodity prices are empirically more 5 suitable for modelling options on gold (in this respect, gold “trades like a currency”). On the other 6 hand, the exponentially dampened type of jumps is shown to be more suitable for modelling most other 7 commodities (especially crude oil, natural gas and electricity). 8 A feature of “primary” and “daughter” commodities is that, it is observed empirically that, when 9 there are jumps in the price of the “primary” commodity, then there are also simultaneous jumps in the 10 price of the “daughter” commodity, albeit, generally of a different magnitude. 11 In this paper, we consider two commodities which we will label Commodity 1 and Commodity 2. 12 We consider how we can adapt the Crosby (2005) model to realistically handle the case of two 13 different commodities. Heuristically, we suppose that there are background (for example, economic) 14 factors which influence the dynamics of futures commodity prices. These background factors are 15 represented mathematically as Brownian motions and Poisson processes. To provide some heuristic 16 intuition as toFor how the PoissonPeer processes Review relate to the dynamics ofOnly futures commodity prices, we 17 consider the following: One could imagine there being factors which caused the futures prices of both 18 natural gas and electricity to jump simultaneously whilst there could also be factors (an outage, for 19 example) which caused electricity prices to jump but did not cause jumps in the futures prices of 20 natural gas. Equally there could be factors which always caused simultaneous jumps in the futures 21 prices of crude oil and the futures prices of a refined petroleum product (although, of course, the 22 magnitudes of the jumps could be different). At the other end of the spectrum, one could imagine 23 modelling the futures prices of two commodities (perhaps a base metal and an energy-related 24 commodity) which would have no simultaneous jumps at all. Of course, our aim in this paper is to 25 price commodity derivatives for which we need to model commodity prices in the risk-neutral measure 26 – it is not to explain price movements in the real-world physical measure. The heuristic intuition above 27 is simply designed to provide an insight into our model. 28 In order to cater for all the different possible cases of modelling the futures prices of two different 29 underlying commodities, we suppose there are M Poisson processes which drive all futures 30 commodity prices. If, in fact, the price of a particular commodity does not jump in response to a jump 31 of a particular Poisson process, we can cater for this by setting the jump size to be identically equal to 32 zero. 33 In addition to Poisson processes, futures commodity prices are also driven by multiple Brownian 34 motions. The diffusion volatilities associated with the Brownian motions are assumed deterministic but 35 otherwise can be specified in a fairly flexible manner (Crosby (2005) provides more details or see 36 Miltersen (2003) for a specification which can model seasonality in the term structure of volatilities, 37 which is an empirically observed feature of the natural gas markets). 38 In this paper, we assume that interest-rates are stochastic and, therefore (Cox et al. (1981)), futures 39 commodity prices and forward commodity prices are not the same. We will work with futures 40 commodity prices but, results in, for example, Jamshidian (1993) and Crosby (2005) show that pricing 41 options involving forward commodity prices is a straightforward extension. 42 The rest of this paper is organised as follows: In section 2, we consider a simple but realistic 43 extension of the Crosby (2005) framework to model two underlying commodities. In section 3, we 44 define the payoff of a simple class of exotic options. In section 4, we derive a generic formula for the 45 price of these options using Fourier Transform methods. In section 5, we provide some numerical 46 examples of our methodology. Section 6 is a short conclusion. 47 48 49 2. Extending the model to two underlying commodities 50 51 In this paper, we will make the standard assumptions that markets are frictionless and arbitrage-free. 52 We will work exclusively in the equivalent martingale measure (EMM), under which 1 futures 53 commodity prices are martingales, which, depending on the form of the model, may not be unique. In 54 essence, in the case of non-uniqueness (which corresponds to market incompleteness) we assume that 55 an EMM has been “fixed” through the market prices of standard (plain vanilla) options and by an abuse 56 of language call this the (rather than an) EMM. Crosby (2005) provides more details. We denote 57 expectations, at time t , with respect to the EMM by Et [ ]. 58 59 60 1 To be precise, the EMM under which futures prices are martingales is defined with respect to the money market account numeraire.

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1 2 3 We denote the (continuously compounded) risk-free short rate, at time , by and we denote the 4 t r(t) 5 price, at time t , of a (credit risk free) zero coupon bond maturing at time T by P(t,T ) . We assume 6 that interest-rates are stochastic and (see Heath et al. (1992)) follow a Gaussian interest-rate model (eg 7 Hull-White, extended Vasicek, Babbs (1990), Hull and White (1993)), which is an arbitrage-free model 8 consistent with any initial term structure of interest-rates. The dynamics of bond prices under the EMM 9 are (Babbs (1990), Heath et al. (1992), Hull and White (1993)): 10 11 dP (t,T ) 12 = r()()()t dt + σ P t,T dz P t , 13 P()t,T 14 15 where σ P (t,T ) is a purely deterministic function of t and T , with σ P (T,T ) = 0 , and dz P (t) 16 denotes standardFor Brownian Peer increments. In Review section 5, we will provide Only numerical examples where we 17 work within a one factor Gaussian (Hull-White, extended Vasicek) model in which we write 18

19 σ P (t,T ) ≡ σ r (1− exp (−α r (T − t))) α r , where σ r and α r are positive constants. However, all 20 results in this paper are extendable to any multi-factor Gaussian HJM (Heath et al. (1992)) interest-rate 21 model without further ado. 22 23 We consider two commodities, labelled Commodity 1 and Commodity 2. We denote the futures 24 price of Commodity i , i = 2,1 , at time t to time T (ie the futures contract, into which Commodity 25 26 i , i = 2,1 , is deliverable, matures at time T ) by H i (t,T ) . Then for each i , i = 2,1 , we assume, 27 following Crosby (2005), that the dynamics of futures commodity prices under the EMM are: 28 29 K dH (t,T ) i 30 i = ∑σ Hi ,k ()()()()t,T dz Hi ,k t −σ P t,T dz P t 31 H i ()t−,T k =1 32 M T M 33      + exp γ exp − b ()()u du  −1dN − e t,T dt , (2.1) 34 ∑  i,m  ∫ i,m   mt ∑ i,m 35 m=1    t   m=1 36 37 where 38    T   39 e ()()()t,T ≡ λ t E exp γ exp − b u du  −1 , (2.2) i,m m i,m   i,m  ∫ i,m   40    t   41 42 where, for each k , k = 2,1 ,..., K , σ t,T are purely deterministic functions of at most t and 43 i Hi ,k ( ) 44 T , dz Hi ,k (t), for each k , are standard Brownian increments (which can be correlated with each other 45 46 and with dz P (t) but we assume the instantaneous correlations between these Brownian motions are 47 constant and form a positive semi-definite correlation matrix), and N mt , for each m , m = 1,..., M , 48 49 are independent Poisson processes whose intensity rates, under the EMM, at time t , are λm (t) which 50 are positive deterministic functions of at most t . The functions b (t), for each m , and for each i , 51 i,m 52 are non-negative deterministic functions which we call jump decay coefficient functions. The 53 parameters γ i,m , for each m , are parameters, which we call spot jump amplitudes. For each m , Ei,m 54 denotes the expectation operator, at time , conditional on a jump occurring in , ie in equations 55 t N mt 56 2.1 and 2.2, it computes the expected impact of jumps in the m th Poisson process on Commodity i . 57 We use the terminology spot jump amplitudes for the parameters γ i,m because it can be seen 58 59 (Crosby (2005)) that γ i,m is the log of the jump amplitude of the futures price for a futures contract 60 with T = t (ie a futures contract for immediate delivery), ie for each m , γ i,m is the (log of the) spot

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1 2 3 jump amplitude for Commodity . We assume that the spot jump amplitudes are one of two 4 i γ i,m 5 possible forms, which we term those of assumption 2.1 and assumption 2.2, which in turn are linked to 6 two possible specifications of the jump decay coefficient functions bi,m (t). 7 8 For each m , m = 1,..., M , we assume that either: 9 Assumption 2.1 : 10 The spot jump amplitudes γ i,m , for each i , are assumed to be constants, which we denote by βi,m . In 11 12 this case, the jump decay coefficient functions bi,m (t) are assumed to be any non-negative 13 deterministic function. • 14 Or: 15 Assumption 2.2 : 16 In this case, For the jump decay Peer coefficient functionsReview b (t), for each Only i , i = 2,1 , are assumed to be 17 i,m

18 identically equal to zero ie bi,m (t) ≡ 0 for all t and for each i . The spot jump amplitudes γ i,m are 19 20 assumed to be normally distributed random variables, with (under the EMM) mean βi,m and standard 21 deviation υ , for each i , each of which is independent of each of the Brownian motions and of each 22 i,m 23 of the Poisson processes. For different m , the spot jump amplitudes γ i,m are assumed to be 24 independent. However, for a given m , we assume that, for this m , the correlation between the spot 25 J J 26 jump amplitudes γ ,1 m and γ ,2 m is ρ12 ,m . In other words, we assume correl (γ ,1 m ,γ ,2 n ) ≡ ρ12 ,m if 27 m = n and correl (γ ,γ ) ≡ 0 otherwise. We assume that, for each i and for each m , β , 28 ,1 m ,2 n i,m J 29 υi,m and ρ12 ,m are all constants. • 30 31 32 Remark 2.3 : Note that, in assumptions 2.1 and 2.2, β ,1 m need not equal β ,2 m and also that one of 33 β or β may be zero (and for assumption 2.2, likewise υ and υ ). This allows us to 34 ,1 m ,2 m ,1 m ,2 m 35 capture the effect where, in response to a jump in N mt at time t , the spot price H1 ( ,tt ) of 36 Commodity 1 and the spot price H ( ,tt ) of Commodity 2 may jump by different magnitudes (and 37 2 38 one may not actually jump at all). 39 40 We define the indicator functions, for each m , m = 1,..., M , 1m()1.2 = 1 if assumption 2.1 is 41 satisfied, for this m , and 1 = 0 otherwise and 1 = 1 if assumption 2.2 is satisfied, for this 42 m()1.2 m()2.2 43 m , and 1m()2.2 = 0 otherwise. Then equation 2.1 and assumptions 2.1 and 2.2 imply that 44 45 M M     T   46 e ()t,T = 1 λ ()()t exp β exp − b u du  −1 ∑ i,m ∑ m()1.2 m   i,m  ∫ i,m   47 m=1 m=1     t   48 M 49    1    t 2  , (where we have used b t ≡ if = ). 50 + ∑1m()2.2 λm ()exp β i,m + υi,m  −1 i,m ( ) 0 1m()2.2 1 m=1  2  51    52 Crosby (2005) provides more information about the consequences of the assumptions above and of 53 equation 2.1. In short, the consequences are that futures commodity prices are martingales in the EMM 54 55 and (with a suitable (see Crosby (2005)) form for σ Hi ,k (t,T )) log of the spot prices of both 56 Commodity 1 and Commodity 2 exhibit mean reversion in the EMM. It is also shown how, when the 57 jumps are of the type of assumption 2.1, jumps can also contribute to the effect of mean reversion and 58 that the speed of this jump-related mean reversion is given by the values of the jump decay coefficient 59 functions. When there are jumps, in the case of assumption 2.1 (and provided the relevant jump decay 60 coefficient functions bi,m (t) are strictly positive), the prices of long-dated futures contracts jump by smaller magnitudes than short-dated futures contracts because of the exponential dampening effect of

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1 2 3 the jump decay coefficient functions in equation 2.1. This is in accordance with stylised empirical 4 observations in most commodities markets (especially for energy-related commodities). In the case of 5 assumption 2.2, jumps cause parallel shifts in the (log of the) futures commodity prices to all tenors 6 because, in this case, the jump decay coefficient functions b t are identically equal to zero. 7 i,m ( ) 8 Stylised empirical observations suggest this is more appropriate for gold. 9 10 We have deliberately worked with very general forms of the diffusion volatility parameters 11 σ Hi ,k (t,T ), the intensity rates λm (t) and the jump decay coefficient functions bi,m (t). The specific 12 13 functional forms and the values of K1 , K 2 and M would be chosen by the trader according to her 14 intuition of the behaviour of the two underlying commodities. To help with this process, we will briefly 15 consider possible specifications of the dynamics of the futures prices of the two commodities. 16 For Peer Review Only 17 2.1 A possible specification for the jumps and the diffusion volatilities 18 19 Suppose that Commodity 1 is WTI grade crude oil. This is the “primary” commodity. It is very 20 actively traded and there are many standard European options traded on it whose prices the trader can 21 observe in the market. Suppose Commodity 2 is heating oil, a refined petroleum product. This is the 22 “daughter” commodity. It is not so actively traded but there are some (but a smaller number than for 23 WTI grade crude oil) standard European options traded on it whose prices she can observe in the

24 market. We suppose K1 = 2 , K 2 = 3 and M = 1. Furthermore, we suppose the two Brownian 25 motions driving Commodity 1 are also precisely the first two Brownian motions driving Commodity 2, 26 with the same volatility parameters. The third Brownian motion driving Commodity 2 is specific to that 27 commodity. More specifically, we assume 28 29 σ H (t,T ) = σ H (t,T ) = η + χ exp (− a (T − t)) (2.3) 30 1,1 1,2 1 1 1 31 σ H 2,1 (t,T ) = σ H 2,2 (t,T ) = χ 2 exp (− a2 (T − t)) (2.4) 32 33 and we assume the diffusion volatility function for the third Brownian motion (driving only 34 Commodity 2) is of the form 35

36 σ H 3,2 (t,T ) = χ 3 exp (− a3 (T − t)), (2.5) 37 38 where η , χ , χ , χ , a , a and a are all constants. 39 1 1 2 3 1 2 3 40 We will drop the first subscripted index for the Brownian motions in this subsection only (ie write 41 42 dz H ,1 k (t) = dz H ,2 k (t) ≡ dz H ,k (t), for k = 2,1 and dz H ,2 k (t) ≡ dz H ,k (t), for k = 3 ). 43 We define the correlations (assumed constant), for i = 3,2,1 and j = 3,2,1 : 44 45 , . 46 ρi, j ≡ correl (dz H ,i (t), dz H ,j (t)) ρ P,i ≡ correl (dz P (t), dz H ,i (t)) 47 48 We assume that Commodity 1 and Commodity 2 both jump in response to increments in the Poisson 49 process N1t which we assume to be of the type of assumption 2.1 and to have a constant intensity rate 50 ie λ t ≡ λ , where λ is a constant. The jump decay coefficient functions are identical for each 51 1 ( ) 1 1 52 commodity and assumed constant ie b 1,1 (t) ≡ b 1,2 (t) ≡ b1 , where b1 is a constant. However, we 53 54 assume the spot jump amplitudes are possibly different ie β 1,1 is not necessarily equal to β 1,2 . 55 56 Then we can write the dynamics of Commodity 1 and Commodity 2 (under the EMM) as: 57 58 dH (t,T ) 1 = ()()()η + χ exp ()− a ()T − t dz t + χ exp ()()− a T − t dz t 59 H t−,T 1 1 1 H 1, 2 2 H 2, 60 1 () − σ P (t,T )dz P (t) + (exp (β 1,1 exp (− b1 (T − t)))−1)dN 1t − e 1,1 (t,T )dt , (2.6)

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1 2 3 4 dH 2 (t,T ) = ()()()η1 + χ1 exp ()− a1 ()T − t dz H 1, t + χ 2 exp ()()− a2 T − t dz H 2, t 5 H 2 ()t−,T 6 + χ exp (− a (T − t))dz (t) 7 3 3 H 3,

8 −σ P (t,T )dz P (t)+ (exp (β 1,2 exp (− b1 (T − t)))−1)dN 1t − e 1,2 (t,T )dt . (2.7) 9 10 H (t,T ) H ( ,tt ) 11 2 2 If we define R 1/2 ()t,T ≡ and C 1/2 ()t ≡ , then by Ito’s lemma, 12 H1 ()t,T H1 (),tt 13 14 dR 1/2 (t,T ) 15 = χ 3 exp ()()()()− a3 ()T − t dz H 3, t + ρ P3σ P t,T χ 3 exp ()− a3 T − t dt 16 R 1/2 ()t−,T For Peer Review Only 17 − χ3 exp (− a3 (T − t))(ρ 1,3 (η1 + χ1 exp (− a1 (T − t)))+ ρ 2,3 (χ2 exp (− a2 (T − t))))dt 18

19 20 + (exp ((β 1,2 − β 1,1 )exp (− b1 (T − t)))−1)dN 1t − (e 1,2 (t,T )− e 1,1 (t,T ))dt . (2.8) 21 22 Note the form of the diffusion volatility term which only depends on the Brownian increments 23 dz (t) . In fact, utilising results in section 3 of Crosby (2005), it is now clear, from equation 2.8, that 24 H 3, 25 the SDE for C 1/2 (t) can be written either in the form 26 27 d(ln C 1/2 (t)) = a3 (Λ J (t)− (ln C 1/2 (t)))dt + χ 3dz H 3, (t)+ (β 1,2 − β 1,1 )dN 1t , (2.9) 28

29 or, equivalently and alternatively, in the form 30 31 d(ln C (t)) = b (Λ (t)− (ln C (t)))dt + χ dz (t)+ (β − β )dN , (2.10) 32 1/2 1 D 1/2 3 H 3, 1,2 1,1 1t 33 34 where Λ J (t) and Λ D (t) are stochastic mean reversion levels whose exact forms can easily be 35 obtained utilising the methodology leading to proposition 3.4 of Crosby (2005), albeit at the expense of 36 some algebra (in fact, Λ (t) is a pure-jump stochastic process and Λ (t) is a pure-diffusion 37 J D 38 stochastic process). 39 We see that the log ratio ln C 1/2 (t) of the spot prices of the two commodities is a mean reverting 40 stochastic process (under the EMM). This is an attractive feature for modelling, for example, the case 41 where Commodity 1 is crude oil (the “primary” commodity) and Commodity 2 is a refined petroleum 42 product (the “daughter” commodity) such as heating oil, because, heuristically, we would expect the 43 price differential (and therefore also the log ratio) in the long-term to not move too far away from a 44 long-run mean level which reflects the cost of the refining process. However, in the short-term, the log 45 price ratio (and therefore also the arithmetic price difference) can go negative in line with the stylised 46 empirical observations made in section 1. 47 We will also briefly mention how this model might be calibrated. Usually, there will be fewer 48 actively traded options on the “daughter” commodity than on the “primary” commodity. One could 49 estimate the parameters of the process for the “primary” commodity by calibrating to the market prices 50 of standard options. In our example above, there would be eleven parameters, namely η , χ , χ , 51 1 1 2 52 a1 , a2 , ρ 2,1 , ρ P 1, , ρ P 2, , λ1 , b1 and β 1,1 . Having determined these eleven parameters, one could 53 take these as given. Then one could estimate the remaining six parameters, namely χ , a , ρ , 54 3 3 1,3 55 ρ 2,3 , ρ P 3, , β 1,2 , from the market prices of standard options on the “daughter” commodity. There 56 would typically, be fewer actively traded options on the “daughter” commodity but, equally, there are 57 fewer parameters to estimate. Of course, it would require an empirical investigation, beyond the scope 58 of this paper, to determine how feasible our suggested calibration mechanism might be. 59 In order to give some intuition about the correlation between the futures prices of Commodity 1 and 60 Commodity 2, we compute the model implied correlation between log of the futures prices of the two

commodities for different tenors S and T , ie correl (ln (H1 ( ,0 S)) ln, (H 2 ( ,0 T ))), given the

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1 2 3 model specification in equations 2.6 and 2.7. The results are in figure 1. For both S and T (plotted on 4 the x and y axes), we used the values 0, 0.5, 1, 1.5, 2, 2.5, 3 (all tenors are in years). Our parameter 5 6 values are exactly as in examples 1 and 2 (see section 5). Note that when S = T , the correlations are 7 lowest for the shortest tenor (ie for S = T = 0 ) but tend to one for the longest tenor ( S = T = 3). 8 This behaviour seems 2 quite intuitive for the case where Commodity 1 is crude oil and Commodity 2 is 9 a refined petroleum product such as heating oil. One would expect the prices of longer-dated futures 10 contracts on crude oil and heating oil to have a higher correlation since the difference between the (log 11 of the) prices would reflect the (average) cost of refining (which one would expect to vary only a little). 12 By contrast, one would expect a lower correlation between the (log of the) prices of shorter-dated 13 futures contracts on crude oil and heating oil because the price movements would reflect additional 14 short-term issues such as supply and demand, inventory and weather conditions. 15 In our example above, we have considered the dynamics of two commodities (“primary” and 16 “daughter”) whereFor intuition Peer suggests they willReview move closely (but not Onlyperfectly) together. Of course, in 17 the case of two seemingly unconnected commodities such as, for example, natural gas and a base 18 metal, a different specification of the jumps and the diffusion volatilities would be chosen. For 19 example, we might consider two Poisson processes, with the first Poisson process N1t only causing 20 jumps in Commodity 1 (by having β ≠ 0 and β = 0 ) and the second Poisson process N only 21 1,1 1,2 2t 22 causing jumps in Commodity 2 (by having β 2,1 = 0 and β 2,2 ≠ 0 ). We would also specify the 23 diffusion terms differently. The example above is just meant for illustration. 24 We have illustrated how the model could be applied in a specific case of interest but, for this rest of 25 this paper, we now return to considering the general case, as we turn our attention to pricing a class of 26 exotic commodity options. 27

29 30 3. A class of exotic commodity options 31 32 Our aim is to price a European-style option whose payoff is the greater of zero and a particular 33 function involving the futures price, at time T 1,1 , of Commodity 1 deliverable at (ie the futures contract 34 on Commodity 1 matures at) time T and the futures price, at time T , of Commodity 2 deliverable 35 1,2 2,1 36 at (ie the futures contract on Commodity 2 matures at) time T 2,2 , where T 2,1 ≤ T 1,1 , T 1,2 ≥ T 1,1 and 37 38 T 2,2 ≥ T 2,1 . The payoff is known at time T 1,1 but is paid at (a possibly later) time Tpay . Note 39 T ≥ T ≥ T . 40 pay 1,1 2,1 41 More mathematically, we price a European-style option whose payoff is: 42 43   H T ,T − K * H T ,T ε     1 ( 1,1 1,2 ) [ 2 ( 2,1 2,2 )]   44 max η 0, , at time Tpay , (3.1)   H T ,T α   45   []2 ()2,1 2,2   46 47 where η = 1 if the option is a call and η = −1 if the option is a put. Note ε and α are constants 48 * 49 and, furthermore, K is a constant which might, for example, account for different units of 50 measurement. The reason for investigating options with this class of payoffs is that it contains as 51 special cases a number of option types of interest, all of which are actively traded in the OTC 52 commodity options markets. 53 54 We will now briefly outline (for the case of call options) some of these special cases: 55 56 57 58 2 Note that the parameter λ is the intensity rate in the risk-neutral EMM. Clearly, this parameter could be 59 1 different from the intensity rate in the real-world physical measure (in addition, had we also considered jumps of 60 the type of assumption 2.2, the mean jump amplitudes may also be different in the two different measures). Hence, whilst the model implied (risk-neutral) correlations shown in figure 1 can provide intuition for traders, they are not directly comparable to historical correlations (implicitly evaluated in the real-world physical measure).

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1 2 3 Spread (crack spread or dark spread or spark spread) options 4 These are options on the difference in price between two different underlying commodities. Their 5 payoffs can be defined (for the “zero strike” case) via equation 3.1 with ε = 1 and α = 0 . In practice, 6 7 we usually have T 1,1 = T 2,1 . If the underlying prices, on which the option payoff is determined, are 8 spot prices, then we also set T = T and T = T . • 9 1,1 1,2 2,1 2,2 10 11 Ratio spread or relative performance options 12 These are options on the ratio of the price of two different underlying commodities. Their payoffs 13 can be defined via equation 3.1 with ε = 1 and α = 1. In practice, we usually have T 1,1 = T 2,1 . If the 14 underlying prices, on which the option payoff is determined, are spot prices, then we also set 15 T = T and T = T . • 16 1,1 1,2 For2,1 Peer2,2 Review Only 17 18 Options on futures commodity price curve spreads 19 These are options on a single underlying physical commodity but with futures commodity contracts 20 of different tenors ie T 1,2 ≠ T 2,2 . Their payoffs can be defined via equation 3.1 with 21 22 H1 (•,•) ≡ H 2 (•,•). Typically, we have T 1,1 = T 2,1 , ε = 1 and, either α = 0 or α = 1. • 23 24 Forward start options 25 These are options on a single underlying commodity in which T is strictly less than T . The 26 2,1 1,1 27 payoff is the greater of zero and the difference between the futures commodity price to a given tenor at 28 some calendar time and the futures commodity price to the same tenor at some earlier calendar time. 29 Their payoffs can be defined via equation 3.1 with H1 (•,•) ≡ H 2 (•,•), ε = 1 and α = 0 . For the 30 case just described, one would have T = T but other variants are possible. For example, forward 31 1,2 2,2

32 start options on the spot commodity price would have T 1,1 = T 1,2 and T 2,1 = T 2,2 . • 33

34 Ratio forward start options 35 Note that these options might also be called single-leg cliquets by analogy with terminology in the 36 37 equity options markets. These are also options on a single underlying commodity in which T 2,1 is 38 strictly less than T . The payoff is the greater of zero and the ratio of the futures commodity price to a 39 1,1 40 given tenor at some calendar time and the futures commodity price to the same tenor at some earlier 41 calendar time (minus a constant strike term). Their payoffs can be defined via equation 3.1 with 42 H1 (•,•) ≡ H 2 (•,•), ε = 1 and α = 1. Again, one could also have ratio forward start options on the 43 spot commodity price with T = T and T = T . • 44 1,1 1,2 2,1 2,2 45 46 Of course, we can also price options which are generalisations or mixtures of the special cases noted 47 above. For example, ε and α need not be integers. 48 We should also make a brief comment about the time Tpay at which the option payoff is paid. The 49 50 most common situation, in practice, is that Tpay would be set equal to T 1,1 . However, occasionally, we 51 observe in the OTC markets that commodity options are traded where the payoff is deferred for a short 52 period of time after T 1,1 (and this is not just the standard two working day spot settlement but might, 53 54 for example, be a period of a few weeks). For example, it might be that Tpay is set equal to the 55 maturity of one of the underlying futures contracts. 56 We will now return, for the rest of the paper, to the completely general case of considering the class 57 of exotic options whose payoff is given by equation 3.1. 58 59 60

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1 2 3 4. Fourier Transform methodology 4

5 In this section, we will use a Fourier transform methodology, to price European-style options whose 6 payoff is defined in equation 3.1. We will proceed along the lines of Sepp (2003) who considers the 7 case of standard European options (on a single underlying asset). 8 9 Define, for times t1 ≥ t and t2 ≥ t : 10 11  H t ,T   H t,T   1 ( 1 1,2 )   1 ( 1,2 )  12 Y ()t1 ,T 1,2 ,t2 ,T 2,2 ;t ≡ log   . (4.1)  H t ,T ε   H t,T ε  13  []2 ()2 2,2   []2 ()2,2  14 15 The price of the European-style option, whose payoff is given by equation 3.1, at time t , (for 16 For Peer Review Only ) is: 17 t ≤ T 2,1 ≤ T 1,1 18 19   Tpay    H (T ,T )− K * [H (T ,T )]ε   20     1 1,1 1,2 2 2,1 2,2   Et exp − r()s ds max η 0,  21   ∫    H T ,T α    t    []2 ()2,1 2,2   22   23 = M 1 (t)+ M 2 (t)− M 3 (t), (4.2) 24 25 T   pay   ()1+η −α 26 where M ()t ≡ E exp − r()s ds H ()()T ,T []H T ,T  (4.3) 27 1 t  ∫  1 1,1 1,2 2 2,1 2,2 2  t  28     T 29 1−η   pay   30 ()   * ε −α and M 2 ()t ≡ Et exp − r()s ds K []H 2 ()T 2,1 ,T 2,2  (4.4) 31 2  ∫    t   32 33   Tpay  34 and M ()()t ≡ E exp − r s ds  3 t  ∫  35   t  36   37 ε −α H1 (t,T 1,2 )  *  . (4.5) 38 []H 2 ()T 2,1 ,T 2,2 min ε exp ()Y ()T 1,1 ,T 1,2 ,T 2,1 ,T 2,2 ;t , K   H t,T  39  []2 ()2,2  40 41 The last set of equations follows from a simple algebraic arrangement. 42 We focus, firstly, on M 3 (t). 43 44  H t,T  45  1 ( 1,2 ) *  Define f ()Y ()T 1,1 ,T 1,2 ,T 2,1 ,T 2,2 ;t ≡ min exp ()Y ()T 1,1 ,T 1,2 ,T 2,1 ,T 2,2 ;t , K 46  []H ()t,T ε  47  2 2,2  48 and then write 49 ˆ 50 f (Y (T 1,1 ,T 1,2 ,T 2,1 ,T 2,2 ;t)) in (inverse) terms of its Fourier Transform f (z) ie write 51 52 1 iz i +∞ 53 f ()Y ()T ,T ,T ,T ;t = exp ()− izY ()T ,T ,T ,T ;t fˆ()z dz , (4.6) 54 1,1 1,2 2,1 2,2 ∫ 1,1 1,2 2,1 2,2 2π iz −∞ 55 i

56 where z is complex. Results in Lewis (2001) and Sepp (2003) show that, by taking the Fourier 57 58 Transform of f (Y (T 1,1 ,T 1,2 ,T 2,1 ,T 2,2 ;t)), which exists provided 0 < zi < 1, where zi is the 59 imaginary part of z , then: 60

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1 2 3 iz +1  * ε  4 H1 (t,T 1,2 )  1  K [H 2 (t,T 2,2 )] 5 fˆ()z =    . (4.7) ε z 2 − iz  H t,T  6 []H 2 ()t,T 2,2   1 ()1,2  7 8 Furthermore, by substituting equation 4.6 into equation 4.5, M 3 (t) is given by: 9 10 T   pay  iz i +∞  11   ε −α 1 ˆ 12 Et exp − r()s ds []H 2 ()T 2,1 ,T 2,2 exp ()− izY ()T 1,1 ,T 1,2 ,T 2,1 ,T 2,2 ;t f ()z dz    ∫  2π ∫  13   t  iz i −∞  14 T iz i +∞   pay   15 1   ε −α ˆ = Et exp − r()s ds []H 2 ()T 2,1 ,T 2,2 exp ()− izY ()T 1,1 ,T 1,2 ,T 2,1 ,T 2,2 ;t  f ()z dz 16 2π ∫For  ∫Peer Review Only  17 iz i −∞   t   18 1 iz i +∞ 19 ≡ Φ()− z t,; T ,T ,T ,T fˆ()z dz , (4.8) 2π ∫ 1,1 1,2 2,1 2,2 20 iz i −∞ 21 22 where Φ(− z t,; T 1,1 ,T 1,2 ,T 2,1 ,T 2,2 )≡ 23 T 24   pay   E exp − r()s ds []H ()T ,T ε −α exp ()− izY ()T ,T ,T ,T ;t  , (4.9) 25 t  ∫  2 2,1 2,2 1,1 1,2 2,1 2,2 26   t   27 28 and where we use Fubini’s theorem to justify the interchange of the integral and the expectation 29 operator. We will call Φ(− z t,; T 1,1 ,T 1,2 ,T 2,1 ,T 2,2 ) the “extended” characteristic function (we have 30 borrowed the terminology from Duffie et al. (2000) but our definition is somewhat different). 31

32 We now collect the equations above into the form of a proposition. 33

34 Proposition 4.1 : The price of the European-style option, at time t , whose payoff is defined in 35 equation 3.1, is: 36 37 1 iz i +∞ 38 M ()()t + M t − Φ()− z t,; T ,T ,T ,T fˆ()z dz . (4.10) 39 1 2 ∫ 1,1 1,2 2,1 2,2 2π iz −∞ 40 i

41 Proof : From equations 4.2 and 4.8. • 42 43 Remark 4.2 : Note that equation 4.10 holds independently of the specific model for futures commodity 44 prices. So, for example, we could consider extensions of the Crosby (2005) model which allow for, for 45 example, stochastic volatility or alternative specifications of the jump processes (in the manner of 46 Heston (1993), Duffie et al. (2000) and Barndorff-Nielsen and Shephard (2001)) and equation 4.10 47 would still be applicable, provided that the “extended” characteristic function can be calculated. 48 Equation 4.10 (with minor modifications) could also be useful for options involving other asset classes 49 such as equities (see Duffie et al. (2000)) or inflation (see Mercurio (2005), where it is shown that the 50 valuation of derivatives on year-on-year inflation involves calculations very similar to valuing ratio 51 forward start (cliquet) options). However, for the sake of brevity, we will not pursue this point further 52 in this paper. 53 54 In the appendix, we write down the “extended” characteristic function when the dynamics of futures 55 commodity prices are given by equation 2.1. From the form of the “extended” characteristic function, 56 we can also easily obtain explicit forms for M (t) and M (t) (see the appendix). 57 1 2 58 We can now calculate the option price, via equation 4.10 provided the integral is well-defined, 59 which requires 0 < zi < 1. One choice (as in Lewis (2001) and Sepp (2003)) is to evaluate the 60 integral along the straight line given by z = u + i 2 , where u is real.

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1 2 3 With this choice, the price of the European-style option at time t , whose payoff is defined in 4 equation 3.1, is: 5 6 1 ∞ 7 M ()()t + M t − Φ()− u − i t,;2 T ,T ,T ,T fˆ()u + i 2 du , (4.11) 8 1 2 ∫ 1,1 1,2 2,1 2,2 π 0 9 10 where we have also changed the lower limit of the integration from − ∞ to zero by using the fact that 11 the option price is real and hence the integrand is odd in its imaginary part and even in its real part. We 12 will not write down the option price formula in its most explicit form as it is rather long and would not 13 greatly enhance intuition. Equations for and are in the appendix and ˆ can 14 M 1 (t) M 2 (t) f (u + i 2) 15 be obtained from equation 4.7. 16 If the “extended”For characteristic Peer function Review were to be completely Only analytic, then it would be 17 straightforward to evaluate the integral in equation 4.11. In particular, we can compute option prices 18 using a single one-dimensional integration irregardless of how many Brownian motions and Poisson 19 Processes drive the futures commodity prices. If all the Poisson processes satisfy assumption 2.2, and 20 provided that λm (s)ds is easily evaluated (and, of course, in practice, one would choose a form for 21 ∫ 22 the intensity rates λ s so that λ s ds can be evaluated analytically), then this would be the m ( ) ∫ m ( ) 23 case in our model. Unfortunately, if any of the Poisson processes satisfy assumption 2.1, then our 24 “extended” characteristic function involves integrals (see the second, third, fourth and fifth lines of 25 equation A.2 in the appendix) which means that evaluating equation 4.11 involves at least a double 26 integral. This is certainly computationally feasible but equally performing a double integral will be 27 considerably slower than a single integral. Crosby (2006) shows how calculation times can be speeded 28 up, when pricing standard (plain vanilla) European options, by using power series expansions of terms 29 appearing in the characteristic function. A similar idea can be used here provided we make some 30 simplifying assumptions. 31 32 As in Crosby (2006), we make the following assumption: 33 Assumption 4.3 : We will henceforth assume that, for each , , and, for 34 m m = 1,..., M λm (s) ≡ λm 35 each i , bi,m (t) ≡ bi,m are constants. Furthermore, we assume that, if for this m , the jumps satisfy 36 37 assumption 2.1, then bi,m > 0 . (This condition is not restrictive since if bi,m were to equal zero, we 38 could treat it as in the case of assumption 2.2 which is much simpler). • 39 40 This means that we can use the power series expansions of Crosby (2006) for the terms on the 41 second, third and fourth lines of the “extended” characteristic function (see equation A.2) (into which 42 we would substitute z = u + i 2 , where u is real). 43 In order to rapidly compute the following term (the fifth line) in equation A.2 (into which, again, we 44 45 would substitute z = u + i 2 ): 46 47  M T 2,1  48 exp  1 λ ()()s exp ε −α + iz ε β φ s,T − iz β φ s,T ds  , ∑ m()1.2 ∫ m ( ,2 m ,2 m ()()2,2 ,1 m ,1 m 1,2 )  49  m=1 t  50 51 (where φ (s,T ) and φ (s,T ) are defined as in equation A.1 in the appendix) we will make 52 ,1 m 1,2 ,2 m 2,2 53 the following additional assumption: 54

55 Assumption 4.4 : We will henceforth assume that, for each m , b ,1 m and b ,2 m are identically equal ie 56 57 that b ,1 m ≡ b ,2 m ≡ bm , say. • 58 59 Remark 4.5 : Crosby (2005) shows that futures commodity prices can be written in terms of a number 60 of Gaussian state variables and M Poisson jump state variables. It can therefore be shown that assumption 4.4 is equivalent to saying (for assumption 2.1) that the futures prices of Commodity 1 and Commodity 2 are driven by the same jump state variables. It is shown in Crosby (2005) that our model

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1 2 3 is consistent with mean reversion, under the EMM. Not only that, but it is also shown that, when the 4 jump processes are of the type of assumption 2.1, then jumps can also contribute to the effect of mean 5 reversion and that the speed of this jump-related mean reversion is equal to the associated jump decay 6 coefficient function. Hence assumption 4.4 is also equivalent to assuming that, after a jump, there is a 7 common speed of jump-related mean reversion in Commodity 1 and Commodity 2. Although it would 8 be an empirical matter, beyond the scope of this paper, to fully justify assumption 4.4, this assumption 9 does, therefore, have some economic intuition. In addition, we note that assumption 4.4 is obviously a 10 non-assumption in the special case when the option is on a single underlying commodity (see section 3, 11 for example, options on futures commodity price curve spreads, forward start options and ratio forward 12 start options), since it must hold. 13 14 With assumption 4.4, we can make a similar type of power series expansion which we specify in the 15 next proposition. 16 For Peer Review Only 17 Proposition 4.6 : Define ψ ≡ exp (− b (T − T )) and ψ ≡ exp (− b (T − T )) , with 18 start m start end m end 19 Tstart ≤ Tend ≤ T . Then: 20 21 Tend

22 λm exp ()()()()iω1 + ω2 exp ()− bm T − s ds = λm Tend − Tstart 23 ∫ Tstart 24 n   2 2 n n  25 ∞ ω + ω ψ −ψ ()cos ()()nθ + isin nθ λm  1 ( 1 2 ) ()end start  26 + ∑  , (4.12) 27 bm  n=1 n n!  28   

29 where ω1 and ω2 are real numbers, independent of s , and where θ , is defined as follows: 30 Firstly, define θ , 0 ≤ θ ≤ π , via cos θ ≡ ω ω 2 + ω 2 , then: 31 2 2 1 2 32 33 If ω1 ≥ 0 and ω2 ≥ 0 , then θ = θ , else if ω1 ≥ 0 and ω2 < 0 , then θ = π −θ , else if ω1 < 0 34 and ω ≥ 0 , then θ = 2π −θ , else if ω < 0 and ω < 0 , then θ = π + θ . 35 2 1 2 36 Proof: This proposition is just a generalisation of proposition 3.3 in Crosby (2006) and can be proved 37 3 38 in an identical fashion. Therefore, the proof is omitted . • 39 40 All the integrals (see the second, third, fourth and fifth lines of equation A.2) which appear in the 41 “extended” characteristic function can be nested in a form which enables them to be evaluated by 42 proposition 4.6, provided assumptions 4.3 and 4.4 hold. Hence, we can quickly and easily evaluate the 43 “extended” characteristic function. We can also evaluate M 1 (t) and M 2 (t) (see appendix) in the 44 same way. We can then very rapidly, using standard one dimensional numerical integration techniques, 45 compute the integral in equation 4.11 and hence also compute the price of the European-style option 46 whose payoff is defined in equation 3.1. 47 48 49 5. Numerical examples and results 50 51 In this section, we will provide four numerical examples, labelled examples 1, 2, 3 and 4, of our 52 methodology, the results of which are in tables 1, 2, 3 and 4 respectively. In all four examples, we 53 value European-style options, whose payoff is defined in equation 3.1, using equation 4.11. 54 We evaluate the integral with respect to u in equation 4.11 using Simpson’s rule with 1024 points. 55 Examining the forms of equations A.2 and 4.11, we see that for large u , the integrand behaves 56 57 58 59 3 It is straightforward to see that the power-series expansion in equation 4.12 will be rapidly convergent. Indeed 60 the modulus of the term appearing in the square brackets is guaranteed to be monotonically declining to zero when 2 2 n > max ( ,2 ω1 + ω2 ).

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1 2 3 4  1  2 1  2   2 1  asymptotically like exp − u + Σ ()t,T 1,1 ,T 2,1 ,T 1,2 ,T 2,2  u +  which clearly tends 5  2  4    4  6 2 7 to zero rapidly as u → ∞ , since Σ (t,T 1,1 ,T 2,1 ,T 1,2 ,T 2,2 ) ≥ 0 . We truncate the upper limit of the 8  1 1  1 9  2  2  2  integral when the value of u is such exp − u + Σ ()t,T 1,1 ,T 2,1 ,T 1,2 ,T 2,2  u +  is 10  2  4    4  11 less than 10 -8. We truncate the infinite series in equation 4.12 when the value of an additional term in 12 the series has converged to less than 10 -12 . 13 In all four examples, we assume that the initial (ie as of the valuation date of the options that we will 14 value in our examples) futures prices of Commodity 1 to all maturities are 40 and the initial futures 15 prices of Commodity 2 to all maturities are 41. We assume that the initial interest-rate yield curve is 16 flat with a continuouslyFor compoundedPeer risk-free Review rate of 0.044 ie we assume Only that the discount factors for 17 all maturities T , as of the valuation date, time t , of the options that we will value, are all of the form 18 . Interest-rates are stochastic and evolve following a one factor Hull-White 19 exp (− .0 044 (T − t)) 20 (extended Vasicek) model in which 21 σ P (t,T ) ≡ σ r (1− exp (−α r (T − t))) α r , where σ r = .0 012 and α r = .0 125 . 22 In all four examples, we use the same form for the diffusion parameters as in equations 2.3 to 2.5 in 23 section 2.1. That is, we suppose K = 2 and K = 3 and, furthermore, we suppose 24 1 2 25 η1 = 12.0 , χ1 = 22.0 , χ 2 = 25.0 , χ 3 = .0 242 , a1 = 9.0 , a2 = 7.0 , a3 = 5.1 . 26 We assume all correlations are 0.05 ie for all i = 3,2,1 and j = 3,2,1 : 27 28 ρi, j = 05.0 and ρ P,i = 05.0 . Note that all these parameters are just for illustration. 29 30 In all four examples, we assume that the maturities of the futures contracts on Commodity 1 are of 31 the form T 1,2 = T 1,1 + (31 365 ) and on Commodity 2 of the form T 2,2 = T 2,1 + (91 365 ) ie the 32 33 futures contracts on Commodity 1 and Commodity 2 mature 31 days and 91 days respectively after 34 T 1,1 and T 2,1 . In all four examples, we set Tpay ≡ T 1,1 . We specify T 1,1 and T 2,1 in the examples. 35 In each example, we value six options and all of them are calls (ie η = 1). For the first three 36 * 37 options, α = 0 , ε = 1 and the values of K are 0.95, 0.975 and 1. The fourth, fifth and sixth options 38 have α = 1 and, again, ε = 1 and the values of K * are 0.95, 0.975 and 1. Thus, we evaluate spread 39 * * 40 options for three different values of K and ratio spread options for the same three values of K in 41 each example. 42 43 Example 1 : 44 In example 1, we assume T 1,1 = 1 and T 2,1 = 1. We assume that there is one Poisson process, 45 M =1 , and it satisfies assumption 2.1 and it has an intensity rate λ = .0 512 . As in the example in 46 1 47 section 2.1, both Commodity 1 and Commodity 2 exhibit jumps of non-zero magnitude in response to 48 jumps in this Poisson process. We assume b 1,1 = b 1,2 = 55.1 , β 1,1 = 55.0 , β 1,2 = 35.0 . We price 49 the six different options and the results are in table 1. • 50 51 Example 2 : 52 In example 2, we assume T = 3 and T = 2 . This means that the maturities of the futures 53 1,1 2,1 54 contracts on Commodity 1 and Commodity 2 are approximately 3.08493151 and 2.24931507 years 55 respectively. Note that because T 2,1 is strictly less than T 1,1 , the options in this example can also be 56 57 viewed as hybrid forward start options (for the first three options where α = 0 ) and ratio forward start 58 options (for the fourth, fifth and sixth options where α = 1) involving two different commodities. We 59 assume that all the jump parameters are exactly the same as in example 1. We price the six different 60 options and the results are in table 2. •

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1 2 3 Example 3 : 4 In example 3, we assume and . We use exactly the same diffusion parameters as in 5 T 1,1 = 1 T 2,1 = 1 6 examples 1 and 2 but to provide a contrast with those examples, we assume that there are two Poisson 7 process, M = 2 , and they both satisfy assumption 2.1 and they have intensity rates λ1 = .0 512 and 8 9 λ2 = 47.0 respectively. Commodity 1 jumps but Commodity 2 does not jump in responses to jumps

10 in this first Poisson process N1t . Conversely, Commodity 2 jumps but Commodity 1 does not jump in 11 12 responses to jumps in this second Poisson process N 2t . We assume b 1,1 = b 1,2 = 55.1 , 13 b = b = 55.1 , β = 55.0 , β = 0 , β = 0 , β = 35.0 . We price the six different 14 2,1 2,2 1,1 1,2 2,1 2,2 15 options and the results are in table 3. • 16 For Peer Review Only 17 Example 4 : 18 In example 4, we assume T 1,1 = 3 and T 2,1 = 2 . We assume that there are two Poisson processes 19 again and that all the jump parameters are exactly the same as in example 3. We assume that all the 20 diffusion parameters are exactly the same as in examples 1, 2 and 3. Note that, as in example 2, 21 because T is strictly less than T , the options in this example can also be viewed as hybrid forward 22 2,1 1,1 23 start and ratio forward start options. We price the six different options and the results are in table 4. • 24 25 Computations were performed on a desk-top p.c., running at 2.8 GHz, with Microsoft Windows XP 26 Professional, with 1 Gb of RAM with a program written in Microsoft C++. The total calculation time 27 for all 24 options in examples 1 to 4 was 0.532 seconds or an average of less than 23 milliseconds per 28 option. By significantly increasing the number of points in the numerical integration and by 29 significantly reducing the tolerances used to truncate the upper limit of the integral (in equation 4.11) 30 and the power series expansions (as in equation 4.12), we were able to confirm that in proportional (ie 31 proportional to the calculated option prices) terms, all the option prices in tables 1 to 4 are accurate to 32 at least one part in 500,000 and, also, that in absolute terms, all the option prices are accurate to at least 33 5 decimal places. So our algorithm is both fast and accurate. 34 Note how the option prices in examples 3 and 4 are higher than the corresponding option prices in 35 examples 1 and 2 respectively. This is intuitive given the different specifications of the jump processes 36 driving futures commodity prices, between, on the one hand, examples 1 and 2, and, on the other hand, 37 examples 3 and 4, and given the arguments we presented after equation 2.10. 38 39 40 6. Conclusions 41 42 We have extended the Crosby (2005) model to simultaneously model the prices of multiple 43 commodities. We then priced a class of simple exotic options which includes those whose payoffs 44 involve two different underlying commodities, or a single underlying commodity but with futures 45 contracts of two different tenors or the price of a single underlying futures contract observed at two 46 different calendar times. This class of exotic options includes common exotics such as (crack, dark or 47 spark) spread options, ratio spread options, forward start options and ratio forward start options (single 48 leg cliquets). We have shown that these exotic options can be priced using Fourier methods in any 49 model in which the relevant “extended” characteristic function is known analytically or can be 50 computed rapidly. The Crosby (2005) model falls into the latter category. We have provided some 51 numerical examples which demonstrate that our methodology is both fast and accurate. 52 Finally, we will briefly mention two possible areas for future research: 53 (i) We have focussed, when pricing spread options in this paper, on the “zero strike” case. Dempster and Hong (2000) show how “non-zero-strike” spread options can be priced using a two-dimensional 54 Fast Fourier Transform methodology combined with an ingenious decomposition of the option payoff 55 analogous to Riemann sums. Their approach (combined with assumptions 4.3 and 4.4 and the power 56 series expansion of proposition 4.6) could be used to price “non-zero-strike” spread options within the 57 framework of this paper. It might also be possible to extend the Dempster and Hong (2000) approach in 58 order to price more exotic variations of some of the option types we discussed in section 3. 59 (ii) In section 2.1, we provided an example of specifying the dynamics of the futures prices of two 60 different commodities based on heuristics and trader-intuition. It might be possible to construct a more

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1 2 3 systematic approach based on suitable extensions of the methodology described in section 3 of 4 Casassus and Collin-Dufresne (2005). However, we leave this for future research. 5 6 7 Appendix 8

9 10 In order to obtain the forms for M 1 (t) and M 2 (t), defined in equations 4.3 and 4.4, we can 11 essentially use the “extended” characteristic function (defined in equation 4.9 and given explicitly in 12 equation A.2 below), into which we substitute z = i and z = 0 respectively, then : 13 14   Tpay   15 ()1+η   −α M 1 ()t ≡ Et exp − r()s ds H1 ()()T 1,1 ,T 1,2 []H 2 T 2,1 ,T 2,2  16 For2 Peer ∫  Review Only   t   17   18 (1+η) H1 (t,T 1,2 ) = Φ()− ti ,; T 1,1 ,T 1,2 ,T 2,1 ,T 2,2 19 2 H t T ε 20 []2 (), 2,2 21 22 and

23   Tpay   24 ()1−η * ε −α 25 M ()t ≡ E exp − r()s ds K []H ()T ,T  2 2 t  ∫  2 2,1 2,2 26   t   27 (1−η) 28 = Φ()t,;0 T 1,1 ,T 1,2 ,T 2,1 ,T 2,2 . 29 2 30 31 We will now proceed to write down the “extended” characteristic function when the dynamics of 32 futures commodity prices are given by equation 2.1, after defining the following notation: 33 C C 34 dH (t,T ) Ki dH (t,T ) For each i , i = 2,1 , i ≡ σ ()()()()t,T dz t −σ t,T dz t , ie i 35 H C ()t,T ∑ Hi ,k Hi ,k P P H C ()t,T 36 i k=1 i 37 denotes the purely continuous martingale component in the SDE for Commodity i . 38 39  T  For each i , i = 2,1 , φ s,T ≡ exp − b u du  . (A.1) 40 i,m ()() ∫ i,m  41  s  42 43 T 2,1 C  dP (s,Tpay ) dH 2 (s,T 2,2 ) 44 Define U ()t,T ,T ≡ −()ε −α cov  , ds 2,1 2,2 ∫  P s T C  45 t  (), pay H 2 ()s,T 2,2 

46 T 2,1 C 47 1  dH 2 (s,T 2,2 ) − (ε −α )(ε −α −1 ) var  ds . 48 2 ∫  H C ()s,T  49 t  2 2,2 

50 51 Define W (t,T 1,1 ,T 2,1 ,T 1,2 ,T 2,2 ) ≡

52 T 1,1 C T 2,1 C  dP (s,Tpay ) dH 1 (s,T 1,2 )  dP (s,Tpay ) dH 2 (s,T 2,2 ) 53 cov  , ds − ε cov  , ds 54 ∫  P s T C  ∫  P s T C  t  (), pay H1 ()s,T 1,2  t  (), pay H 2 ()s,T 2,2  55 T 2,1 C C T 2,1 C 56  dH 1 (s,T 1,2 ) dH 2 (s,T 2,2 ) 1  dH 2 (s,T 2,2 ) −α cov  , ds + ()ε + 2εα − ε 2 var  ds . 57 ∫  H C s,T H C s,T  2 ∫  H C s,T  58 t  1 ()1,2 2 ()2,2  t  2 ()2,2  59 60

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1 2 3 T 1,1 C 4 2  dH 1 (s,T 1,2 ) Define Σ ()t,T ,T ,T ,T ≡ var  ds 5 1,1 2,1 1,2 2,2 ∫  H C ()s,T  6 t  1 1,2  7 T 2,1 C C T 2,1 C  dH 1 (s,T 1,2 ) dH 2 (s,T 2,2 ) 2  dH 2 (s,T 2,2 ) 8 − 2ε cov  , ds + ε var  ds . ∫  C C  ∫  C  9 t  H1 ()s,T 1,2 H 2 ()s,T 2,2  t  H 2 ()s,T 2,2  10 11 In order to compute the “extended” characteristic function, we will use the fact that Brownian 12 motions and Poisson processes have independent increments. Then by direct calculation: 13 14  M T 1,1  15   Φ(− z t,; T 1,1 ,T 1,2 ,T 2,1 ,T 2,2 ) = exp − ∑ λm ()s ds 16 For Peer Review∫  Only  m=1 t  17 18  M T 1,1  exp  1 λ s exp − iz β φ s,T ds  19 ∑ m()1.2 ∫ m () (),1 m ,1 m ()1,2  m=1  20  T 2,1  21 T 1,1 22  M  exp iz 1 λ ()s exp β φ s,T −1 ds  23  ∑ m()1.2 ∫ m ()(),1 m ,1 m ()1,2  24  m=1 t  25  M T 2,1  26   exp − ()ε −α + iz ε ∑1m()1.2 λm ()s ()exp ()β ,2 mφ ,2 m ()s,T 2,2 −1 ds 27  ∫   m=1 t  28 T 29  M 2,1  exp  1 λ ()()s exp ε −α + iz ε β φ s,T − iz β φ s,T ds  30 ∑ m()1.2 ∫ m ( ,2 m ,2 m ()()2,2 ,1 m ,1 m 1,2 )  31  m=1 t  32  M  T 1,1   33   1 2 2  exp 1  λ ()s exp − iz β − υ z 34 ∑ m()2.2 ∫ m  ,1 m ,1 m  m=1    2  35   T 2,1   36 T  M  1,1   1   37 exp iz 1  λ ()s ds exp  β + υ 2  −1 38  ∑ m()2.2  ∫ m  ,1 m ,1 m  m=1 t   2   39     T 40  M  2,1  1  41     2   exp − ()ε − α + iz ε ∑1m()2.2 λm ()s ds exp β ,2 m + υ ,2 m  −1 42   ∫  2   m=1  t     43 44  M  T 2,1      1 2 2 45 exp 1m 2.2 λm ()()s ds exp  ε − α + iz ε β ,2 m + ()ε −α + iz ε υ ,2 m ∑ ()  ∫  2 46  m=1  t   47 48 1 2 2 J  − iz β ,1 m − υ ,1 m z − ()ε −α + iz ε iz ρ12 ,mυ ,1 mυ ,2 m  49 2  50 51  1 2 2  exp  ()iz − z Σ ()()t,T 1,1 ,T 2,1 ,T 1,2 ,T 2,2 exp ()− izW t,T 1,1 ,T 2,1 ,T 1,2 ,T 2,2 52  2  53 ε −α 54 P(t,Tpay )exp (−U (t,T 2,1 ,T 2,2 ))(H (t,T 2,2 )) . (A.2) 55 56 57 58 59 60

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1 2 3 Figure 1 : 4 5 Graph of the model implied instantaneous correlation correl ln H ,0 S ln, H ,0 T between 6 ( ( 1 ( )) ( 2 ( ))) 7 the futures prices of Commodity 1 and Commodity 2, for different tenors S and T , given the model 8 specification in equations 2.6 and 2.7. For both S and T (plotted on the x and y axes), we used the 9 values 0, 0.5, 1, 1.5, 2, 2.5, 3 (all tenors are in years). Our parameter values are as in examples 1 and 2 10 (see section 5). 11 12 13 14 Model implied instantaneous correlations 15 16 For Peer Review Only 17 1.000 18 19 0.950 20 0.900 21 22 0.850 23 24 0.800 25 0.750 26 27 0.700 28 0.650 29 30 0.600 31 3 2.5 32 2 1.5 33 1 0.5 34 0 Futures tenor (in years) 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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1 2 3 Table 1 : 4 5 There is one Poisson process. T , T , T , T . 6 1,1 = 1 2,1 = 1 1,2 = 1+ (31 365 ) 2,2 = 1+ (91 365 ) 7 The values of K * are across the first row and the option prices are in bold across the second row. 8 9 α = 0 , ε = 1 (spread options) α = 1, ε = 1 (ratio spread options) 10 0.95 0.975 1.0 0.95 0.975 1.0 11 2.64579 2.19204 1.80901 0.05799 0.04737 0.03852 12 13 ______

14 Table 2 : 15

16 For Peer Review Only 17 There is one Poisson process. T 1,1 = 3 , T 2,1 = 2 , T 1,2 = 3 + (31 365 ) , T 2,2 = 2 + (91 365 ) . 18 * The values of K are across the first row and the option prices are in bold across the second row. 19

20 21 α = 0 , ε = 1 (spread options) α = 1, ε = 1 (ratio spread options) 0.95 0.975 1.0 0.95 0.975 1.0 22 23 6.04522 5.66903 5.31508 0.17500 0.16471 0.15498 24 ______25 26 Table 3 : 27

28 There are two Poisson processes. T 1,1 = 1 , T 2,1 = 1, T 1,2 = 1+ (31 365 ), T 2,2 = 1+ (91 365 ) . 29 * 30 The values of K are across the first row and the option prices are in bold across the second row. 31 32 α = 0 , ε = 1 (spread options) α = 1, ε = 1 (ratio spread options) 33 0.95 0.975 1.0 0.95 0.975 1.0 34 4.02340 3.63361 3.28715 0.10248 0.09258 0.08379 35 ______36 37 Table 4 : 38 39 There are two Poisson processes. T = 3 , T = 2 , T = 3 + (31 365 ) , T = 2 + (91 365 ) . 40 1,1 2,1 1,2 2,2 * 41 The values of K are across the first row and the option prices are in bold across the second row. 42 43 α = 0 , ε = 1 (spread options) α = 1, ε = 1 (ratio spread options) 44 0.95 0.975 1.0 0.95 0.975 1.0 45 6.17001 5.79409 5.43994 0.17959 0.16926 0.15949 46 ______47 48 49 50 51 References 52 53 Babbs S. (1990) “The term structure of interest-rates: Stochastic processes and Contingent Claims” Ph.D thesis, 54 Imperial College, London 55 56 Barndorff-Nielsen O. and N. Shephard (2001) “Non-Gaussian Ornstein-Uhlenbeck-based models and some of 57 their uses in financial economics” Journal of the Royal Statistical Society B 63 p167-241

58 Casassus J. and P. Collin-Dufresne (2005) “Stochastic Convenience Yield Implied from Commodity Futures and 59 Interest Rates” Journal of Finance Vol LX (5) p2283-2331 60 Clewlow L. and C. Strickland (2000) “Energy Derivatives: Pricing and Risk Management” Lacima Group

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1 2 3 Cox J., J. Ingersoll and S. Ross (1981) “The relation between Forward prices and Futures prices” Journal of 4 Financial Economics 9 p321-346 5 6 Crosby J. (2005) “A multi-factor jump-diffusion model for commodities” accepted for publication – forthcoming 7 in Quantitative Finance. Also available on the website of the Centre for Financial Research, Cambridge University at: http://mahd-pc.jbs.cam.ac.uk/seminar/2005-6.html 8 9 Crosby J. (2006) “Commodity options optimised” Risk Magazine May 2006 Vol 19 p72-77 10 11 Dempster M.A.H. and S.S.G. Hong (2000) “Spread option valuation and the fast fourier transform” Judge 12 Business School, Cambridge University Working Paper WP26/2000. Also published in: Proceedings of the First 13 World Congress of the Bachelier Finance Society, Paris (2000).

14 Deng S. (Oct 1998) “Stochastic Models of Energy Commodity Prices and Their Applications: Mean-reversion 15 with Jumps and Spikes” IE & OR Dept, University of California at Berkeley, Berkeley, CA 94720 16 For Peer Review Only 17 Duffie D., J. Pan and K. Singleton (2000) “Transform Analysis and Asset Pricing for Affine Jump Diffusions” 18 Econometrica vol 68 p1343-1376 19 Garman M. (1992) “Spread the load” Risk Magazine 1992 Vol 5 (11) p68-84 20 21 Geman H. (2005) “Commodities and Commodity derivatives” John Wiley, Chichester, United Kingdom 22 23 Heath D., R. Jarrow and A. Morton (1992) “Bond pricing and the term structure of Interest Rates: A new 24 methodology for contingent claim valuation” Econometrica Vol 60 p77-105 25 Heston S. (1993) “A closed-form solution for options with stochastic volatility with applications to bond and 26 currency options” Review of Financial Studies 6 p327-343 27 28 Hilliard J. and J. Reis (1998) “Valuation of commodity futures and options under stochastic convenience yields, 29 interest rates and jump diffusions in the spot” Journal of Financial and Quantitative Analysis 33 (1) p61-86 30 Hull J. and A. White (1993) “One-factor Interest-Rate Models and the valuation of Interest-Rate Derivative 31 Securities” Journal of Financial and Quantitative Analysis 28 (2) p235-254 32 33 Jamshidian F. (April 1993) “Option and Futures Evaluation with deterministic volatilities” Mathematical Finance 34 vol 3 no 2 p149-159 35 36 Lewis A. (2001) “A simple option formula for general jump-diffusion and other exponential Levy processes” Working paper (available at www.optioncity.net ). 37 38 Margrabe W. (1978) “The value of an option to exchange one asset for another” Journal of Finance vol 33 p177- 39 187 40 41 Mercurio F. (2005) “Pricing Inflation-Indexed Derivatives” Quantitative Finance 5 (3) p289-302 42 Miltersen K. (Feb 2003) “Commodity price modelling that matches current observables: A new approach” 43 Quantitative Finance 3 p51-58 (also available at www.nhh.no/for/cv/miltersen/2602.pdf) 44 45 Rubenstein M. (1991a) “Two in one” Risk Magazine May 1991 Vol 4 p49 46 47 Rubenstein M. (1991b) “One for another” Risk Magazine July 1991 Vol 4 p30-32

48 Sepp A. (2003) “Pricing European-Style Options under Jump Diffusion Processes with Stochastic Volatility: 49 Applications of Fourier Transform” Institute of Mathematical Statistics, University of Tartu, Estonia (available at 50 http:// math.ut.ee/~spartak/ ) or Acta et Commentationes Universitatis Tartuensis de Mathematica vol. 8 p.123-133 51 52 ______53 54 The author, John Crosby, is Global Head of Quantitative Analytics and Research at Lloyds TSB Financial Markets in London and can be contacted at: [email protected] 55 56 57 58 59 60

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1 2 3 4 5 alpha = 0, epsilon = 1 (spread options) alpha = 1, epsilon = 1 (ratio spread options) 6 TABLE 1 (Example 1) 7 K* -> 0.95 0.975 1.0 0.95 0.975 8 Option Price 2.64579 2.19204 1.80901 0.05799 0.04737 9 10 TABLE 2 (Example 2) 11 K* -> 0.95 0.975 1.0 0.95 0.975 12 Option Price 6.04522 5.66903 5.31508 0.17500 0.16471 13 14 TABLEFor 3 (Example Peer 3) Review Only 15 K* -> 0.95 0.975 1.0 0.95 0.975 16 Option Price 4.02340 3.63361 3.28715 0.10248 0.09258 17 18 TABLE 4 (Example 4) 19 K* -> 0.95 0.975 1.0 0.95 0.975 20 Option Price 6.17001 5.79409 5.43994 0.17959 0.16926 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 E-mail: [email protected] URL://http.manuscriptcentral.com/tandf/rquf Quantitative Finance Page 23 of 46

1 2 3 4 alpha = 1,5 epsilon = 1 (ratio spread options) 6 7 1.0 8 0.03852 9 10 11 1.0 12 0.15498 13 14 For Peer Review Only 15 1.0 16 0.08379 17 18 19 1.0 20 0.15949 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 E-mail: [email protected] URL://http.manuscriptcentral.com/tandf/rquf Page 24 of 46 Quantitative Finance

1 2 3 4 5 6 7 8 9 10 11 12 13 14 For Peer Review Only 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 correl(H1(0,S),H2(0,T)) from examples 1 and 2 (but with different tenors) 30 31 0 0.5 1 1.5 2 2.5 32 0 0.881268 0.903977 0.884259 0.843201 0.797556 0.754378 33 0.5 0.874569 0.93563 0.94799 0.92918 0.896973 0.860878 34 1 0.840726 0.932046 0.972136 0.974822 0.957701 0.931548 35 1.5 0.796371 0.907933 0.968926 0.989949 0.987536 0.97259 36 2 0.75201 0.875233 0.950585 0.986126 0.996853 0.993006 37 2.5 0.711487 0.840206 0.924717 0.971182 0.992926 0.999117 38 3 0.675965 0.806303 0.896266 0.950632 0.981143 0.995792 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 E-mail: [email protected] URL://http.manuscriptcentral.com/tandf/rquf Quantitative Finance Page 25 of 46

1 2 3 4 5 6 7 8 9 10 11 12 13 14 For Peer Review Only 15 16 17 18 19 20 21 Model implied instantaneous correlations 22 23 24 1 25 26 0.95 27 0.9 28 29 0.85 30 31 3 0.8 32 0.716001 0.75 33 0.825543 34 0.902457 0.7 35 0.951732 0.65 36 0.981041 0.6 37 0.995654 38 3 0.999772 2.5 2 39 1.5 40 1 0.5 Futures tenor (in years) 41 0 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 E-mail: [email protected] URL://http.manuscriptcentral.com/tandf/rquf Page 26 of 46 Quantitative Finance

1 2 3 4 5 6 7 8 9 10 11 12 13 14 For Peer Review Only 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 E-mail: [email protected] URL://http.manuscriptcentral.com/tandf/rquf Quantitative Finance Page 27 of 46

12 th th 13 28 March 2006, revised 17 January 2007 14

22 23 Abstract 24 A recent paper, Crosby (2005), introduced a multi-factor jump-diffusion model which would allow 25 futures (or forward) commodity prices to be modelled in a way which captured empirically observed 26 features of the commodity and commodity options markets. However, the model focused on modelling 27 a single individual underlying commodity. In this paper, we investigate an extension of this model which would allow the prices of multiple commodities to be modelled simultaneously in a simple but 28 realistic fashion. We then price a class of simple exotic options whose payoff depends on the difference 29 (or ratio) between the prices of two different commodities (for example, spread options), or between 30 the prices of two different (ie with different tenors) futures contracts on the same underlying 31 commodity, or between the prices of a single futures contract as observed at two different calendar 32 times (for example, forward start or cliquet options). We show that it is possible, using a Fourier 33 Transform based algorithm, to derive a single unifying form for the prices of all these aforementioned 34 exotic options and some of their generalisations. Although we focus on pricing options within the 35 model of Crosby (2005), most of our results would be applicable to other models where the relevant 36 “extended” characteristic function is available in analytical form. 37 38 39 40 1. Introduction 41 Our aim, in this paper, is to price a class of simple European-style exotic commodity options within 42 an extension of the Crosby (2005) model. One of the features of the commodities markets is that 43 options which are considered “exotic” for other asset classes are very common in the commodities 44 markets. Consider an option which pays the greater of zero and the difference between the prices of 45 two commodities minus a fixed strike (which might in practice, be zero). These options are very 46 actively traded. When the commodities are crude oil and a refined oil product (such as heating oil or jet 47 fuel), an option on the price difference is called a crack spread option. These crack spread options are 48 actively traded, not only in the OTC market but also, on NYMEX, the New York futures exchange. 49 When one of the commodities is coal, spread options are called dark spread options and when one of 50 the commodities is electricity, spread options are called spark spread options. Phraseology apart, all 51 these options are options on the difference between the prices of two commodities. The prices in 52 question might be the futures prices to some given tenors or the spot prices of two different 53 commodities. In this paper, we will focus on the case when the prices in question are futures 54 commodity prices because, we can easily include the case of spot prices as a special case of the former 55 (ie as a futures contract which matures at the same time as the option maturity). 56 Another phraseology that is also used for spread options is that of “primary” commodity and 57 “daughter” commodity. A “primary” commodity might be, for example, a very actively traded blend of 58 crude oil (in practice, either Brent or WTI) and a “daughter” commodity would then be either a much 59 less actively traded blend (eg Bonny Light from Nigeria or Dubai) of crude oil or a refined petroleum 60 product such as heating oil, jet fuel or gasoline. The price movements of the “daughter” commodity would closely, but not perfectly, follow those of the “primary” commodity. In practice, many spread

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1 2 3 options involve a “primary” commodity and a “daughter” commodity although, clearly, spread options 4 on two seemingly unrelated commodities, such as natural gas and a base metal, are possible. 5 It would be possible to approximate the prices of spread options by making ad-hoc assumptions such 6 as assuming the price spread is normally or log-normally distributed. However, such assumptions are 7 ad-hoc and are inconsistent with the assumptions typically made about the dynamics of the individual 8 commodities. The disadvantages of these approaches are discussed in, for example, Dempster and 9 Hong (2000) and Garman (1992). We will briefly mention one disadvantage of modelling price spreads 10 ie (arithmetic) price differences as log-normal. Because, of the way that crude oil is refined, through 11 fractional distillation, into a basket of refined products, one would expect the basket of refined products 12 to be always worth more than the same quantity of crude oil and that the difference is the (positive) 13 cost of refining. One might therefore be tempted to expect that the price of a particular refined 14 petroleum product (for example, heating oil or aviation fuel) is always higher (when measured in the 15 same units) than the price of crude oil. In fact, whilst a positive price differential is the more common 16 situation, it isFor empirically Peerobserved (see, forReview example, Geman (2005)) Only that sometimes an imbalance of 17 supply and demand in the international markets results in a negative price differential, albeit usually for 18 just short periods of time. It is also observed that, over a period of time, the spot price of a benchmark 19 grade of crude oil (such as Brent or WTI) can trade both more expensively and, at different times, more 20 cheaply than a given, less actively traded grade of crude oil (such as Bonny Light or Dubai). Clearly, it 21 would not be appropriate, therefore, to model (arithmetic) price differences as log-normal. So therefore, 22 in this paper, we will look at pricing spread options without ad-hoc assumptions about the price spread 23 and consistent with each of the two commodities following the dynamics of the model of Crosby 24 (2005). 25 Quite often the fixed strike of the spread option is, in fact, zero and we will call these “zero strike” 26 spread options. These are the type we will focus on, in this paper. The “zero strike” type of spread 27 option (an option to exchange one asset for another) was first considered by Margrabe (1978) for the 28 case of log-normally distributed asset prices. See also Rubinstein (1991a), (1991b) and Geman (2005) 29 and the references therein. Duffie et al. (2000) consider the pricing of some simple types of exotic 30 options for assets (bonds (both risk-free and defaultable), foreign exchange rates and equities) 31 following affine jump-diffusion processes. Deng (1998) considers the pricing of spread options on spot 32 commodity prices where the underlying spot commodity prices follow affine jump-diffusion processes. 33 In addition, Dempster and Hong (2000) have considered spread options (including the more difficult 34 case of “non-zero-strike”) on options where the underlying assets can follow more general stochastic 35 processes, including processes with stochastic volatility. Duffie et al. (2000), Deng (1998) and 36 Dempster and Hong (2000) all use Fourier Transform methods. 37 There are other actively traded variants on spread options, including options on the price ratio (rather 38 than the price difference). Another variant is that the underlying is actually a single physical 39 commodity but the spread involves the price difference (or ratio) between two futures contracts on that 40 same commodity but with two different tenors. These could be viewed as options on the slope of the 41 futures commodity curve. A somewhat different variant again is that a single commodity futures 42 contract is observed at two different calendar times. This gives rise to forward start and ratio forward 43 start (single leg cliquet) options. Using Fourier Transform methods, we will derive a single unifying 44 form for all these exotic options and some of their generalisations. 45 It is well-known (see Geman (2005) and Crosby (2005) and the references therein) that jumps are an 46 important feature of the commodities and commodity options markets, being both more frequent and 47 larger in magnitude than in, for example, the equity and foreign exchange markets. 48 In Crosby (2005), we introduced a multi-factor jump-diffusion model for commodities and 49 commodity options. It is an arbitrage-free model consistent with any initial term structure of futures 50 commodity prices. The model incorporates multiple jump processes into the dynamics of futures 51 commodity prices. It also allows for a specific empirically observed feature, common in the 52 commodities markets (especially for energy related commodities such as crude oil, natural gas and 53 electricity), that when there are jumps in futures commodity prices, the short end of the futures 54 commodity price curve jumps by a larger magnitude than the long end of the futures commodity price 55 curve. This is a feature that did not seem to have appeared in the literature before. In fact, Deng (1998), and several other papers, such as Hilliard and Reis (1998) and Clewlow and Strickland (2000), include 56 jumps in models for spot commodity prices. None of these models are consistent with any initial term 57 structure of futures commodity prices but even if time-dependent drift terms were introduced to allow 58 for this, they are only able to produce jumps which cause parallel shifts in the term structure of (log) 59 futures commodity prices. We also allow for these latter types of jumps (see Assumption 2.2 in section 60 2) but, in addition, through an exponential dampening feature, we also allow for jumps (see Assumption 2.1 in section 2) which cause long-dated futures commodity prices to jump by smaller

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1 2 3 magnitudes than short-dated futures commodity prices. In Crosby (2005), we explain how jumps which 4 cause parallel shifts in the term structure of (log) futures commodity prices are empirically more 5 suitable for modelling options on gold (in this respect, gold “trades like a currency”). On the other 6 hand, the exponentially dampened type of jumps is shown to be more suitable for modelling most other 7 commodities (especially crude oil, natural gas and electricity). 8 A feature of “primary” and “daughter” commodities is that, it is observed empirically that, when 9 there are jumps in the price of the “primary” commodity, then there are also simultaneous jumps in the 10 price of the “daughter” commodity, albeit, generally of a different magnitude. 11 In this paper, we consider two commodities which we will label Commodity 1 and Commodity 2. 12 We consider how we can adapt the Crosby (2005) model to realistically handle the case of two 13 different commodities. Heuristically, we suppose that there are background (for example, economic) 14 factors which influence the dynamics of futures commodity prices. These background factors are 15 represented mathematically as Brownian motions and Poisson processes. To provide some heuristic 16 intuition as toFor how the PoissonPeer processes Review relate to the dynamics ofOnly futures commodity prices, we 17 consider the following: One could imagine there being factors which caused the futures prices of both 18 natural gas and electricity to jump simultaneously whilst there could also be factors (an outage, for 19 example) which caused electricity prices to jump but did not cause jumps in the futures prices of 20 natural gas. Equally there could be factors which always caused simultaneous jumps in the futures 21 prices of crude oil and the futures prices of a refined petroleum product (although, of course, the 22 magnitudes of the jumps could be different). At the other end of the spectrum, one could imagine 23 modelling the futures prices of two commodities (perhaps a base metal and an energy-related 24 commodity) which would have no simultaneous jumps at all. Of course, our aim in this paper is to 25 price commodity derivatives for which we need to model commodity prices in the risk-neutral measure 26 – it is not to explain price movements in the real-world physical measure. The heuristic intuition above 27 is simply designed to provide an insight into our model. 28 In order to cater for all the different possible cases of modelling the futures prices of two different 29 underlying commodities, we suppose there are M Poisson processes which drive all futures 30 commodity prices. If, in fact, the price of a particular commodity does not jump in response to a jump 31 of a particular Poisson process, we can cater for this by setting the jump size to be identically equal to 32 zero. 33 In addition to Poisson processes, futures commodity prices are also driven by multiple Brownian 34 motions. The diffusion volatilities associated with the Brownian motions are assumed deterministic but 35 otherwise can be specified in a fairly flexible manner (Crosby (2005) provides more details or see 36 Miltersen (2003) for a specification which can model seasonality in the term structure of volatilities, 37 which is an empirically observed feature of the natural gas markets). 38 In this paper, we assume that interest-rates are stochastic and, therefore (Cox et al. (1981)), futures 39 commodity prices and forward commodity prices are not the same. We will work with futures 40 commodity prices but, results in, for example, Jamshidian (1993) and Crosby (2005) show that pricing 41 options involving forward commodity prices is a straightforward extension. 42 The rest of this paper is organised as follows: In section 2, we consider a simple but realistic 43 extension of the Crosby (2005) framework to model two underlying commodities. In section 3, we 44 define the payoff of a simple class of exotic options. In section 4, we derive a generic formula for the 45 price of these options using Fourier Transform methods. In section 5, we provide some numerical 46 examples of our methodology. Section 6 is a short conclusion. 47 48 49 2. Extending the model to two underlying commodities 50 51 In this paper, we will make the standard assumptions that markets are frictionless and arbitrage-free. 52 We will work exclusively in the equivalent martingale measure (EMM), under which1 futures 53 commodity prices are martingales, which, depending on the form of the model, may not be unique. In 54 essence, in the case of non-uniqueness (which corresponds to market incompleteness) we assume that 55 an EMM has been “fixed” through the market prices of standard (plain vanilla) options and by an abuse 56 of language call this the (rather than an) EMM. Crosby (2005) provides more details. We denote 57 expectations, at time t , with respect to the EMM by Et [ ]. 58 59 60 1 To be precise, the EMM under which futures prices are martingales is defined with respect to the money market account numeraire.

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1 2 3 4 We denote the (continuously compounded) risk-free short rate, at time t , by (tr ) and we denote the 5 price, at time t , of a (credit risk free) zero coupon bond maturing at time T by (,TtP ). We assume 6 that interest-rates are stochastic and (see Heath et al. (1992)) follow a Gaussian interest-rate model (eg 7 Hull-White, extended Vasicek, Babbs (1990), Hull and White (1993)), which is an arbitrage-free model 8 consistent with any initial term structure of interest-rates. The dynamics of bond prices under the EMM 9 are (Babbs (1990), Heath et al. (1992), Hull and White (1993)): 10 11 (),TtdP 12 ()+= σ P (, )P ()tdzTtdttr , 13 (),TtP 14 15 where σ P (,Tt ) is a purely deterministic function of t and T , with σ P ( TT ) = 0, , and P (tdz ) 16 denotes standardFor Brownian Peer increments. In Reviewsection 5, we will provide Only numerical examples where we 17 work within a one factor Gaussian (Hull-White, extended Vasicek) model in which we write 18 19 σ P ()Tt σ r ()exp1, ()α r ()−−−≡ tT α r , where σ r and α r are positive constants. However, all 20 results in this paper are extendable to any multi-factor Gaussian HJM (Heath et al. (1992)) interest-rate 21 model without further ado. 22 23 We consider two commodities, labelled Commodity 1 and Commodity 2. We denote the futures 24 price of Commodity i , i = 2,1 , at time t to time T (ie the futures contract, into which Commodity 25 26 i , i = 2,1 , is deliverable, matures at time T ) by H i ( ,Tt ) . Then for each i , i = 2,1 , we assume, 27 following Crosby (2005), that the dynamics of futures commodity prices under the EMM are: 28 29 K ,TtdH i 30 i () = ∑σ ,kHi (), ,kHi ()()(−σ P , P tdzTttdzTt ) 31 i ()−,TtH k =1 32 T 33 M ⎛ ⎛ ⎛ ⎞⎞ ⎞ M + ⎜ ⎜γ expexp ⎜− () ⎟⎟ −1⎟ − (), dtTtedNduub , (2.1) 34 ∑⎜ ⎜ ,mi ⎜ ∫ ,mi ⎟⎟ ⎟ mt ∑ ,mi 35 m=1 ⎝ ⎝ ⎝ t ⎠⎠ ⎠ m=1 36 37 where 38 ⎛ ⎛ ⎛ T ⎞⎞ ⎞ 39 ⎜ ⎜ ⎟ ⎟ ,mi (), ≡ λ ()EtTte ,mim γ ,mi expexp ⎜− ,mi ()duub ⎟ −1 , (2.2) 40 ⎜ ⎜ ⎜ ∫ ⎟⎟ ⎟ ⎝ ⎝ ⎝ t ⎠⎠ ⎠ 41

42 43 where, for each k , = ,...,2,1 Kk i , σ ,kHi ( ,Tt ) are purely deterministic functions of at most t and 44 T , ()tdz , for each k , are standard Brownian increments (which can be correlated with each other 45 ,kHi 46 and with P ()tdz but we assume the instantaneous correlations between these Brownian motions are 47 constant and form a positive semi-definite correlation matrix), and N , for each m , = ,...,1 Mm , 48 mt 49 are independent Poisson processes whose intensity rates, under the EMM, at time t , are λm ()t which 50 are positive deterministic functions of at most t . The functions (tb ), for each m , and for each i , 51 ,mi 52 are non-negative deterministic functions which we call jump decay coefficient functions. The

53 parameters γ ,mi , for each m , are parameters, which we call spot jump amplitudes. For each m , E ,mi 54 55 denotes the expectation operator, at time t , conditional on a jump occurring in N mt , ie in equations 56 2.1 and 2.2, it computes the expected impact of jumps in the m th Poisson process on Commodity i . 57 We use the terminology spot jump amplitudes for the parameters γ because it can be seen 58 ,mi 59 (Crosby (2005)) that γ ,mi is the log of the jump amplitude of the futures price for a futures contract 60 with = tT (ie a futures contract for immediate delivery), ie for each m , γ ,mi is the (log of the) spot

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1 2 3 jump amplitude for Commodity . We assume that the spot jump amplitudes are one of two 4 i γ ,mi 5 possible forms, which we term those of assumption 2.1 and assumption 2.2, which in turn are linked to 6 two possible specifications of the jump decay coefficient functions ,mi (tb ). 7 8 For each m , = ,...,1 Mm , we assume that either: 9 Assumption 2.1 :

10 The spot jump amplitudes γ ,mi , for each i , are assumed to be constants, which we denote by β ,mi . In 11 12 this case, the jump decay coefficient functions ,mi (tb ) are assumed to be any non-negative 13 deterministic function. • 14 Or: 15 Assumption 2.2 : 16 In this case, Forthe jump decay Peer coefficient functionsReview (tb ), for each Only i , i = 2,1 , are assumed to be 17 ,mi 18 identically equal to zero ie ,mi ()tb ≡ 0 for all t and for each i . The spot jump amplitudes γ ,mi are 19 assumed to be normally distributed random variables, with (under the EMM) mean β and standard 20 ,mi

21 deviation υ ,mi , for each i , each of which is independent of each of the Brownian motions and of each 22 23 of the Poisson processes. For different m , the spot jump amplitudes γ ,mi are assumed to be 24 independent. However, for a given m , we assume that, for this m , the correlation between the spot 25 J J jump amplitudes γ and γ is ρ . In other words, we assume correl , ≡ ργγ if 26 ,1 m ,2 m ,12 m ( ,2,1 nm ) ,12 m

27 = nm and correl(γ γ ,2,1 nm ) ≡ 0, otherwise. We assume that, for each i and for each m , β ,mi , 28 J 29 υ ,mi and ρ ,12 m are all constants. • 30 31 Remark 2.3 : Note that, in assumptions 2.1 and 2.2, β need not equal β and also that one of 32 ,1 m ,2 m

33 β ,1 m or β ,2 m may be zero (and for assumption 2.2, likewise υ ,1 m and υ ,2 m ). This allows us to 34 35 capture the effect where, in response to a jump in N mt at time t , the spot price 1 (),ttH of 36 Commodity 1 and the spot price 2 (),ttH of Commodity 2 may jump by different magnitudes (and 37 one may not actually jump at all). 38

39 40 We define the indicator functions, for each m , = ,...,1 Mm , m()1.2 = 11 if assumption 2.1 is 41 satisfied, for this m , and m()1.2 = 01 otherwise and m()2.2 = 11 if assumption 2.2 is satisfied, for this 42 43 m , and m()2.2 = 01 otherwise. Then equation 2.1 and assumptions 2.1 and 2.2 imply that 44 T 45 M M ⎛ ⎛ ⎛ ⎛ ⎞⎞ ⎞⎞ ()Tte = ⎜1, λ ()t ⎜exp⎜ β exp⎜− ()duub ⎟⎟ −1⎟⎟ 46 ∑ ,mi ∑⎜ ()1.2 mm ⎜ ⎜ ,mi ⎜ ∫ ,mi ⎟⎟ ⎟⎟ 47 m=1 m=1 ⎝ ⎝ ⎝ ⎝ t ⎠⎠ ⎠⎠ 48 M ⎛ ⎞ 49 ⎛ ⎛ 1 2 ⎞ ⎞ + ⎜1 ()2.2 mm ()t ⎜exp⎜ ,mi + υβλ ,mi ⎟ −1⎟⎟ , (where we have used ,mi (tb ) ≡ 0 if m()2.2 = 11 ). 50 ∑⎜ ⎜ ⎟⎟ m=1 ⎝ ⎝ ⎝ 2 ⎠ ⎠⎠ 51 52 Crosby (2005) provides more information about the consequences of the assumptions above and of 53 equation 2.1. In short, the consequences are that futures commo dity prices are martingales in the EMM 54 and (with a suitable (see Crosby (2005)) form for σ ( ,Tt )) log of the spot prices of both 55 ,kHi 56 Commodity 1 and Commodity 2 exhibit mean reversion in the EMM. It is also shown how, when the 57 jumps are of the type of assumption 2.1, jumps can also contribute to the effect of mean reversion and 58 that the speed of this jump-related mean reversion is given by the values of the jump decay coefficient 59 functions. When there are jum ps, in the case of assumption 2.1 (and provided the relevant jump decay 60 coefficient functions ,mi ()tb are strictly positive), the prices of long-dated futures contracts jump by smaller magnitudes than short-dated futures contracts because of the exponential dampening effect of

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11 σ ,kHi (),Tt , the intensity rates λm ()t and the jum p decay coe fficient functions ,mi ()tb . The specific 12 13 functional forms and the values of K1 , K 2 and M would be chosen by the trader according to her 14 intuition of the behaviour of the two underlying commodities. To help with this process, we will briefly 15 consider possible specifications of the dynamics of the futures prices of the two commodities. 16 For Peer Review Only 17 2.1 A possible specification for the jumps and the diffusion volatilities 18 19 Suppose that Commodity 1 is WTI grade crude oil. This is the “primary” commodity. It is very 20 actively traded and there are many standard European options traded on it whose prices the trader can 21 observe in the market. Suppose Commodity 2 is heating oil, a refined petroleum product. This is the 22 “daughter” commodity. It is not so actively traded but there are some (but a smaller number than for 23 WTI grade crude oil) standard European optio ns trad ed on it whose prices she can observe in the

24 market. We suppose K1 = 2 , K 2 = 3 and M = 1. Furthermore, we suppose the two Brownian 25 motions driving Commodity 1 are also precisely the first two Brownian motions driving Commodity 2, 26 with the same volatility parameters. The third Brownian motion driving Commodity 2 is specific to that 27 commodity. More specifically, we assume 28 29 σ (), = σ ( ,TtTt ) η += χ exp(− ( − tTa )) (2.3) 30 H 1,1 H 1,2 11 1 31 σ H 2,1 (), = σ H 2,2 ( ,TtTt ) = χ 2 exp(− 2 ( − tTa )) (2.4) 32 33 and we assume the diffusion volatility function for the third Brownian motion (driving only 34 Commodity 2) is of the form 35

36 σ H 3,2 (),Tt χ 3 exp()3 (−−= tTa ), (2.5) 37 38 where η , χ , χ , χ , a , a and a are all constants. 39 1 1 2 3 1 2 3 40 We will drop the first subscripted index for the Brownian motions in this subse ction onl y (ie write 41 42 ,1 kH ()= ,2 kH ()≡ ,kH ()tdztdztdz , for k = 2,1 and dz H 2,k (t) ≡ ,kH (tdz ), for k = 3). 43 We define the correlations (assumed constant), for i = 3,2,1 and j = 3,2,1 : 44 45 ρ ≡ , tdztdzcorrel , ρ ≡ , tdztdzcorrel . 46 , ji ( ,iH ( )(, jH )) ,iP ( P ( ) ,iH ( )) 47 48 We assume that Commodity 1 and Commodity 2 both jump in response to increments in the Poisson 49 process N1t which we assume to be of the type of assumption 2.1 and to have a constant intensity rate 50 ie λ ()t ≡ λ , where λ is a constant. The jump decay coefficient functions are identical for each 51 1 1 1 52 commodity and assumed constant ie 1,1 ( ) ≡ 1,2 (tbtb ) ≡ b1 , where b1 is a constant. Howe ver, we 53 assume the spot jump ampl itudes are possibly different ie is not n ecessarily equal to . 54 β 1,1 β 1,2 55 56 Then we can write the dynamics of Commodity 1 and Commodity 2 (under the EMM) as: 57 58 (),TtdH 1 ()χη exp()()−−+= (tdztTa )(χ exp()−−+ )()tdztTa 59 ()−,TtH 11 1 H 1, 2 2 H 2, 60 1 − σ P ( , ) P (tdzTt ) + ( (β 1,1 expexp (− 1 ( − )))−1) t − 1,11 ( , )dtTtedNtTb , (2.6)

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1 2 3 4 2 (),TtdH ()χη 11 exp()1 ()−−+= H 1, (tdztTa )(χ 2 exp()2 −−+ )H 2, ()tdztTa 5 2 ()−,TtH 6 χ exp( ()−−+ ) (tdztTa ) 7 3 3 H 3,

8 − σ P ()(), P tdzTt + ( (β 1,2 expexp (− 1 ( − )))−1) t − 1,21 ( , )dtTtedNtTb . (2.7) 9 10 ,TtH ,ttH 11 2 () 2 ( ) If we define 1/2 (),TtR ≡ and 1/2 ()tC ≡ , then by Ito’s lemma, 12 1 (),TtH 1 (),ttH 13 14 1/2 (),TtdR 15 χ 3 exp()()()(3 ()−−= H 3, + 3 PP TttdztTa χσρ 3 exp, ()3 −− )dttTa 16 1/2 ()−,TtR For Peer Review Only 17 χ exp()()−−− tTa (ρ ()η + χ exp(− ( − tTa )) + ρ (χ exp(− ( − ))))dttTa 18 3 3 111,3 1 22,3 2

19 20 + (exp((β β 1,11,2 )exp()1 ()−−− )−1) t − ( 1,21 ( )− 1,1 ( ,, ))dtTteTtedNtTb . (2.8) 21 22 Note the form of the diffusion volatility term which only depends on the Brownian increments 23 ()tdz . In fact, utilising results in section 3 of Crosby (2005), it is now clear, from equation 2.8, that 24 H 3, 25 the SDE for 1/2 ()tC can be written either in the form 26 27 ()()ln 1/2 () = 3 ()Λ J − ()ln 1/2 ( ) + χ H 3,3 (tdzdttCtatCd )+ (β − β )dN11,11,2 t , (2.9) 28

29 or, equivalently and alternatively, in the form 30 31 ()()ln () ()−Λ= ()ln ( ) + χ (tdzdttCtbtCd )+ (β − β )dN , (2.10) 32 1/2 1 D 1/2 H 3,3 11,11,2 t 33 34 where Λ J ()t and Λ D ()t are stochastic mean reversion levels whose exact forms can easily be 35 obtained utilising the methodology leading to proposition 3.4 of Crosby (2005), albeit at the expense of 36 some algebra (in fact, Λ ()t is a pure-jump stochastic process and Λ (t) is a pure-diffusion 37 J D 38 stochastic process). 39 We see that the log ratio ln 1/2 ()tC of the spot prices of the two commodities is a mean reverting 40 stochastic process (under the EMM). This is an attractive feature for modelling, for example, the case 41 where Commodity 1 is crude oil (the “primary” commodity) and Commodity 2 is a refined petroleum 42 product (the “daughter” commodity) such as heating oil, because, heuristically, we would expect the 43 price differential (and therefore also the log ratio) in the long-term to not move too far away from a 44 long-run mean level which reflects the cost of the refining process. However, in the short-term, the log 45 price ratio (and therefore also the arithmetic price difference) can go negative in line with the stylised 46 empirical observations made in section 1. 47 We will also briefly mention how this model might be calibrated. Usually, there will be fewer 48 actively traded options on the “daughter” commodity than on the “primary” commodity. One could 49 estimate the parameters of the process for the “primary” commodity by calibrating to the market prices 50 of standard options. In our example above, there would be eleven parameters, namely η , χ , χ , 51 1 1 2 52 a1 , a2 , ρ 2,1 , ρ P 1, , ρ P 2, , λ1 , b1 and β 1,1 . Having determined these eleven parameters, one could 53 take these as given. Then one could estimate the remaining six parameters, namely χ , a , ρ , 54 3 3 1,3

55 ρ 2,3 , ρ P 3, , β 1,2 , from the market prices of standard options on the “daughter” commodity. There 56 would typically, be fewer actively traded options on the “daughter” commodity but, equally, there are 57 fewer parameters to estimate. Of course, it would require an empirical investigation, beyond the scope 58 of this paper, to determine how feasible our suggested calibration mechanism might be. 59 In order to give some intuition about the correlation between the futures prices of Commodity 1 and 60 Commodity 2, we compute the model implied correlation between log of the futures prices of the two

commodities for different tenors S and T , ie ( ( 1 ( ))(( 2 ,0ln,,0ln THSHcorrel ))), given the

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1 2 3 model specification in equations 2.6 and 2.7. The results are in figure 1. For both S and T (plotted on 4 the x and y axes), we used the values 0, 0.5, 1, 1.5, 2, 2.5, 3 (all tenors are in years). Our parameter 5 6 values are exactly as in examples 1 and 2 (see section 5). Note that when = TS , the correlations are 7 lowest for the shortest tenor (ie for = TS = 0 ) but tend to one for the longest tenor ( S T == 3). 8 This behaviour seems2 quite intuitive for the case where Commodity 1 is crude oil and Commodity 2 is 9 a refined petroleum product such as heating oil. One would expect the prices of longer-dated futures 10 contracts on crude oil and heating oil to have a higher correlation since the difference between the (log 11 of the) prices would reflect the (average) cost of refining (which one would expect to vary only a little). 12 By contrast, one would expect a lower correlation between the (log of the) prices of shorter-dated 13 futures contracts on crude oil and heating oil because the price movements would reflect additional 14 short-term issues such as supply and demand, inventory and weather conditions. 15 In our example above, we have considered the dynamics of two commodities (“primary” and 16 “daughter”) whereFor intuition Peer suggests they willReview move closely (but not Only perfectly) together. Of course, in 17 the case of two seemingly unconnected commodities such as, for example, natural gas and a base 18 metal, a different specification of the jumps and the diffusion volatilities would be chosen. For

19 example, we might consider two Poisson processes, with the first Poisson process N1t only causing 20 jumps in Commodity 1 (by having and ) and the second Poisson process only 21 β 1,1 ≠ 0 β 1,2 = 0 N 2t 22 causing jumps in Commodity 2 (by having β 2,1 = 0 and β 2,2 ≠ 0 ). We would also specify the 23 24 diffusion terms differently. The example above is just meant for illustration. 25 We have illustrated how the model could be applied in a specific case of interest but, for this rest of this paper, we now return to considering the general case, as we turn our attention to pricing a class of 26 exotic commodity options. 27

42 43 ⎛ ⎛ , − * ,TTHKTTH ε ⎞ ⎞ ⎜ ()1,21,11 [] (2,22,12 )⎟ 44 max η⎜ ⎟ 0, , at time T , (3.1) ⎜ ⎜ ,TTH α ⎟ ⎟ pay 45 ⎝ ⎝ []()2,22,12 ⎠ ⎠ 46 47 where η = 1 if the option is a call and η = −1 if the option is a put. Note ε and α are constants 48 * 49 and, furthermore, K is a constant which might, for example, account for different units of 50 measurement. The reason for investigating options with this class of payoffs is that it contains as 51 special cases a number of option types of interest, all of which are actively traded in the OTC 52 commodity options markets. 53 54 We will now briefly outline (for the case of call options) some of these special cases: 55 56 57 58 2 Note that the parameter λ is the intensity rate in the risk-neutral EMM. Clearly, this parameter could be 59 1 different from the intensity rate in the real-world physical measure (in addition, had we also considered jumps of 60 the type of assumption 2.2, the mean jump amplitudes may also be different in the two different measures). Hence, whilst the model implied (risk-neutral) correlations shown in figure 1 can provide intuition for traders, they are not directly comparable to historical correlations (implicitly evaluated in the real-world physical measure).

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20 of different tenors ie ≠ TT 2,21,2 . Their payoffs can be defined via equation 3.1 with 21 22 1 ()HH 2 (,, ••≡•• ). Typically, we have = TT 2,11,1 , ε = 1 and, either α = 0 or α = 1. • 23 24 Forward start options 25 These are options on a single underlying commodity in which T is strictly less than T . The 26 2,1 1,1 27 payoff is the greater of zero and the difference between the futures commodity price to a given tenor at 28 some calendar time and the futures commodity price to the same tenor at some earlier calendar time. 29 Their payoffs can be defined via equation 3.1 with 1 (• •) ≡ HH 2 (•,, •), ε = 1 and α = 0 . For the 30 case just described, one would have = TT but other variants are possible. For example, forward 31 2,21,2 32 start options on the spot commodity price would have = TT 1,21,1 and = TT 2,22,1 . • 33 34 Ratio forward start options 35 Note that these options might also be called single-leg cliquets by analogy with terminology in the 36 equity options markets. These are also options on a single underlying commodity in which T is 37 2,1 38 strictly less than T 1,1 . The payoff is the greater of zero and the ratio of the futures commodity price to a 39 given tenor at some calendar time and the futures commodity price to the same tenor at some earlier 40 calendar time (minus a constant strike term). Their payoffs can be defined via equation 3.1 with 41 42 1 ()HH 2 (),, ••≡•• , ε = 1 and α = 1. Again, one could also have ratio forward start options on the 43 spot commodity price with = TT 1,21,1 and = TT 2,22,1 . • 44 45 Of course, we can also price options which are generalisations or mixtures of the special cases noted 46 above. For example, and need not be integers. 47 ε α 48 We should also make a brief comment about the time Tpay at which the option payoff is paid. The 49 most common situation, in practice, is that T would be set equal to T . However, occasionally, we 50 pay 1,1 51 observe in the OTC markets that commodity options are traded where the payoff is deferred for a short

52 period of time after T 1,1 (and this is not just the standard two working day spot settlement but might, 53 54 for example, be a period of a few weeks). For example, it might be that Tpay is set equal to the 55 maturity of one of the underlying futures contracts. 56 We will now return, for the rest of the paper, to the completely general case of considering the class 57 of exotic options whose payoff is given by equation 3.1. 58 59 60

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1 2 3 4. Fourier Transform methodology 4

5 In this section, we will use a Fourier transform methodology, to price European-style options whose 6 payoff is defined in equation 3.1. We will proceed along the lines of Sepp (2003) who considers the 7 case of standard European options (on a single underlying asset). 8 9 Define, for times 1 ≥ tt and 2 ≥ tt : 10 11 ⎡⎛ (),TtH ⎞ ⎛ ( ,TtH ) ⎞⎤ 12 ⎜ 1,211 ⎟ ⎜ 1,21 ⎟ ()2,221,21 tTtTtY ≡ log;,,, ⎢ ε ε ⎥ . (4.1) 13 ⎜ ,TtH ⎟ ⎜ ,TtH ⎟ ⎣⎢⎝ []()2,222 ⎠ ⎝ []()2,22 ⎠⎦⎥ 14 15 The price of the European-style option, whose payoff is given by equation 3.1, at time t , (for 16 For Peer Review Only ) is: 17 ≤≤ TTt 1,12,1 18 19 ⎡ ⎛ Tpay ⎞ ⎛ ⎛ (), − * [ ( ,TTHKTTH )]ε ⎞ ⎞⎤ 20 ⎜ ⎟ ⎜ ⎜ 1,21,11 2,22,12 ⎟ ⎟ Et ⎢exp − ()dssr max η 0, ⎥ 21 ⎢ ⎜ ∫ ⎟ ⎜ ⎜ ,TTH α ⎟ ⎟⎥ ⎝ t ⎠ ⎝ ⎝ []()2,22,12 ⎠ ⎠ 22 ⎣ ⎦ 23 1 ()2 ()−+= 3 ()tMtMtM , (4.2) 24 25 T ⎡ ⎛ pay ⎞ ⎤ ()1+η −α 26 where ()tM ≡ E ⎢exp⎜− () ⎟ ()(), [,TTHTTHdssr ]⎥ (4.3) 27 1 t ⎜ ∫ ⎟ 2,22,121,21,11 2 ⎢ t ⎥ 28 ⎣ ⎝ ⎠ ⎦ T 29 1−η ⎡ ⎛ pay ⎞ ⎤ 30 () ⎜ ⎟ * −αε and 2 ()≡ EtM t ⎢exp − () [(),TTHKdssr 2,22,12 ]⎥ (4.4) 31 2 ⎜ ∫ ⎟ ⎢ ⎝ t ⎠ ⎥ 32 ⎣ ⎦ 33 ⎡ ⎛ Tpay ⎞ 34 and () EtM ⎢exp⎜−≡ ()dssr ⎟ 3 t ⎜ ∫ ⎟ 35 ⎣⎢ ⎝ t ⎠ 36 37 ⎛ ,TtH ⎞⎤ −αε ⎜ ( 1,21 ) * ⎟ 38 [](),TTH 2,22,12 min ((2,22,11,21,1 )),;,,,exp KtTTTTY ⎥ . (4.5) ⎜ ,TtH ε ⎟ 39 ⎝ []()2,22 ⎠⎦⎥ 40 41 The last set of equations follows from a simple algebraic arrangement. 42 We focus, firstly, on 3 ()tM . 43 44 ⎛ ,TtH ⎞ 45 ⎜ ( 1,21 ) * ⎟ Define ()()2,22,11,21,1 tTTTTYf ≡ min;,,, ()()2,22,11,21,1 ,;,,,exp KtTTTTY 46 ⎜ ,TtH ε ⎟ 47 ⎝ []()2,22 ⎠ 48 and then write 49 50 ˆ ( ( 2,22,11,21,1 ;,,, tTTTTYf )) in (inverse) terms of its Fourier Transform (zf ) ie write 51

52 izi ∞+ 53 1 ()();,,, tTTTTYf = exp(− (;,,, ))ˆ()dzzftTTTTizY , (4.6) 54 2,22,11,21,1 2π ∫ 2,22,11,21,1 55 izi ∞− 56 where z is complex. Results in Lewis (2001) and Sepp (2003) show that, by taking the Fourier 57 58 Transform of ( ( 2,22,11,21,1 ;,,, tTTTTYf )), which exists provided < zi < 10 , where zi is the 59 imaginary part of z , then: 60

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1 2 3 iz+1 ⎛ * ε ⎞ 4 (),TtH 1,21 ⎛ 1 ⎞ [ ( ,TtHK 2,22 )] 5 ˆ()zf = ⎜ ⎟⎜ ⎟ . (4.7) ε 2 − izz ⎜ ,TtH ⎟ 6 [](),TtH 2,22 ⎝ ⎠⎝ ()1,21 ⎠ 7 8 Furthermore, by substituting equation 4.6 into equation 4.5, 3 (tM ) is given by: 9 10 T ⎡ ⎛ pay ⎞ izi ∞+ ⎤ 11 ⎜ ⎟ −αε 1 ˆ 12 Et ⎢exp − () [](),TTHdssr 2,22,12 exp()− ()2,22,11,21,1 ;,,, ()dzzftTTTTizY ⎥ ⎢ ⎜ ∫ ⎟ 2π ∫ ⎥ 13 ⎣ ⎝ t ⎠ izi ∞− ⎦ 14 T izi ∞+ ⎡ ⎛ pay ⎞ ⎤ 15 1 ⎜ ⎟ −αε ˆ = Et ⎢exp − () [](),TTHdssr 2,22,12 exp()− ()2,22,11,21,1 ;,,, ⎥ ()dzzftTTTTizY 16 2π ∫∫For⎢ ⎜ Peer⎟ Review Only ⎥ 17 izi ∞− ⎣ ⎝ t ⎠ ⎦ 18 1 izi ∞+ 19 (−Φ≡ ,,,,; )ˆ()dzzfTTTTtz , (4.8) 2π ∫ 2,22,11,21,1 20 izi ∞− 21 22 where (−Φ ,,,,; TTTTtz 2,22,11,21,1 )≡ 23 24 ⎡ ⎛ Tpay ⎞ ⎤ E ⎢exp⎜− () ⎟[](),TTHdssr −αε exp()− (;,,, tTTTTizY )⎥ , (4.9) 25 t ⎜ ∫ ⎟ 2,22,12 2,22,11,21,1 26 ⎣⎢ ⎝ t ⎠ ⎦⎥ 27 28 and where we use Fubini’s theorem to justify the interchange of the integral and the expectation 29 operator. We will call (−Φ ,,,,; TTTTtz 2,22,11,21,1 ) the “extended” characteristic function (we have 30 borrowed the terminology from Duffie et al. (2000) but our definition is somewhat different). 31

32 We now collect the equations above into the form of a proposition. 33

34 Proposition 4.1 : The price of the European-style option, at time t , whose payoff is defined in 35 equation 3.1, is: 36 37 1 izi ∞+ 38 () ()tMtM (−Φ−+ ,,,,; )ˆ()dzzfTTTTtz . (4.10) 39 1 2 2π ∫ 2,22,11,21,1 40 izi ∞−

41 Proof : From equations 4.2 and 4.8. • 42

43 Remark 4.2 : Note that equation 4.10 holds independently of the specific model for futures commodity 44 prices. So, for example, we could consider extensions of the Crosby (2005) model which allow for, for 45 example, stochastic volatility or alternative specifications of the jump processes (in the manner of 46 Heston (1993), Duffie et al. (2000) and Barndorff-Nielsen and Shephard (2001)) and equation 4.10 47 would still be applicable, provided that the “extended” characteristic function can be calculated. 48 Equation 4.10 (with minor modifications) could also be useful for options involving other asset classes 49 such as equities (see Duffie et al. (2000)) or inflation (see Mercurio (2005), where it is shown that the 50 valuation of derivatives on year-on-year inflation involves calculations very similar to valuing ratio 51 forward start (cliquet) options). However, for the sake of brevity, we will not pursue this point further 52 in this paper. 53 54 In the appendix, we write down the “extended” characteristic function when the dynamics of futures 55 commodity prices are given by equation 2.1. From the form of the “extended” characteristic function, 56 we can also easily obtain explicit forms for tM and tM (see the appendix). 57 1 ( ) 2 ( ) 58 We can now calculate the option price, via equation 4.10 provided the integral is well-defined, 59 which requires 0 zi << 1. One choice (as in Lewis (2001) and Sepp (2003)) is to evaluate the 60 integral along the straight line given by z = + iu 2 , w here u is real.

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1 2 3 With this choice, the price of the European-style option at time t , whose payoff is defined in 4 equation 3.1, is: 5 6 1 ∞ 7 () ()tMtM ()−−Φ−+ ,,,,;2 ˆ()+ 2 duiufTTTTtiu , (4.11) 8 1 2 ∫ 2,22,11,21,1 π 0 9 10 where we have also changed the lower limit of the integration from − ∞ to zero by using the fact that 11 the option price is real and hence the integrand is odd in its imaginary pa rt and even in its real part. We 12 will not write down the option price formula in its most explicit form as it is rather long and would not 13 greatly enhance intuition. Equations for and are in the appendix and ˆ can 14 1 (tM ) 2 (tM ) ()+ iuf 2 15 be obtained from equation 4.7. 16 If the “extended”For character Peeristic function Review were to be completely Only analytic, then it would be 17 straightforward to evaluate the integral in equation 4.11. In particular, we can compute option prices 18 using a single one-dimensional integration irregardless of how many Brownian motions and Poisson 19 Processes drive the futures commodity prices. If all the Poisson processes satisfy assumption 2.2, and 20 provided that λm ()dss is easily evaluated (and, of course, in practice, one would choose a form for 21 ∫ 22 the intensity rates λ s so that λ dss can be evaluated analytically), then this would be the m ( ) ∫ m ( ) 23 case in our model. Unfo rtunately, if any of the Poisson processes satisfy assumption 2.1, then our 24 “extended” characteristic function involves integrals (see the second, third, fourth and fifth lines of 25 equation A.2 in the appendix) which means that evaluating equation 4.11 involves at least a double 26 integral. This is certainly computationally feasible but equally performing a double integral will be 27 considerably slower than a single integral. Crosby (2006) shows how calculation times can be speeded 28 up, when pricing standard (plain vanilla) European options, by using power series expansions of terms 29 appearing in the characteristic function. A similar idea can be used here provided we make some 30 simplifying assumptions. 31 32 A s in Crosby (2006), we make the following assumption: 33 Assumption 4.3 : We will henceforth assume that, for each , , and, for 34 m = ,...,1 Mm λm ()s ≡ λm

35 each i , ,mi ()≡ btb ,mi are constants. Furthermore, we assume t hat, if for this m , the jumps satisfy 36 37 assump tion 2.1, then b ,mi > 0 . (This condition is not restrictive since if b ,mi were to equal zero, we 38 could treat it as in the case of a ssumption 2.2 which is much simpler). • 39 40 This means that we can use the power ser ie s expansions of Crosby (2006) for the terms on the 41 second, third and fourth lines of the “extended” characteristic function (see equation A.2) (into which 42 we would substitute = + iuz 2 , where u is real). 43 In order to rapidly compute the followin g term (the fifth line) in equation A.2 (into which, again, we 44 45 would substitute += iuz 2 ): 46 47 ⎛ M T 2,1 ⎞ 48 ⎜ 1exp ()s exp (+− iz ) φβεαελ , − φβ , dsTsizTs ⎟ , ⎜∑ m()1.2 ∫ m ()mm ()2,2,2,2 mm ()1,2,1,1 ⎟ 49 ⎝ m=1 t ⎠ 50 51 (where φ ( ,Ts ) and φ ( ,Ts ) are defined as in equation A.1 in the appendix) we will make 52 m 1,2,1 m 2,2,2 53 the following additional ass umption: 54 55 Assumption 4.4 : We will henceforth assume that, for each m , b ,1 m and b ,2 m are identically equal ie 56 that ≡≡ bbb , say. • 57 m ,2,1 mm 58 59 Remark 4.5 : Crosby (2005) shows that futures commodity prices can be written in terms of a number 60 of Gaussian state variables and M Poisson jump state variables. It can therefore be shown that assumption 4.4 is equivalent to saying (for assumption 2.1) that the futures prices of Commodity 1 and Commodity 2 are driven by the same jump state variables. It is shown in Crosby (2005) that our model

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1 2 3 is consistent with mean reversion, under the EMM. Not only that, but it is also shown that, when the 4 jump processes are of the type of assumption 2.1, then jumps can also contribute to the effect of mean 5 reversion and that the speed of this jump-related mean reversion is equal to the associated jump decay 6 coefficient function. Hence assumption 4.4 is also equivalent to assuming that, after a jump, there is a 7 common speed of jump-related mean reversion in Commodity 1 and Commodity 2. Although it would 8 be an empirical matter, beyond the scope of this paper, to fully justify assumption 4.4, this assumption 9 does, therefore, have some economic intuition. In addition, we note that assumption 4.4 is obviously a 10 non-assumption in the special case when the option is on a single underlying commodity (see section 3, 11 for example, options on futures commodity price curve spreads, forward start options and ratio forward 12 start options), since it must hold. 13 14 With assumption 4.4, we can make a similar type of power series expansion which we specify in the 15 next proposition. 16 For Peer Review Only 17 Proposition 4.6 : Define ψ ≡ exp(− ( − TTb )) and ψ ≡ exp(− ()− TTb ) , with 18 start m start end m end 19 start end ≤≤ TTT . Then: 20 21 Tend

22 m exp()()i ωωλ 21 exp()m ()λ (endm −=−−+ TTdssTb start ) 23 ∫ Tstart 24 n ⎛ ⎡ 2 2 n n ⎤⎞ 25 ∞ + −ψψωω ()cos()+ sin ()nin θθ λm ⎜ ⎢1 ( 1 2 ) ()end start ⎥⎟ 26 + ⎜∑ ⎟ , 27 m ⎜ n=1 ⎢nb n! ⎥⎟ 28 ⎝ ⎣ ⎦⎠ 29 (4.12)

30 where ω1 and ω2 are real numbers, independent of s , and where θ , is defined as follows: 31 Firstly, de fine θ , 0 θ ≤≤ π , via cos ≡ + ωωωθ 22 , then: 32 2 212 33 34 If ω1 ≥ 0 and ω2 ≥ 0 , then = θθ , else if ω1 ≥ 0 and ω2 < 0 , then −= θπθ , else if ω1 < 0 35 and ω ≥ 0 , then θ = 2π −θ , else if ω < 0 and ω < 0 , then θ = + θπ . 36 2 1 2 37 Proof: This proposition is just a generalisation of proposition 3.3 in Crosby (2006) and can be proved 38 3 39 in an identical fashion. Therefore, the proof is omitted . • 40 41 All the integrals (see the second, third, fourth and fifth lines of equation A.2) which appear in the 42 “extended” characteristic function can be nested in a form which enables them to be evaluated by 43 proposition 4.6, provided assumptions 4.3 and 4.4 hold. Hence, we can quickly and easily evaluate the 44 “extended” characteristic function. We can also evaluate 1 (tM ) and 2 (tM ) (see appendix) in the 45 same way. We can then very rapidly, using standard one dimensio nal num erical integration techniques, 46 compute the integral in equation 4.11 and hence also compute the price of the European-style option 47 whose payoff is defined in equation 3.1. 48 49 50 5. Numerical examples and results 51 52 In this section, we will provide four numerical examples, labelled examples 1, 2, 3 and 4, of our 53 methodology, the results of which are in tables 1, 2, 3 and 4 respectively. In all four examples, we 54 value European-style options, whose payoff is defined in equation 3.1, using equation 4.11. 55 We evaluate the integral with respect to u in equation 4.11 using Simpson’s rule with 1024 points. 56 Examining the forms of equations A.2 and 4.11, we see that for large u , the integrand behaves 57 58 59 3 It is straightforward to see that the power-series expansion in equation 4.12 will be rapidly convergent. Indeed 60 the modulus of the term appearing in the square brackets is guaranteed to be monotonically declining to zero when 2 2 n > ( ,2max 1 + ωω 2 ).

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1 2 3 4 ⎛ 1 ⎛ 2 1 ⎞ 2 ⎞ ⎛ 2 1 ⎞ asymptotically like exp⎜ ⎜ +− ⎟Σ (),,,, 2,21,22,11,1 ⎟ ⎜uTTTTtu + ⎟ which clearly tends 5 ⎝ 2 4 ⎠⎝ ⎠ 4 ⎠⎝ 6 2 7 to zero rapidly as u ∞→ , since Σ ( TTTTt 2,21,22,11,1 ) ≥ 0,,,, . We truncate the upper limit of the 8 ⎛ 1 1 ⎞ 1 9 ⎛ 2 ⎞ 2 ⎛ 2 ⎞ integral when the value of u is such exp⎜ ⎜ +− ⎟Σ (),Ttu ,,, 2,21,22,11,1 ⎟ ⎜uTTT + ⎟ is 10 ⎝ 2 ⎝ 4 ⎠ ⎠ 4 ⎠⎝ 11 less than 10-8. We truncate the infinite series in equation 4.12 when the value of an additional term in 12 the series has converged to less than 10-12. 13 In all four examples, we assume that the initial (ie as of the valuation date of the options that we will 14 value in our examples) futures prices of Commodity 1 to all maturities are 40 and the initial futures 15 prices of Commodity 2 to all maturities are 41. We assume that the initial interest-rate yield curve is 16 flat with a continuouslyFor compoundedPeer risk-free Review rate of 0.044 ie we assume Only that the discount factors for 17 all maturities T , as of the valuation date, time t , of the options that we will value, are all of the form 18 19 ()044.0exp ()−− tT . Interest-rates are stochastic and evolve following a one factor Hull-White 20 (extended Vasicek) model in which

21 σ P ()Tt σ r (−≡ e1, xp()α r ()−− tT )α r , where σ r = 012.0 and α r = 125.0 . 22 In all four examples, we use the same form for the diffusion parameters as in equations 2.3 to 2.5 in 23 section 2.1. That is, we suppose K = 2 and K = 3 and, furthermore, we suppose 24 1 2 25 η1 = 12.0 , χ1 = 22.0 , χ 2 = 25.0 , χ 3 = 242.0 , a1 = 9.0 , a2 = 7.0 , a3 = 5.1 . 26 We assume all correlations are 0.05 ie for all i = 3,2,1 and j = 3,2,1 : 27 28 ρ , ji = 05.0 and ρ ,iP = 05.0 . Note that all these parameters are just for illu stration . 29 30 In all four exam ples, we assume that the maturities of the futures contracts on Commodity 1 are of 31 the form = TT + ()36531 and on Commodity 2 of the form = TT + ()36591 ie the 32 1,11,2 2,12,2 33 futures contracts on Commodity 1 and Commodity 2 mature 31 days and 91 days respectivel y after 34 T 1,1 and T 2,1 . In all four examples, we set pay ≡ TT 1,1 . We specify T 1,1 and T 2,1 in the examples. 35 In each example, we value six options and all of them are calls (ie η = 1). For the first three 36 * 37 options, α = 0 , ε = 1 and the values of K are 0.95, 0.975 and 1. The fourth, f ifth and sixth options 38 have α = 1 and, again, ε = 1 and the values of K * are 0.95, 0.975 and 1. Thus, we evaluate spread 39 * * 40 options for t hree different values of K and ratio spread options for the same three values of K in 41 each examp le. 42 43 Example 1 : 44 In example 1, we assume T 1,1 = 1 and T 2,1 = 1. We assume that there is one Poisson process, 45 M = 1 , and it satisfies assumption 2. 1 and i t has an intensity rate λ = 512.0 . As in the example in 46 1 47 section 2.1, both Commodity 1 and Co mmodi ty 2 exhi bit jumps of non-zero magnitude in response to 48 jumps in this Poisson process. We assume = bb 1,21,1 = 55.1 , β 1,1 = 55.0 , β 1,2 = 35.0 . We price 49 the six different options and the results are in table 1. • 50 51 Example 2 : 52 In example 2, we assume T = 3 and T = 2 . This means that the maturities of the futures 53 1,1 2,1 54 contracts on Commodity 1 and Commodity 2 are ap proximately 3.08493151 and 2.24931507 years

55 respectively. Note that because T 2,1 is strictly less than T 1,1 , the options in this example can also be 56 57 viewed as hybrid forward start options (for the first three optio ns where α = 0 ) and ratio forward start 58 options (for the fourth, fifth and sixth options where α = 1) involving two different commodities. We 59 assume that all the jump parameters are exactly the same as in example 1. We price the six different 60 options and the results are in table 2. •

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1 2 3 Example 3 : 4 In example 3, we assume T 1 and T 1. We use exactly the same diffusion parameters as in 5 1 1, = ,1 2 = 6 examples 1 and 2 but to provide a contrast with those examples, we assume that there are two Poisson 7 process, M = 2 , and they both satisfy assumption 2.1 and t hey have intensity rates λ1 = 512.0 and 8 9 λ2 = 47.0 respectively. Commodity 1 jumps but Commodity 2 does not jum p in responses to jumps 10 in this first Poisson process N1t . Conversely, Commodity 2 jumps but Commodity 1 does not jump in 11 12 responses to jumps in this second Poisson process N 2t . We assume = bb 1,21,1 = 55.1 , 13 b b == 55.1 , β = 55.0 , β = 0 , β = 0 , β = 35.0 . We price the six different 14 2,22,1 1,1 1,2 2,1 2,2 15 options and the results are in table 3. • 16 For Peer Review Only 17 Example 4 : 18 In example 4, we assume T 1,1 = 3 and T 2,1 = 2 . We assume that there are two Poisson processes 19 again and that all the jump parameters are exactly the same as in example 3. We assume that all the 20 diffusion parameters are exactly the same as in examples 1, 2 and 3. Note that, as in example 2, 21 because T is strictly less than T , the options in this example can also be viewed as hybrid forward 22 2,1 1,1 23 start and ratio forward start options. We price the six different options and the results are i n table 4. • 24 25 Computations were performed on a desk-top p.c., running at 2.8 GHz, with Microsoft Windows XP 26 Professional, with 1 Gb of RAM with a program written in Microsoft C++. The total calculation time 27 for all 24 options in examples 1 to 4 was 0.532 seconds or an average of less than 23 milliseconds per 28 option. By significantly increasing th e number of points in the numerical integration and by 29 significantly reducing the tolerances used to truncat e the upper limit of the integral (in equation 4.11) 30 and the power series expansions (as in equation 4.12), we were able to confirm that in proportional (ie 31 proportional to the calculated option prices) terms, all the option prices in tables 1 to 4 are accurate to 32 at least one part in 500,000 and, also, that in absolute terms, all the option prices are accurate to at least 33 5 decimal places. So our algorithm is both fast and accurate. 34 Note how the option prices in examples 3 and 4 are higher than the corresponding option prices i n 35 examples 1 and 2 respectively. This is intuitive given the different specifications of the jump processes 36 driving futures commodity prices, between, on the one hand, examples 1 and 2, and, on the other hand, 37 examples 3 and 4, and given the arguments we presented after equation 2.10.

39 40 6. Conclusions 41 42 We have extended the Crosby (2005) model to simultaneously model the prices of multiple 43 commodities. We then priced a class of simple exotic options which includes those whose payoffs 44 inv olve two different underlying commodities, or a single underlying commodity but with futures 45 contracts of two different tenors or the price of a single underlying futures contract observed at two 46 different calendar times. This class of exotic options includes common exotics such as (crack, dark or 47 spark) spread options, ratio spread options, forward start options and ratio forward start options (single 48 leg cliquets). We have shown that these exotic options can be priced using Fourier methods in any 49 model in which the relevant “extended” characteristic function is known analytically or can be 50 computed rapidly. The Crosby (2005) model falls into the latter category. We have provided some 51 numerical examples which demonstrate that our methodology is both fast and accurate. Finally, we will briefly mention two possible areas for future research: 52 (i) We have focussed, when pricing spread options in this paper, on the “zero strike” case. Dempster 53 and Hong (2000) show how “non-zero-strike” spread options can be priced using a two-dimensional 54 Fast Fourier Transform methodology combined with an ingenious decomposition of the option payoff 55 analogous to Riemann sums. Their approach (combined with assumptions 4.3 and 4.4 and the power 56 series expansion of proposition 4.6) could be used to price “non-zero-strike” spread options within the 57 framework of this paper. It might also be possible to extend the Dempster and Hong (2000) approach in 58 order to price more exotic variations of some of the option types we discussed in section 3. 59 (ii) In section 2.1, we provided an example of specifying the dynamics of the futures prices of two 60 different commodities based on heuristics and trader-intuition. It might be possible to construct a more

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1 2 3 systematic approach based on suitable extensions of the methodology described in section 3 of 4 Casassus and Collin-Dufresne (2005). However, we leave this for future research. 5 6 7 Appendix 8

9 10 In order to obtain the forms for 1 (tM ) and 2 (tM ), defined in equations 4.3 and 4.4, we can 11 essentially use the “extended” characteristic function (defined in equation 4.9 and given explicitly in 12 equation A.2 below), into which we substitute = iz and z = 0 respectively, then : 13 14 ⎡ ⎛ Tpay ⎞ ⎤ ()1+η −α 15 M ()t ≡ E ⎢exp⎜− () ⎟ ()(), [],TTHTTHdssr ⎥ 16 1 Fort Peer⎜ ∫ ⎟ Review 2,22,121,21,11 Only 2 ⎢ t ⎥ 17 ⎣ ⎝ ⎠ ⎦ 18 ()1+η ( ,TtH 1,21 ) = ()−Φ ,T ,,,; TTTti 19 2 ε 2,22,11,21,1 20 [](),TtH 2,22 21 22 and 23 24 ⎡ ⎛ Tpay ⎞ ⎤ ()1−η * −αε 25 ()≡ EtM ⎢exp⎜− () ⎟ [(),TTHKdssr ]⎥ 2 2 t ⎜ ∫ ⎟ 2,22,12 26 ⎣⎢ ⎝ t ⎠ ⎦⎥ 27 1−η 28 () = Φ(),,,,;0 TTTTt 2,22,11,21,1 . 29 2 30 31 We will now proceed to write down the “extended” characteristic function when the dynamics of 32 futures commodity prices are given by equation 2.1, after defining the following notation: 33 C C 34 (),TtdH Ki (),TtdH For each i , i = 2,1 , i ≡ σ , −σ , tdzTttdzTt , ie i 35 C ∑ ,kHi (),kHi ()()(P P ) C 36 i (),TtH k =1 i (),TtH 37 denotes the purely continuous martingale compo nent in the SDE for Commodity i . 38 39 ⎛ T ⎞ For each i , i = 2,1 , φ Ts exp, ⎜−≡ duub ⎟ . (A.1) 40 ,mi ()⎜ ∫ ,mi ()⎟ 41 ⎝ s ⎠ 42 43 T 2,1 C ⎛ ( ,TsdP pay ) ( ,TsdH 2,22 )⎞ 44 Define (),TtU ,T ≡ ()−− αε cov⎜ , ⎟ds 2,22,1 ∫ ⎜ ,TsP C ⎟ 45 t ⎝ ()pay (),TsH 2,22 ⎠

46 T 2,1 C 47 1 ⎛ ( ,TsdH 2,22 )⎞ ()()αεαε −−−− var1 ⎜ ⎟ds . 48 2 ∫ ⎜ C (),TsH ⎟ 49 t ⎝ 2,22 ⎠ 50 51 Define ( ,,,, TTTTtW 2,21,22,11,1 ) ≡

52 T 1,1 C T 2,1 C ⎛ (),TsdP pay (),TsdH 1,21 ⎞ ⎛ ( ,TsdP pay ) ( ,TsdH 2 2,2 )⎞ 53 cov⎜ , ⎟ds − ε cov⎜ , ⎟ds 54 ∫ ⎜ ,TsP C ⎟ ∫ ⎜ ,TsP C ⎟ t ⎝ ()pay (),TsH 1,21 ⎠ t ⎝ ()pay (),TsH 2,22 ⎠ 55 T 2,1 C C T 2,1 C 56 ⎛ (),TsdH 1,21 (),TsdH 2,22 ⎞ 1 ⎛ dH (),Ts 2,22 ⎞ − α cov⎜ , ⎟ds ()2 −++ εεαε 2 var⎜ ⎟ds . 57 ∫ ⎜ C ,TsH C ,TsH ⎟ 2 ∫ ⎜ C ,TsH ⎟ 58 t ⎝ ()1,21 ()2,22 ⎠ t ⎝ ()2,22 ⎠ 59 60 T 1,1 C ⎛ ( ,TsdH 1,21 )⎞ Define Σ 2 ,,,, TTTTt ≡ var⎜ ⎟ds ()2,21,22,11,1 ∫ ⎜ C ⎟ t ⎝ (),TsH 1,21 ⎠

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1 2 3 T 2,1 C C T 2,1 C 4 ⎛ (),TsdH 1,21 (),TsdH 2,22 ⎞ 2 ⎛ ( ,TsdH 2,22 )⎞ − ε cov2 ⎜ , ⎟ds + ε var⎜ ⎟ds . 5 ∫ ⎜ C (),TsH C (),TsH ⎟ ∫ ⎜ C (),TsH ⎟ 6 t ⎝ 1,21 2,22 ⎠ t ⎝ 2,22 ⎠ 7 8 In order to compute the “extended” characteristic function, we will use the fact that Brown ian 9 motions and Poisson processes have independent increments. Then by direct calculation: 10 T 11 ⎛ M 1,1 ⎞ −Φ z; ,,,, TTTTt = exp⎜− λ ()dss ⎟ 12 ( 2,22,11,21,1 ) ⎜ ∑ ∫ m ⎟ 13 ⎝ m=1 t ⎠ 14 ⎛ M T 1,1 ⎞ 15 ⎜ 1exp λ ()exp − φβ , dsTsizs ⎟ 16 ∑ m()For1.2 ∫ m Peer() mm Review()1,2,1,1 Only ⎜ m=1 ⎟ 17 ⎝ T 2,1 ⎠ 18 ⎛ M T 1,1 ⎞ 19 ⎜ ⎟ iz∑1exp m()1.2 m ()s ()exp()φβλ mm ()1,2,1,1 −1, dsTs 20 ⎜ ∫ ⎟ ⎝ m=1 t ⎠ 21 T 22 ⎛ M 2,1 ⎞ exp⎜ ()+−− izεαε 1 ()s exp φβλ s −1, dsT ⎟ 23 ⎜ ∑ m()1.2 ∫ m ()()mm ()2,2,2,2 ⎟ 24 ⎝ m=1 t ⎠ 25 T ⎛ M 2,1 ⎞ 26 ⎜ 1exp ()s exp() (+− iz ) φβεαελ (), − φβ (), dsTsizTs ⎟ 27 ⎜∑ m()1.2 ∫ m mm 2,2,2,2 mm 1,2,1,1 ⎟ m=1 t 28 ⎝ ⎠ T 29 ⎛ M ⎛ 1,1 ⎞ ⎛ 1 ⎞⎞ 30 ⎜ 1exp ⎜ λ ⎟exp izs −− υβ z 22 ⎟ ⎜∑ m()2.2 ∫ m () ⎜ ,1 m ,1 m ⎟⎟ 31 m=1 ⎜ ⎟ ⎝ 2 ⎠ ⎝ ⎝ T 2,1 ⎠ ⎠ 32 33 ⎛ M ⎛ T 1,1 ⎞ ⎞ ⎛ ⎛ 1 2 ⎞ ⎞ 34 ⎜iz 1exp ⎜ λ ()dss ⎟⎜exp⎜ + υβ ⎟ −1⎟⎟ ⎜ ∑ m()2.2 ⎜ ∫ m ⎟⎜ ,1 m 2 ,1 m ⎟⎟ 35 ⎝ m=1 ⎝ t ⎠⎝ ⎝ ⎠ ⎠⎠ 36 T 37 ⎛ M ⎛ 2,1 ⎞⎛ ⎛ 1 ⎞ ⎞⎞ exp⎜ ()+−− izεαε 1 ⎜ λ ()dss ⎟⎜exp⎜ + υβ 2 ⎟ −1⎟⎟ 38 ⎜ ∑ m()2.2 ⎜ ∫ m ⎟⎜ ,2 m 2 ,2 m ⎟⎟ 39 ⎝ m=1 ⎝ t ⎠⎝ ⎝ ⎠ ⎠⎠ 40 ⎛ M ⎛ T 2,1 ⎞ 41 ⎧ 1 2 ⎜ 1exp ⎜ λ ()dss ⎟exp (iz ) ()+−++− iz υεαεβεαε 2 42 ∑ m()2.2 ⎜ ∫ m ⎟ ⎨ ,2 m ,2 m ⎜ m=1 ⎩ 2 43 ⎝ ⎝ t ⎠ 44 1 22 J ⎫⎞ 45 iz ,1 m ,1 m z ()+−−−− iziz m υυρεαευβ ,2,1,12 mm ⎬⎟ 46 2 ⎭⎠ 47 ⎛ 1 22 ⎞ 48 exp⎜ ()Σ− ()()TTTTtziz 2,21,22,11,1 ⎟ ()− ,,,,exp,,,, TTTTtizW ,21,22,11,1 2 49 ⎝ 2 ⎠ −αε 50 ()TtP ()(− ()(),,,exp, TtHTTtU ). (A.2) 51 pay 2,22,1 2,2 52 53 54 55 56 57 58 59 60

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1 2 3 Figure 1 : 4 5 Graph of the model implied instantaneous correlation between 6 ( ( 1 ( ))(( 2 ,0ln,,0ln THSHcorrel ))) 7 the futures prices of Commodity 1 and Commodity 2, for different tenors S and T , given the model 8 specification in equations 2.6 and 2.7. For both S and T (plotted on the x and y axes), we used the 9 values 0, 0.5, 1, 1.5, 2, 2.5, 3 (all tenors are in years). Our parameter values are as in examples 1 and 2 10 (see section 5). 11 12 13 14 Model implied instantaneous correlations 15 16 For Peer Review Only 17 1.000 18 19 0.950 20 0.900 21 22 0.850 23 24 0.800 25 0.750 26 27 0.700 28 0.650 29 30 0.600 31 3 2.5 32 2 1.5 33 1 0.5 Futures tenor (in years) 34 0 35

36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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1 2 3 Table 1 : 4 5 There is one Poisson process. T = 1 , T = 1, T = + 365311 , T = + 365911 . 6 1,1 2,1 1,2 ( ) 2,2 () 7 The values of K * are across the first row and the option prices are in bold across the second row. 8 9 α = 0 , ε = 1 (spread options) α = 1, ε = 1 (ratio spread options) 10 0.95 0.975 1.0 0.95 0.975 1.0 11 2.64579 2.19204 1.80901 0.05799 0.04737 0.03852 12 13 ______14 15 Table 2 :

16 For Peer Review Only 17 There is one Poisson process. T 1,1 = 3 , T 2,1 = 2 , T 1,2 = + ( 365313 ) , T 2,2 = + ()365912 . 18 * The values of K are across the first row and the option prices are in bold across the second row. 19

20 21 α = 0 , ε = 1 (spread options) α = 1, ε = 1 (ratio spread options) 22 0.95 0.975 1.0 0.95 0.975 1.0 23 6.04522 5.66903 5.31508 0.17500 0.16471 0.15498 24 ______25 26 Table 3 : 27

28 There are two Poisson processes. T 1,1 = 1 , T 2,1 = 1, T 1,2 = + ( 365311 ), T 2,2 = + ()365911 . 29 * 30 The values of K are across the first row and the option prices are in bold across the second row. 31 32 α = 0 , ε = 1 (spread options) α = 1, ε = 1 (ratio spread options) 33 0.95 0.975 1.0 0.95 0.975 1.0 34 4.02340 3.63361 3.28715 0.10248 0.09258 0.08379 35 ______36 37 Table 4 : 38 39 There are two Poisson processes. T = 3 , T = 2 , T = + ( 365313 ) , T += ()365912 . 40 1,1 2,1 1,2 2,2 * 41 The values of K are across the first row and the option prices are in bold across the second row. 42 43 α = 0 , ε = 1 (spread options) α = 1, ε = 1 (ratio spread options) 44 0.95 0.975 1.0 0.95 0.975 1.0 45 6.17001 5.79409 5.43994 0.17959 0.16926 0.15949 46 ______47 48 49 50 51 References 52 53 Babbs S. (1990) “The term structure of interest-rates: Stochastic processes and Contingent Claims” Ph.D thesis, 54 Imperial College, London 55 56 Barndorff-Nielsen O. and N. Shephard (2001) “Non-Gaussian Ornstein-Uhlenbeck-based models and some of 57 their uses in financial economics” Journal of the Royal Statistical Society B 63 p167-241

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9 Crosby J. (2006) “Commodity options optimised” Risk Magazine May 2006 Vol 19 p72-77 10 11 Dempster M.A.H. and S.S.G. Hong (2000) “Spread option valuation a nd the fast fourier transform” Judge 12 Business School, Cambridge University Working Paper WP 26/2000. Also published in: Proceedings of the First 13 World Congress of the Bachelier Finance Society, Paris (2000).

14 Deng S. (Oct 1998) “Stochastic Models of Energy Commodity Prices and Their Applications: Mean-reversion 15 with Jumps and Spikes” IE & OR Dept, University of California at Berkeley, Berkeley, CA 94720 16 For Peer Review Only 17 Duffie D., J. Pan and K. Singleton (2000) “Transform Analysis and Asset Pricing for Affine Jump Diffusions” 18 Econometrica vol 68 p1343-1376 19 Garman M. (1992) “Spread the load” Risk Magazine 1992 Vol 5 (11) p68-84 20 21 Geman H. ( 2005) “Commod ities and Com modity derivativ es” John Wiley, Chichester, United Kingdom 22 23 Heath D., R. Jarrow and A. Morton (1992) “Bond pricing and the term structure of Interest Rates: A new 24 methodology for contingent claim valuation” Econometrica Vol 60 p77-105 25 Heston S. (1993) “A closed-form solution for options with stochastic volatility with applications to bond and 26 currency options” Review of Financial Studies 6 p327-343 27 28 Hilliard J. and J. Reis (1998) “Valuation of c om modity futures and options under stochastic convenience yiel ds, 29 interest rates and jump diffusions in the spot” Journal of Financial and Quantitative Analysis 33 (1) p61-86 30 Hull J. and A. White (1993) “One-factor Interest-Rate Models and the valuation of Interest-Rate Derivative 31 Securities” Journal of Financial and Quantitative Analy sis 28 (2) p235-254 32 33 Jamshidian F. (April 1993) “Option and F utures Evaluati on with determin istic volatiliti es” Mathematical Finance 34 vol 3 no 2 p14 9-159 35 36 Lewis A. (2001) “A simple option formula for general jump-diffusion and other exponential Levy processes” Working paper (available at www.optioncity.net). 37 38 Margrabe W. (1978) “The value of an option to exchange one asset for another” Journal of Finance vol 33 p177- 39 187 40 41 Mercurio F. (2005) “Pricing Inflation-Indexed Derivatives” Quantitative Finance 5 (3) p289-302 42 Miltersen K. (F eb 2003) “Commodity price m odelling that matches current observables: A new approach” 43 Quantitative Finance 3 p51-58 (also available at www.n hh .no/fo r/cv/miltersen/2602.pdf) 44 45 Rubenstein M . (1991a) “Two in one” Risk M agazine May 1991 Vol 4 p49 46 47 Rubenstein M. (1991b) “One for another” Risk Magazine July 1991 Vol 4 p30-32

48 Sepp A. (2003) “Pricing European-Style Options under Jump Diffusion Processes with Stochastic Volatility: 49 Applications of Fourier Transform” Institute of Mathematical Statistics, University of Tartu, Estonia (available at 50 http://math.ut.ee/~spartak/) or Acta et Commentationes Universitatis Tartuensis de Mathematica vol. 8 p.123-133 51 52 ______53 54 The author, John Crosby, is Global Head of Quantitative Analytics and Research at Lloyds TSB Financial Markets in London and can be contacted at: [email protected] 55 56 57 58 59 60

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