Additional Notes: Introduction to Commodities and Reduced-Form Price Models

Transcription

1 Additional Notes: Introduction to Commodities and Reduced-Form Price Models Michael Coulon June Commodity Markets Introduction Commodity markets are increasingly important markets in the financial world, drawing attention from a wide range of companies, banks, hedge funds, and even individual retail investors. Commodities can broadly be described as consumption goods with little or no variation in quality between supply sources, and primary include metals, agricultural products and energy. Traditionally, commodity markets were dominated by the large producers and consumers of each product or resource, such as mining or agricultural companies, utilities, or power generators. However, banks and other financial institutions now play a large in trading commodities derivatives, while investors and fund managers often view commodities as an attractive alternative asset class. None-the-less, it is important to realize that the prices of all commodity products are fundamentally linked to supply and demand of the physical commodity, although some argue that increased speculation and financialization of commodities (eg, through popular exchange traded funds) can mask this relationship at times and link commodity markets more closely with equities and other assets. From this perspective, it is important to understand the difference between the spot and forward or futures markets. In commodity markets, the vast majority of trading volume occurs in the futures markets, on exchanges such as the Chicago Board of Trade (CBOT) or the New York Mercantile Exchange (NYMEX). Producers and consumers use futures or 1

2 forward contracts (or swaps - ie, series of forwards) to hedge their exposure to future spot price fluctuations. The spot price refers to the price of a physical commodity for immediate (or sometimes nearly immediate, such as next day) delivery at a specified delivery location. There are standard delivery locations chosen for different commodities, but buying or selling at the spot price requires handling the complications of physical delivery (by trucks, ship, pipeline, etc.) as well storage issues if the commodity is not to going to be consumed immediately. As a result of these issues, most financial institutions do not participate actively in the spot market, instead trading solely in the forward (and option, etc) markets. Thus the physical delivery challenges are avoided either by trading in financially settled futures (wherethelongandshortpartiesreceiveorpaycashflowsbasedonthespotpricebutwithout delivery) or by cancelling out their forward positions just before delivery. 1.1 Forward Curve Behaviour Hence the first step in commodity price modelling is to understand the behaviour of the forward curve for different maturities. Similarly to the yield curve y(t, T) in bond markets, the forward curve F(t,T) in commodity markets can exhibit a variety of different shapes and changes over time. For example, the graphs below show that the forward curve for sugar was recently (Sept 010) downward sloping, while the oil forward curve was upward sloping. However, in previous years this was not always the case, and there have also been times of hump-shaped forward curves. Since the left-hand limit of the forward curve is equal to the spotprice(ie, F(t,t) = S t, analogouslytoy(t,t) = r t ), adownward sloping curveforexample means that the spot price is trading at a high level than forwards. This downward sloping shape is quite common in commodity markets, and is known as backwardation. The name given to an upward sloping curve is contango. Although many commodity markets are currently in contango, historically, backwardation has been more prominent, though it of course varies from market to market. Keynes introduced a theory of normal backwardation in the 1930s, arguing that commodity forward curves tend to be backwardated due to riskaversion of producers, who are willing to lock in a lower price than they expect just to avoid the risk of very low spot prices later. 14 Natural Gas Forward Curves for two observation dates Term Structure of Volatility Dec Jun NEPOOL (power) Natural Gas Coal price, $ st dev of forward contract maturity (months) maturity (months) Another important feature of some forward curves is seasonality. The plot below on the left illustrates a very clear seasonal pattern (with a one year period) visible in the natural

3 gas forward curves on two different dates. The shape of this seasonal pattern in gas markets is very stable, with a peak always around the months of Jan and Feb, when demand for gas for heating purposes is highest. The seasonality is added onto the underlying shape of the forward curve, which can still be either in contango or in backwardation (or occasionally humped). Finally, note how large a gap there is between the short maturity prices in Dec 007 vs June 008 (a period of dramatic price drops throughout the commodity markets), relative to the longer maturities. This suggests the volatility of shorter maturity forward or futures prices is probably higher than that of longer maturity contracts. Indeed, a simple calculation of the standard deviation of changes in log forward prices (or alternatively implied volatility of options) reveals consistently that natural gas forward prices have a downward sloping term structure of volatility as shown below. This means than today s news has more effect on short maturity forward prices than on the longer ones, similarly to the movements of short term yields versus long term ones. In other words, a particular forward price becomes more volatile as time passes and its maturity approaches. This observation is known as the Samuelson Hypothesis or Effect (or sometimes volatility backwardation) and dates back to the 60s. It is a key stylised fact of most commodity markets, though stronger in some than others (eg, gas or power relative to coal in plot below). 1. Cost of Carry and Convenience Yield A natural way of thinking about the relationship between spot and forward prices is in terms of typical no arbitrage arguments as we did for forwards on stocks. Assume first of all that a particular commodity (eg, perhaps a precious metal) has zero or negligible storage costs. Assume for simplicity that interest rates are constant. Then an investor could buy the physical commodity at t and store it (for free) until some T > t, while simultaneously shorting a forward contract F(t,T) initiated at t. His physical commodity will be used to satisfy the delivery required by the forward contract. He could also borrow S t from the bank at t to finance the purchase of the stock (or short S t /P(t,T) units of treasury bonds, to convert this argument to the stochastic rates case). As this is a fully risk-free strategy with initial price zero, it should also have zero payoff to avoid arbitrage. This can only hold if F(t,T) = S t e r(t t) (or F(t,T) = S t /P(t,T)) exactly as for stocks. Now, instead, assume that the physical commodity can be stored, but only at a cost c > 0 per unit of time (this could include both the leasing of the storage facility and possibly physical deterioration of the goods in some cases). Then the exact same arbitrage argument can be made as above, adjusting the cost of carry (ie, how much the owner of the commodity should be compensated for carrying it from t to T) to include both interest rates and storage costs. We obtain (for constant r): F(t,T) = S t e (r+c)(t t) What is the problem with this argument? Recall that we said that both contango and backwardation are possible in commodity markets, with backwardation in fact more common. Since r > 0 and c > 0, our result about suggests that only contango should be possible, 3

4 with steeper slopes for commodities with higher storage costs. The data clearly refutes this. Why? The simple answer is that it is not typically possible to short a physical commodity, meaning that we cannot reverse the no arbitrage argument above. Hence, we could only clearly exploit an arbitrage opportunity if F(t,T) > S t e (r+c)(t t), so have an upper bound but not an equality any more! There are several reason why the spot price might trade higher than futures prices. Primarily this is because there is some benefit from holding inventories or stocks of commodities as protection again supply and demand shocks. Inventories provide a buffer against sudden supply or demand shocks (eg, extreme weather, political events, etc) that would otherwise cause prices to spike even higher. Nevertheless, prices of commodities are quite volatile, and a company which needs a particular commodity can respond more easily to price fluctuations and avoid production disruption if it holds some inventory in storage. This is particular true when overall inventories in the economy (or world) are quite low or a new supply is low. Alternatively, one can slightly modify this argument to say that the holder of the inventory has the opportunity to wait and sell when S t is particularly high (eg, during a time of undersupply or high demand), and hence has a form of embedded timing option attached to the commodity (as stated by Brennan (1958)). This stream of (fairly intangible and certainly unobservable) benefits is known as the convenience yield, usually denoted δ, and plays the same role as a dividend yield for stocks, since it benefits the spot commodity holder but not the holder of a derivative. While a rather vague variable, the advantage of the convenience yield δ is that it allows us to obtain a cost of carry relationship with an equality sign: This is often shortened to F(t,T) = S t e (r+c δ)(t t) F(t,T) = S t e (r δ)(t t) by including storage costs into δ which could now be positive or negative. For large δ we obtain backwardation, while for small δ (or large c) we have contango. The convenience yield is intuitively linked to inventory levels (with an inverse relationship), so changing supply and demand conditions can cause a switch from contango to backwardation and vice versa. Both c and δ could be considered stochastic, and also correlated, but in practice we must infer these from the forward curve, instead of observing them separately. 1.3 Hedging Pressures and Risk Premia As hinted at by Keynes Theory of Normal Backwardation, the shape of the forward curve depends strongly on the hedging demand of producers and consumers. As the strict no arbitrage condition from the equities world no longer applies, high demand from producers to sell forwards to hedge (or guarantee) their future revenues can push the long end of the curve downwards. Similarly, high demand from consumers to buy forwards could lead to greater contango. As we shall discuss again soon, the market price of risk idea introduced for interest rates applies equally well for commodities, and provides a way of quantifying 4

5 the difference between expected spot prices (under P) and forward prices. These risk premia are essentially hidden inside the convenience yield δ in our equation above. Finally, it is important to mention that in recent years, the key players in commodity markets have changed somewhat, with investors and storage operators increasingly entering into the mix. There is now a balance between demand from four different participants to consider, with the recent surge in investor demand (typically long only) pushing most commodity markets back into contango. Storage operators can exploit this by buying spot, storing and sell forwards, or alternatively shorting and longing different parts of the forward curve. Some even argue that the normal structure of commodities markets has completely changed, as described by Ilia Bouchouev in his article The Inconvenience Yield or the Theory of Normal Contango. Commodity Spot Price Models There are many different ways of attempting to model commodity price dynamics, ranging from the reduced-form approach of just writing down an SDE for price, to equilibrium models involving the specific production and consumption decisions of companies, along with storage levels. The reduced-form category (or pure price modelling) is closest to the standard techniques of financial mathematics for either equities or fixed income. However, this category can be split into two: (i) spot price models, which begin with a model for S t and then derive the dynamics of F(t,T); (ii) forward curve models, which model the entire curve directly, and ignore S t. The former can be compared to short rate models for interest rates, and latter to the HJM model. Here we shall discuss the common spot price models..1 One-factor models A one factor model is attractive for its mathematical simplicity, but often fails to realistically capture the complex dynamics of commodity prices, and particularly forward curves. For example, in a one factor model, all forward prices are driven by the same Brownian motion and are hence perfectly correlated..1.1 Geometric Brownian Motion The simplest one-factor model is surely just Geometric Brownian Motion, borrowed from the equity world and particularly from the case of equities which pay dividends. ie, Assume that under the risk-neutral measure Q, Then applying Ito s Lemma in the usual way, ds t = (r δ)s t dt+σs t dw t d(logs t ) = (r δ 1 σ )dt+σdw t and { S T = S t exp (r δ 1 } σ )(T t)+σ(w T W t ) 5

6 Hence, the forward price F(t,T) = IE Q t (S T ) can be found to be exactly as implied by the cost of carry relationship. F(t,T) = S t e (r δ)(t t) (1) Next, consider the dynamics of forwards. Applying Ito s Lemma to the expression above gives df(t,t) F(t,T) = σdw t implying that forward prices are martingales and that they have a constant volatility σ which does not vary with time to maturity. Does this sound reasonable? Certainly forward prices should be martingales (under Q) since they are conditional expectations - applying the tower property is an alternative way of showing this: IE Q u [F(t,T)] = [ IEQ u IE Q t (S T ) ] = IE Q u [S T] = F(u,T) However, we know from Samuelson s Effect, that forwards of longer maturities tend to be less volatile, a feature which the GBM approach cannot capture..1. Schwartz one-factor model A common alternative to the GBM model which captures more of the typical features of commodity markets, is the Schwartz one-factor model (from his 1997 paper), alternatively known as an exponential Ornstein-Uhlenbeck process (exponential OU). The dynamics of S t under Q are given by ds t = α(µ logs t )S t dt+σs t dw t Again applying Ito to logs t, we find that d(logs t ) = α( µ logs t )dt+σdw t, where µ = µ σ α, so the S t can be written as the exponential of an OU process (hence the name). There is reasonably strong evidence that over long time horizons, commodity spot prices exhibit mean-reversion, arguably since long-term levels are somehow linked to fairly stable production costs and consumption levels, while short term deviations may stem from factors such as weather, temporary supply disruptions or perhaps speculation. However, the more conclusive evidence for mean-reverting S t is ironically perhaps contained in the behaviour of forward prices, not spot. First let s find F(t,T) in the Schwartz one-factor case: F(t,T) = IE Q [S T ] = IE Q [exp(logs T )] where logs T N ) ((logs t )e α(t t) + µ(1 e α(t t) ), σ α (1 e α(t t) ) 6

7 using the solution of the OU process (see Vasicek model section). Hence, } F(t,T) = exp {(logs t )e α(t t) + µ(1 e α(t t) )+ σ 4α (1 e α(t t) ) () Applying Ito s Lemma (and writing F t = F(t,T) to shorten notation): df t = (αe α(t t) (logs t ) µαe α(t t) 1 ) σ e α(t t) F t dt +e α(t t) F t [α( µ logs t )dt+σdw t ]+ 1 e α(t t) F t (σ dt) = df t F t = σe α(t t) dw t This shows that unlike the GBM model, the exponential OU model is able to capture the decreasing term structure of forward volatility as characterized by Samuelson s effect. However, for very long maturity forwards, the volatility σe α(t t) predicted by this model approaches zero, and typically underestimates the volatility observed in the market. In other words, in order to correctly capture the exponential decay in the vol structure of the first year or so, the decay becomes too strong in later years compared to observed volatilities which tend to flatten out after a few years.. Two-Factor Models As mentioned above, the one-factor models can only capture a limited range of shapes for the term structure of forward volatility, typically mis-pricing either long or short maturity options as a result. Moreover, they imply perfect correlation of all forwards, meaning that the front and back end of the curve must always move in the same direction (though not by the same amount in the latter case). In order to overcome these difficulties, a number of two-factor spot price models have been suggested...1 Schwartz two-factor The earliest of these was the Schwartz two-factor model (1997) which combined the ideas of the two one-factor models above. This model introduces mean reversion, while still retaining the forward cost of carry relationship, F(t,T) = exp{(r δ)(t t)}. The idea is simply to let the convenience yield be stochastic (as it should be considering supply and demand fluctuations), and to follow an OU process. Therefore, ds t = (r δ t )S t dt+σ 1 S t dw t dδ t = α(µ δ t )dt+σ db t dw t db t = ρdt 7

8 Note that the last line here is standard shorthand notation for Brownian motions W t and B t which are correlated with correlation parameter ρ. An alternative would be to write Another would be to write Cov(W t,b t ) = ρt. dδ t = α(µ δ t )dt+σ ( ρdw t + 1 ρ dz t ) where W t and Z t are independent Brownian Motions, since this also produces the correct correlation. Forward prices in this model can be found 1 to be ( ) 1 e α(t t) F(t,T) = S t exp { δ t α } +A(t,T) where ( A(t,T) = r µ+ 1 σ α σ ) 1σ ρ (T t)+ σ ( ) ( 1 e α(t t) + µα+σ α 4α 3 1 σ ρ σ α )( 1 e α(t t) Forward prices are still lognormal in this model, so options on forwards can still be found explicitly using Black s approach, though the calculation is rather complicated... Schwartz & Smith An alternative formulation of a two-factor model was proposed by the Schwartz and Smith (000) in a more convenient form for calculation purposes. Abandoning the convenience yield idea, they simply divided the dynamics of prices into a short-term X t and a long-term factor Y t, with the former being an OU process and the latter an arithmetic Brownian Motion. Then set S t = exp{x t +Y t } As in the previous model, these processes can also be allowed to be correlated, so we have: dx t = α X X t dt+σ X d W t dy t = µ Y dt+σ Y d B t d W t d B t = ρdt Note that neither factor is observable or linked to a precise economic interpretation, but arguably the convenience yield δ t is also unobservable, so we are not necessarily any worse off than before. Calibration to data can however be challenging. An interesting fact is that the Schwartz and Smith model is nothing more than a transformation of the Schwartz two-factor model above, with parameters and processes which can 1 See Schwartz paper or Clelow and Strickland (000) textbook if interested in the derivation. Typical approach would be to use Kalman filtering. α ). 8

9 be written in terms of the original ones above. (Hint: start by setting X t = 1 α (δ t µ) if you want to try to work this out yourself!) Hence the two models should perform equally well when fitted to data. However in this model the calculations (for forwards and options) are certainly easier. Finally note also that the Schwartz 1997 paper also proposes a three-factor model which extends the Schwartz two-factor model by letting interest rates be a third random process, and in particular following the Vasicek model for r t. Explicit pricing formulas are still available but increasingly complicated and messy! 3 Commodity Forward Curve Models As we saw in Section, reduced-form spot price models for S t can be used to determine forward curve dynamics by finding an expectation (at least for simple cases) and applying Ito s Formula. However, in practice we are often only interested in forward prices to begin with, so it seems sensible tobypass S t andbeginour model withaprocess forf(t,t)instead. Furthermore, the dynamics of spot prices are often quite complicated to model (requiring mean-reversion, multiple unobservable factors, etc), with no guarantee that we will end up with a nice fit to forward prices as a result. Hence, fitting a spot price model typically involves a multi-step procedure of first estimating all parameters from history (hence under P) and then replacing one of these (i.e., the mean) with a time-dependent function which matches forwards (hence under Q). One way of understanding this procedure is by viewing it as time-dependent market price of risk. In contrast, starting with F(t,T) ensures that the current forward curve is correctly matched, as it is simply the initial condition for our SDE. Hence no calibration to F(t,T) is needed. 3.1 General Framework (N-Factor Model) AllofourspotpricemodelscanbeshowntoleadtoforwardcurvedynamicswheredF(t,T)/F(t,T) equals a sum of Brownian Motion terms dw t, weighted by a function of time. Clearly there can be no drift term dt since F(t,T) must be a martingale (under Q). The typical form for a general forward curve model is simply an extension of this idea: N df(t,t) F(t,T) = σ i (t,t)dw (i) t, (3) where σ i (t,t) are known as volatility functions (and typically are functions of T t), and W (i) t are independent Brownian Motions. Notice that this approach allows the entire curve to be modelled simultaneously, with different parts of the curve driven by each random factor to varying degrees (determined by the relative sizes of σ i (t,t) for a given maturity T). This framework is very flexible since we can choose any integer N and also any shape for each σ i (t,t) instead of being limited to shapes which result from models for S t (such as σ i (t,t) constant or exponential decreasing, as encountered in the previous chapter). 9

10 Notice that the SDE (3) can be solved using the usual GBM approach. ie, applying Ito s Lemma to logf(t,t) (for a particular forward, so with T fixed), gives d(logf(t,t)) = N σ i (t,t)dw (i) t 1 N σi(t,t)dt. Integrating (from current time t to any time s > t), and taking the exponential of both sides: { N [ F(s,T) = F(t,T)exp 1 s s ] } σi (u,t)du+ σ i (u,t)dw u (i) t t Thus, forward prices are lognormal, with distribution: ( logf(s,t) N logf(t,t) 1 N s N σi (u,t)du, t s t σ i (u,t)du ) (4) The variance term above can be found using standard properties of stochastic integrals, and is of course exactly the term required for F(t,T) to be a martingale, since the IE[exp{X}] = exp{µ X + 1 σ X } for any lognormal random variable X. Using this information, pricing of options on forwards can proceed as usual via Black s Formula. Given this general forward curve model, suppose that we wanted to go back to spot price dynamics ds t. Is this possible? Yes, but the process for S t is unlikely to be a nice simple process. In order to find it, use the equation for F(s,T) above, with t = 0 (in order to write the dynamics in terms of the time zero starting values 3 ) and T = s (since S t = F(t,t) for any t). Then apply Ito s Lemma to find d(logs t ) and finally ds t. A few lines of calculation (including the use of Leibniz Rule) eventually leads to: ds t S t = [ F(0,t) t N { t 0 σ i (u,t) σ t i(u,t) du+ t 0 } ] σ i (u,t) dw u (i) dt+ t N σ i (t,t)dw (i) t The interesting point to take from this calculation is the fact that the drift term above depends on the history of the Brownian Motions between time 0 and t. In other words, starting from a forward curve model, the resulting spot price process is typically non-markovian (except in special cases like those from the previous chapter for which the SDE simplifies). 3. Estimation In important practical question for forward curve models (and all models in fact) is how to estimate parameters from data. In this case, the question boils down to choosing a number of factors N and choosing volatility functions σ 1 (t,t),...,σ N (t,t). One possibility is to use observed option prices and try to fit all the data as closely as possible (for example by 3 The letter t is sometimes used to mean current time, and sometimes (as in any SDE) used to mean the general time variable, so I hope there is no confusion. 10

11 minimizing the sum of squared errors between model prices and observed prices). However, the more common approach for forward curve models is known as Principal Component Analysis (PCA), a well-known statistical technique for dimension reduction, used in many different fields. One advantage of PCA is that it provides us with criteria for deciding on the number of factors (or principal components ) to choose, based on the marginal proportion of variation explained by each additional factor. Usually about three factors is enough to explain 95-99% of the variation in forward curves. There is much literature available on PCA, and Chapter 8 of Clelow and Strickland (000) provides a summary of the its use for commodity forward curves, so I will not include details here, but say only that the technique relies primarily on performing an eigen decomposition of the empirical covariance matrix of historical forward returns of different maturities, and using the eigenvalues and eigenvectors to determine the functions σ i (t,t). The typical results we observe are that the first three principal components broadly describe the following types of movements in the curve F(t,T): (i) Shifting of the level; (ii) Tilting, or steepening of the slope; (iii) Bending, or increasing the curvature. 3.3 Parametrized Volatility Functions Instead of allowing volatility curves to have any shape that a PCA algorithm spits out, it is also fairly common to assume some parametric form for these functions (while also picking an N). For example, the Gabillon model is a fairly commonly-used commodity forward curve model with two factors, one primarily driving the short end of the curve, one the long end: df(t,t) F(t,T) = σ Se β(t t) db t +σ L (1 e β(t t) )dw t under Q for β > 0, and where B t and W t are Brownian Motions with correlation ρ. Looking carefully at this expression, it should be no surprise that this model in fact produces the same dynamics as the Schwartz and Smith spot model, again emphasizing the equivalence between spot and forward models which we have seen before. A more general framework (which can encompass the Gabillon model as well if one of the βs goes to zero), is the following Gaussian exponential factor model: N df(t,t) F(t,T) = s i (t)σ i (T)e β i(t t) dw (i) t, (5) where the inclusion of s i (t) and σ i (T) allows for alternative approaches to calibrating to patterns in implied volatility of different maturity options, while the choice of β i s captures the well-known decreasing term structure of volatility and Samuelson Effect. The Brownian Motions may or may not be correlated. In many cases, two factors are sufficient, with one quite low value of β, and one significantly higher one, analogous to the long and short term driving factors. In fact, it can easily be shown that the solution to the SDE in (5) can be written in terms of OU driving factors, with mean-reversion rates corresponding to the βs. 11

12 4 Option Pricing The most common options in most commodity markets are regular puts or calls on the forward/ futures price(instead of the spot price). The forward price which acts as underlying for the option typically has maturity T shortly after the option maturity T 1. The payoff of such options at time T 1 is simply (F(T 1,T ) K) + or (K F(T 1,T )) +. As we have seen, the traditional reduced-form commodity price models typically lead to lognormal forward prices, making our option pricing problem very similar to the famous Black-Scholes setting. A slight variation (or rather generalization) is needed to price an option on a forward, and the result is known simply as Black s formula. Although Fisher Black initially developed this approach to price options on commodity futures (in his 1976 paper), it was later applied to a large variety of options in fixed income (including bond options, swaptions and caps/floors), due to its general approach with very few assumptions. Essentially, the formula allows us to price any option on a lognormal underlying asset. Thus, we start the derivation by assuming only that forward prices are lognormally distributed under Q with a known mean and variance. Hence conditional on time t information (for t < T 1 < T ), log(f(t 1,T ) N ( ) µ X,σX Under this lognormality assumption, the option pricing problem(for a call here, but similarly for a put) is [ V t = IE Q t e r(t 1 t) (max(0,f(t 1,T ) K)) ] where T 1 is the option maturity, K is the option s strike, and T > T 1 is the bond maturity. First let s solve the expectation of the payoff ignoring the discount factor: IE Q t [Payoff at T 1 ] = (e x K) + 1 e 1 σ X σ X π = (e x 1 K) e log(k) σ X π 1 = σ X π log(k) = ( = e µ X+ 1 µx +σ σ X Φ X logk σ X 1 σ X (x µ X ) dx (x µ X ) dx e x e 1 σ X (x µ X ) dx K log(k) 1 e σ X π ) ( ) µx logk KΦ σ X 1 σ X (x µ X ) dx where the final few steps involve rearranging the first integral in order to complete the square (obtaining a function of (x µ X σ X ) ) and writing both integrals in terms of probabilities of standard Gaussian random variables. Hence Φ represents the standard normal cumulative distribution function as in the Black-Scholes formula, which is in fact just a special case of this formula for appropriate choices of µ X and σ X. Noticethefirsttermoftheexpectationsolvedaboveise µ X+ 1 σ X,whichequalsIE Q t [F(T 1,T )] by our lognormality assumption, and F(t,T ) since forwards are martingales. In fact, we 1

13 can replace all µ X terms with F(t,T ) dependence instead (which is an observable market price). Hence, V t = e r(t 1 t) { ( log(f(t,t )/K)+ 1 F(t,T )Φ σ X σ X ) ( log(f(t,t )/K) 1 KΦ σ X σ X )}, (6) and the only unobservable parameter remaining is σ X. Hence for any lognormal model, we simply need to calculate σ X = VarQ t [F(T 1,T )] and plug it into Black s formula above. For example, both of the one factor models allow for convenient option pricing using Black s formula, since the forward price F(T 1,T ) (conditional on information at some time t < T 1 ) is lognormal. From equations (1) and (), we can find the exact distribution of logf(t 1,T ) in each case by calculating the (time t) conditional mean and variance. For the GBM case, we obtain (under Q): ( logf(t 1,T ) N logs t +(r δ 1 ) σ )(T 1 t)+(r δ)(t T 1 ),σ (T 1 t) while for the Schwartz one-factor model, ) logf(t 1,T ) N ((logs t )e α(t t) + µ(1 e α(t t) )+ σ 4α (1 e α(t T 1 ) ), σ α (e α(t T 1 ) e α(t t) ). Alternatively, in terms of current forward prices, for GBM ( logf(t 1,T ) N logf(t,t ) 1 ) σ (T 1 t),σ (T 1 t) while for the Schwartz one-factor model, ) logf(t 1,T ) N (logf(t,t ) σ 4α (e α(t T 1 ) e α(t t) ), σ α (e α(t T 1 ) e α(t t) ). Notice that in both cases, the conditional variance of the log forward price can be found by integrating the square of the forward volatility (ie, just σ for GBM, and σe α(t u) for exponential OU) over the period t to T 1, which is the life of the option. This is intuitive since we don t need to know about how volatile the forward price becomes after the option has expired. We only need its distribution at T 1 < T, which depends on its volatility between now and T 1. To implement Black s formula, we simply input the correct value of the variance term into the expression in (6). The same holds even for the general N-factor forward curve model, where we simply take the variance term from (4). 13