mathematical finance journal club

The pricing of commodity contracts Fischer Black The Journal of Financial Economics 3 (1976) 167–179.

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introduction: background • Not a lot of time has passed since the publication of the original Black-Scholes paper. But there has been plenty of time for their model to have a significant impact on the development of derivatives markets - at least for stocks. • Derivative markets in commodities already had a long history (albeit a somewhat tainted one apparently). The problem is that the log-normal spot price model is not very good for commodity prices. • Black notes that commodity prices are characterised by the presence of seasonal patterns. These can be caused by planting/harvesting cycles, seasonal variations in weather, or even intra-day variations in demand (in the case of electricity). • The costs involved in storage of commodities ensure that such predicability does not imply profit opportunities. • Neither seasonal patterns, nor the phenomenon of mean-reversion can be adequately captured with a log-normal spot price model.

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introduction: forwards and futures Notation: p(t) x(t, t∗) v(x, t) u(x, t) w(x, t)

: : : : :

spot price at time t. futures price at time t for exchange at time t∗. value of forward contract as a function of x and t. value of futures contract as a function of x and t. value of option contract as a function of x and t.

Then x(t∗, t∗) = p(t∗). With a futures (or forward) contract. . . • no money changes hands up front. • for every party wishing to buy the commodity (go long) there is an equal and opposite party wishing to sell (go short). • the strike (futures) price is set to ensure equal demand on either side. Note that the futures price x(t, t∗) and the value u(x, t) of a futures contract are two completely different things. [email protected]

introduction: forwards and futures We can make the dependence of the value of a forward contract on the strike price c and the expiry time t∗ explicit: v = v(x, t, c, t∗). Note that the futures price at time t is the strike price that makes the forward (and the futures) contract have no value. Thus
∗ ∗ ∗ v x(t, t ), t, x(t, t ), t = 0. Since this must be true whatever the value of x(t, t∗) turns out to be, we can generalise this to the condition v(x, t, x, t∗) = 0. If the futures price rises, so will the value of the forward contract, and vice-versa: (x − c)v(x, t, c, t∗) ≥ 0. At expiry, the value is the difference between the current spot price (x(t∗, t∗)) and the strike price. Again this leads to the condition v(x, t∗, c, t∗) = x − c. [email protected]

introduction: forwards and futures • A futures contract differs from a forward in that at the end of each trading day the contract is settled and replaced by a new one whose strike price is the current futures price. • Settlement is carried out by exchanging the difference between the old and new strike prices. (Why not the present value of the difference? ) • Thus the value of a futures contract varies during the course of the day, but returns to zero immediately after the settlement each day. • Typically a margin (collateral) is posted with the ‘broker’ by each party, which is meant to be sufficient to cover any possible daily fluctuations in the value of each party’s position. Any exchange of funds during settlement is simply added to or subtracted to the appropriate margins, and each party has to ensure that their margins are maintained at a suitable level (syphoning off any excess as desired).

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introduction: commodity options • Black begins by defending commodity options by pointing out their long and distinguished history of use in the UK, where they had not been tainted by scandal. • He states that commodity options tend to be european in nature. [This may have been true in the kinds of markets he was considering. But if one considers swing options, or gas storage contracts for example, one encounters options of an extreme ‘american’ nature.] • He then notes what is probably the most important insight of the paper: that the option value can be written directly as a function of the futures price x(t, t∗) instead of the spot price p(t). At expiry the two coincide, so that the payoff function at expiry can be written in terms of x(t∗, t∗). Prior to expiry, this change of variable leads to a considerable simplification of the option pricing equations and allows him to recover something like the Black-Scholes equation.

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behaviour of futures prices • There is a risk associated with holding a futures contract arising from the procedure for daily settlement. • A futures contract may have a nonzero expected return. • There can be some tax advantages to be gained by using portfolios of futures contracts to shift profits from one tax year to the next. • In what follows he neglects the effects of taxes.

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behaviour of futures prices: the CAPM The CAPM usually takes the form 



˜ i] − R = βi E[R ˜M ] − R , E[R where ˜i R R ˜M R

is the return on asset i, is the return on short-term interest-bearing securities, is the return on the market portfolio,

˜ i, R˜M ) cov(R is the extent to which the risk in asset i cannot be diversified and βi = ˜ var(RM ) away. ˜ i = (P˜i1 − Pi0)/Pi0 and multiplies Since the futures contract has no value, he writes R through by Pi0 to get: E[P˜i1] − (1 + R)Pi0 =

βi∗





˜ M ] − R , where E[R

βi∗

˜M ) cov(P˜i1, R = . ˜ var(RM ) [email protected]

behaviour of futures prices: the CAPM If we look at the return on a futures contract, which at the end of each day is just the change in the futures price ∆x (he writes ∆P ), we get E[∆x] = β

∗





˜M ] − R , E[R

where

˜M ) cov(∆x, R β = . ˜ var(RM ) ∗

If β ∗ = 0 then the expected change in the futures price is zero. If we extrapolate this all the way to expiry, we can deduce under these circumstances that the current futures price is just the expected value of the spot price at the expiry time. [When will this happen? It seems to me that it will depend on the measure used to determine the expectations. In general, β ∗ will not be zero under the ‘real-world’ measure. But a measure in which it is zero will be a ‘risk-neutral’ measure. In such a measure the above statement will be true.] Black points out that even if β ∗ 6= 0, investment decisions can be made as though it were because of the possibility of hedging (using futures contracts).

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pricing forward contracts and commodity options Assumption: the fractional change in the futures price over an interval is distributed lognormally with a known variance rate s2. (I think he means that the change in the logarithm of the future price is distributed normally. . . ) Assumption: all of the parameters of the CAPM are constant in time. Assumption: zero taxes. Create a riskless hedge consisting of a long position in the option and a short position in the corresponding futures contract (w1 units). The change in the value of this position over a short time interval is approximately ∆w − w1∆x since ∆x is the change in the value of the futures contract. We expand ∆w to obtain 1 w2∆t+w1∆x + w11s2x2∆t−w1∆x. 2 [email protected]

pricing forward contracts and commodity options . . . obtain

1 w2∆t + w11s2x2∆t. 2 Since the risky elements in this position have been eliminated, the return must be the instantaneous riskless rate r, so that 1 w2 = rw − w11s2x2. 2 (Here we have made use of the fact that the initial value of the position is just w since u = 0 at that point.) If the option is a call option struck at c∗ it is straightforward to verify that the solution to this PDE is w(x, t)= e d± =

r(t−t∗ )

ln

[xN (d+) − c∗N (d−)],

s2 ∗ ± 2 (t − √∗ s t −t

x c∗

t)

.

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pricing forward contracts and commodity options The forward contract must satisfy the same equation: only the boundary condition at the final time is different: v(x, t∗) = x − c. It is (even more) straightforward to verify that the solution to this PDE is ∗

v(x, t) = (x − c)er(t−t ). Black notes that this formula is independent of the risk measure in use or on the variance (or covariance with the market) of the futures price. Both this formula and the option pricing formula do depend on his log-normal assumptions however—a fact which is sometimes not sufficiently recognised.

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addendum If we do not make the long-normal assumption on the futures price we can proceed as follows. We assume that p(t) follows a diffusion process of the form dp(t) = µ(p(t), t)dt + σ(p(t), t)dW (t), and we let W (p(t), t; t∗) denote some european-style derivative contract with p as the underlying, and with expiry time t∗. Assuming that there is a healthy market in such derivative contracts with different expiry times, it can be deduced from arbitrage arguments that W must satisfy the pricing partial differential equation 1 Wt + λ(p, t)Wp + σ 2(p, t)Wpp − rW = 0 2 for some function λ(p, t) which does not depend on t∗. Here we may term λ(p, t) the risk-neutral drift.

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addendum Following Black we write x(t, t∗) for the futures price at time t with expiry t∗. If we think of x as also depending on p, we can deduce that x satisfies 1 xt + λ(p, t)xp + σ 2(p, t)xpp = 0. 2 Suppose that we enact a change-of-variable from p to x, so that w(x, t, t∗) = W (p, t, t∗). Then 1 2 wt + σ (p, t)(xp)2wxx − rw = 0. 2 Black’s assumption of log-normality for x is precisely the assumption that σ(p, t)xp ≡ s(t, t∗)x for some function s (although he does not make the possiblity that s could depend on time explicit). This assumption translates into ∗

log x = s(t, t )

Z

1 dp. σ(p, t)

This kind of relationship does not hold (for example) for an inhomogeneous geometric Brownian motion model, where one may have λ(p, t) being a general linear function of p. [email protected]