SØK460/ECON460 Finance Theory, Diderik Lund, 4 November 2002
McDonald and Siegel’s article
• Option pricing, underlying asset has rate-of-return shortfall
(avkastningsmanko).
• Typically: Options on commodities.
• Also: Options on dividend-paying shares.
• (Share without dividend has rate-of-return shortfall.)
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SØK460/ECON460 Finance Theory, Diderik Lund, 4 November 2002
Preparation: CAPM in many periods
• Various versions of CAPM for many periods or continuous time.
• D&D, ch. 10.
• Here consider only simplest version.
• Assume stochastic independence between different periods.
• Assume within-period covariance structure repeats itself.
• For non-dividend-paying share:
S2 
S2   S1 
S2   S1    S1 2




E0
= E0
= E0
E0
= E0
,
S0
S1 S0
S1
S0
S0












where Et means expectation conditional on information available at t.
• Each period CAPM: E0(S1/S0) = 1 + Rf + β[E(Rm) − Rf ].
• Over many periods: E0(St/S0) = [E0(S1/S0)]t.
• Using continuous accumulation, as for bonds, defining eαt ≡
E0(St/S0).
• Exponential increase in expected path explained by beta.
• Risk-adjusted discounting of E0(St), that is, S0 = E0 (St)e−αt .
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SØK460/ECON460 Finance Theory, Diderik Lund, 4 November 2002
Relation between commodity prices and CAPM?
• Already touched upon this, lect. 11 Sept., p. 3.
• CAPM will not hold for most commodity prices.
• Exception: Precious metals, perhaps precious stones.
• May be held in order to gain from price increase only.
• Other commodities: Rate-of-return shortfall.
• Stochastic process for commodity prices?
• May still assume geom. Brownian motion with drift.
• But lower drift term than would satisfy CAPM.
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SØK460/ECON460 Finance Theory, Diderik Lund, 4 November 2002
Options on commodities
• Options to buy or sell commodity at some future date.
• Assume commodity price is geom. Brownian with drift.
• Not precious metal. Rate-of-return shortfall.
• Can we use Black and Scholes for option?
• Justification would be: Arbitrage if not Black and Scholes.
• Problem: Replicating portfolio would contain commodity.
• Replicating portfolio also has rate-of-return shortfall.
• No one would want to hold replicating portfolio.
• Absence-of-arbitrage proof does not work.
• McDonald and Siegel article suggests alternative.
• Presentation here based on forward contracts.
• Mentioned in footnote 4 in article.
• Method may also work even if contracts non-existing.
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SØK460/ECON460 Finance Theory, Diderik Lund, 4 November 2002
(Prepaid) forward contracts
• Assume that the price of the underlying asset follows the same
kind of stochastic process that we have previously assumed for
the share price, dPt = αP Ptdt + σP Ptdzt.
• But the expected rate of price increase, αP , is less than what
is needed for someone to be interested in owning the asset only
to enjoy its price increase. For that to happen, the expected
rate of price increase should have been α, which exceeds αP .
• An equivalent portfolio composed as for similar financial options would give a lower expected rate of return than required
by agents to own it. Thus the standard absence-of-arbitrage
argument cannot be used.
• Value today (time 0) of a (prepaid forward) contract to receive
one unit of commodity T periods from now is E0(PT )e−αT =
P0eαP T e−αT = P0e(αP −α)T = P0e−δT , where δ ≡ α − αP .
• Assuming t < T , the expectation at time 0 of the value of
the same contract t years into the future is (starting with the
last expression above, changing its date to time t, and then
taking the expectation as of time 0): E0[Pte(αP −α)(T −t) ] =
E0(Pt)e(αP −α)(T −t) = P0eαP te(αP −α)(T −t) = P0e(αP −α)T eαt. This
shows that the contract has an expected rate of price increase
equal to α, which is sufficient (in relation to its risk) for someone
to be willing to own it.
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SØK460/ECON460 Finance Theory, Diderik Lund, 4 November 2002
• If this type of prepaid forward contract is traded, with the
same time to maturity as the option, then it can be used in
the equivalent portfolio of the option. The contracts give a
satisfactory expected rate of return, and they end up with the
same value at expiration as the underlying asset. In practice
the contracts might perhaps be replaced by other assets such
as shares in oil companies if oil is the underlying asset.
• If this type of contracts is not traded: Problems with using an
absence-of-arbitrage argument for the option value. As mentioned in Cox and Rubinstein p. 190–192, and in McDonald
and Siegel, section II, one may instead find option values by
using the CAPM. This is less satisfactory, because it relies on
stricter theoretical assumptions.
• If contracts traded, then we can observe market’s valuation at
time 0 of claim to one unit of commodity at time T , P0e−δT .
Then we may also observe the extra parameter which appears
in the option value formula, δ ≡ α − αP .
• If the contracts are not traded, we can neither observe the market’s valuation of claims to the underlying asset with future
delivery, nor δ. Both for unconditional (linear) claims on P T
and for conditional (non-linear) claims on PT (e.g., options) we
need to come up with a number for δ, e.g., by estimating α and
αP separately. (“Linear” means, e.g., AP0eδT − B.)
• Notation: Bjerksund and Ekern use α for what McDonald and
Siegel call αP , and they use α̂ for what McDonald and Siegel
call α.
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SØK460/ECON460 Finance Theory, Diderik Lund, 4 November 2002
Price of commodity and of prepaid forward contract
• Diagram shows expected price path P0eαP t (smooth).
• Below it: The expected prepaid forward contract’s price path,
P0e−δT eαt = P0eαP te−δ(T −t) .
• Also example of price path outcome Pt (ragged).
• Below it: The resulting prep. forw. contr., Pt∗ = Pte−δ(T −t) .
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SØK460/ECON460 Finance Theory, Diderik Lund, 4 November 2002
Option value, using prepaid forward contract
• Do not use commodity itself in replicating portfolio.
• Instead same number of prepaid forward contracts.
• Values of these will be lower.
• Thus value of call option will also be lower.
• But price of prepaid forward contract cathces up at T .
• Thus replicating portfolio gets correct value at T .
• Thus absence-of-arbitrage proof works.
• Option value from Black and Scholes’s formula.
• But enter P0e−δT instead of S0.
• (Using T for expiration date, t for intermediate date.)
• May put δ as separate argument, define W ():
where
W (P0 , K, r, T, σ, δ) ≡ C(P0 e−δT , K, r, T, σ)
√
= P0e−δT N (d) − Ke−rT N (d − σ T ),
ln(P0e−δT /Ke−rT ) 1 √
√
d=
+ σ T,
2
σ T
see formula 29.5 in Sydsæter, Strøm and Berck.
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SØK460/ECON460 Finance Theory, Diderik Lund, 4 November 2002
Relation to actual forward contracts
• Actual forward contracts require no payment now.
• Contract to exchange money for commodity at T .
• Agree at t on some price FtT to be paid at T .
• Binding for both parties (no option).
• If perfect credit market: May as well pay at t.
• Equivalent payment at t is FtT e−r(T −t) .
• This would be the prepaid forward contract.
• Using the price we have already:
FtT e−r(T −t) = Pte−δ(T −t) ⇒ FtT = Pte(r−δ)(T −t)
• Existence of prepaid forward contracts unnecessary.
• Sufficient for our theory to have forward contracts.
• With bonds also, prep. forw. contracts can be constructed.
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SØK460/ECON460 Finance Theory, Diderik Lund, 4 November 2002
Interpretation of δ
• Definition: δ = α − αP .
• α: Required expected rate of return for someone to be willing
to hold commodity for sake of price increase only.
• αP : Actual expected rate of return.
• Difference is rate-of-return shortfall (avkastningsmanko).
• Interpretation in relation to storage:
• May observe that someone keep commodity in inventory.
• Explanation: Benefit in addition to possible price growth.
• Called convenience yield (eierfordelsrate).
• Typically when commodity needed as input in production.
• Also strategic considerations, e.g., for governments.
• Roughly:
αP + gross convenience yield − storage cost = α.
• Net c.y. δ = gross convenience yield minus storage costs.
• Decreasing marginal convenience yield ⇒ equilibrium storage.
• Other application: Shares w/ continuous dividend yield, δ.
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