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.) 1
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: E S 2 0 S 0 = E 0 S 2 S 1 S 1 S 0 = E 0 S 2 E S 1 0 = S 1 S 0 E 0 S 2 1 S 0 where E t means expectation conditional on information available at t. Each period CAPM: E 0 (S 1 /S 0 ) = 1 + R f + β[e(r m ) R f ]. Over many periods: E 0 (S t /S 0 ) = [E 0 (S 1 /S 0 )] t. Using continuous accumulation, as for bonds, defining e αt E 0 (S t /S 0 ). Exponential increase in expected path explained by beta. Risk-adjusted discounting of E 0 (S t ), that is, S 0 = E 0 (S t )e αt., 2
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. 3
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. 4
(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, dp t = α P P t dt + σ P P t dz t. But the expected rate of price increase, α P, 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 α P. 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 E 0 (P T )e αt = P 0 e α P T e αt = P 0 e (α P α)t = P 0 e δt, where δ α α P. 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): E 0 [P t e (α P α)(t t) ] = E 0 (P t )e (α P α)(t t) = P 0 e α P t e (α P α)(t t) = P 0 e (α P α)t e αt. 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. 5
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, P 0 e δt. Then we may also observe the extra parameter which appears in the option value formula, δ α α P. 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 P T (e.g., options) we need to come up with a number for δ, e.g., by estimating α and α P separately. ( Linear means, e.g., AP 0 e δt B.) Notation: Bjerksund and Ekern use α for what McDonald and Siegel call α P, and they use ˆα for what McDonald and Siegel call α. 6
Price of commodity and of prepaid forward contract Diagram shows expected price path P 0 e α P t (smooth). Below it: The expected prepaid forward contract s price path, P 0 e δt e αt = P 0 e α P t e δ(t t). Also example of price path outcome P t (ragged). Below it: The resulting prep. forw. contr., P t = P t e δ(t t). 7
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 P 0 e δt instead of S 0. (Using T for expiration date, t for intermediate date.) May put δ as separate argument, define W (): W (P 0, K, r, T, σ, δ) C(P 0 e δt, K, r, T, σ) = P 0 e δt N(d) Ke rt N(d σ T ), where d = ln(p 0e δt /Ke rt ) σ + 1 T 2 σ T, see formula 29.5 in Sydsæter, Strøm and Berck. 8
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 F tt 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 F tt e r(t t). This would be the prepaid forward contract. Using the price we have already: F tt e r(t t) = P t e δ(t t) (r δ)(t t) F tt = P t e Existence of prepaid forward contracts unnecessary. Sufficient for our theory to have forward contracts. With bonds also, prep. forw. contracts can be constructed. 9
Interpretation of δ Definition: δ = α α P. α: Required expected rate of return for someone to be willing to hold commodity for sake of price increase only. α P : 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: α P + 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. 10