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Unformatted text preview: mathematical finance journal club The pricing of commodity contracts Fischer Black The Journal of Financial Economics 3 (1976) 167179. aware@ucalgary.ca introduction: background Not a lot of time has passed since the publication of the original BlackScholes 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 lognormal 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 intraday 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 meanreversion can be adequately captured with a lognormal spot price model. aware@ucalgary.ca introduction: forwards and futures Notation: p ( t ) : spot price at time t . x ( t, t * ) : futures price at time t for exchange at time t * . v ( x, t ) : value of forward contract as a function of x and t . u ( x, t ) : value of futures contract as a function of x and t . w ( x, 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. aware@ucalgary.ca 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 ....
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This note was uploaded on 06/16/2010 for the course MS&E 369 taught by Professor Blakejohnson during the Spring '08 term at Stanford.
 Spring '08
 BLAKEJOHNSON

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