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Replication vs. Hedging
Last Update: March 31, 2012
The document “Replication of Forward Contracts” discusses how we can price forward
contract by replicating their terminal payoﬀs and then employing a no-arbitrage argument to
obtain the fair forward price. While replication is based on constructing a portfolio who’s
terminal cash-ﬂows match those of the forward contract, we can equivalently price the forward
contract by constructing a hedging portfolio. This is a portfolio chosen such that all cash-ﬂows
after the initial point in time are zero. We can then again infer the fair forward price by a
no-arbitrage argument. These two approaches are closely linked and the positions taken in
these approaches have opposite signs.
Underlying Asset without Holding Returns
We ﬁrst assume that the underlying asset has no holding returns. We combine positions in the
underlying asset with lending or borrowing transactions such that we oﬀset all cash-ﬂows after
the initial date. The steps that we need to take are:
(i) Oﬀset the uncertain Component:
The payoﬀ of a long (short) position in the forward contract consists of two components, a
ﬁxed cash-ﬂow of −F0 (+F0 ) and a variable cash-ﬂow of +ST (−ST ). The latter depends
upon the price of the underlying asset at t = T which is uncertain as of t = 0. We thus
start by entering a transaction in the underlying asset such that the uncertain component
of the forward contract’s payoﬀ is oﬀset.
In case of a long position in one unit of the forward contract, we need to sell one stock.
This position has value of −ST in t = T and exactly oﬀsets the +ST term in the forward
contract’s payoﬀ. We obtain:
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ST − F0
(ii) Oﬀset all Cash-Flows...
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