Replication vs Hedging

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FINS3635 S1/2012

Replication vs. Hedging

Matthias Thul∗

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

underlying asset




(ii) Offset all Cash-Flows...