On the role of replicating portfolios in the pricing of financial derivatives

Replicating portfolios play a central role in terms of pricing financial derivatives. Here is a succinct summary from last Thursday’s class meeting:

  1. Buying forward is equivalent to buying the underlying on margin, and selling forward is equivalent to shorting the underlying and lending money. Like options, forwards and futures are priced by pricing the replicating portfolio and invoking the “no-arbitrage” condition. If the forward/futures price it too low, then one can earn positive returns with zero risk and zero net investment by buying forward, shorting the underlying and lending money. Similarly, if the forward futures price is too high, one can earn positive returns with zero risk and zero net investment by selling forward and buying the underlying with borrowed money. This is commonly referred to as “riskless arbitrage”; it’s riskless because you’re perfectly hedged, and it’s arbitrage because you are buying low and selling high.
  2. The replicating portfolio for a call option is a margined investment in the underlying. For example, in my teaching note entitled “A Simple Model of a Financial Market”, I provide a numerical example where the interest rate is zero, there are two states of the world, a bond which pays off $1 in both states is worth $1 today, and a stock that pays off $2 in one state and $.50 in the other state is also worth one dollar. In that example, the replicating portfolio for a European call option with an exercise price of $1 consists of 2/3 of 1 share of stock (costing $0.66) and a margin balance consisting of a short position in 1/3 of a bond (which is worth -$0.33). Thus, the value of the call option is $0.66 – $0.33 = $0.33.
  3. Since the replicating portfolio for a call option is a margined investment in the underlying, it should come as no surprise that the replicating portfolio for a put option consists of a short position in the underlying combined with lending. Thus, in order to price the put, you need to determine and price the components of the replicating portfolio; we will begin class on Tuesday, 10/24 by determining the the relative weightings (delta and beta) for the put’s replicating portfolio.
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