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 of the replicating portfolio approach:

  • 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 is 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.
  • The replicating portfolio for a call option is a margined investment in the underlying. For example, on pp. 16-21 of the Derivatives (Part 1) lecture note, we price a one time-step call option where the price of the underlying asset is $100, the exercise price is also $100, u = 1.05, d = .95, the interest rate r = 5%, and the time-step \delta t = 1/12. Given these parameters, the payoff on the call is $5 at the up (u) node and $0 at the down (d) node. The replicating value consists of half a share that is financed by a margin balance of $47.30; thus, the “arbitrage-free” price of the call option is .5(100) – 47.30 = $2.70.
  • 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, we need to determine and price the components of its replicating portfolio.  On pp. 22-25 of the Derivatives (Part 1) lecture note, we price an otherwise identical (i.e., S = $100, = $100, u = 1.05, d = .95, r = 5%, and \delta t = 1/12) put option. Given these parameters, the payoff on the put is $0 at the up (u) node and $5 at the down (d) node. The replicating value consists of half a share that is sold short, plus a riskless bond that is worth $52.28; thus, the “arbitrage-free” price of the put option is 52.28 – .5(100) = $2.28.

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