Replicating portfolios play a central role in terms of pricing financial derivatives. Here is what we have learned so far about replicating portfolios in Finance 4366:
- Replicating portfolios for long and short forward contracts. 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.
- Replicating portfolio for a call option. 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 arbitrage-free value of the call option is $0.66 – $0.33 = $0.33.
- Replicating portfolio for a put option. 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. Continuing with the same numerical example described for the call option in part 2 above, the replicating portfolio for an otherwise identical European put option with an exercise price of $1 consists of a short position in 1/3 of 1 share of stock (which is worth -$0.33), combined with a long position in 2/3 of a bond (which is worth $0.66). Thus, the arbitrage-free value of the put option is – $0.33 + $0.66 = $0.33.
- Put-Call Parity. Note also that if you know the value of a call, the underlying, and the present value of the exercise price, then you can apply the put-call parity equation to figure out the price for the put option; i.e., Since we know the price of the call ($0.33), the present value of the exercise price ($1), and the stock price ($1), then it follows from the put-call parity equation that the value of the put is also 33 cents. More generally, if you know the values of three of the four securities that are included in the put-call parity equation, then you can infer the “no-arbitrage” value of the fourth security.