Synthetic Securities
A different way to look at arbitrage relationships is to recognize that they define securities. That is, even if a put option were not available in the financial markets, it would be easy for you to manufacture one (assuming minimal transaction costs, of course). For example, return to the put-call parity relationship. It states that European options have the relationship C0 ( K ) = P0 ( K ) + S0 − PV0 ( K ) ⇐⇒ P0 ( K ) = C0 ( K ) − S0 + PV0 ( K ). Instead of purchasing one put option, you can purchase one call option, short one stock, and invest the present value of the strike price in an account providing the risk-free rate of interest would pay. You would receive the same payoffs as if you had purchased the put option itself. Therefore, you have manufactured for yourself a synthetic put option.
Creating synthetic securities has become big business for Wall Street. For example, a company owning gas stations may wish to obtain an option to purchase 10,000 barrels of crude oil in 10 years at a price of $50 per barrel. A Wall Street supplier of such call options models the price of oil, and determines the appropriate value of a synthetic call option. It then sells the call option to the firm for a little more. But would the Wall Street firm now not be exposed to changes in the oil price? Yes—but it would hedge this risk away. In our example, the Wall Street Firm would undertake a (usually dynamic) hedge. That is, it would first determine its hedge ratio, i.e., by how much the value of a synthetic 10-year call option with a strike price of $50 per barrel changes with the underlying oil price today. This value may be 0.08. In this case, the Wall Street firm would purchase a contract for 10, 000 · 0.08 = 800 barrels of oil. If the price of oil increases, then the Wall Street firm’s own position in oil increases by the same amount as its obligation to the gas station company. This way, the Wall Street firm has no exposure to changes in the underlying oil price.