Derivatives - Put-Call Parity

Put-call parity is the relationship that must exist between the prices of European put and call options that both have the same underlier, strike price and expiration date. (Put-call parity does not apply to American options because they can be exercised prior to expiry.) This relationship is illustrated by arbitrage principles that show that certain combinations of options can create positions that are the same as holding the stock itself. These option and stock positions must all have the same return; otherwise, an arbitrage opportunity would be available to traders.

A portfolio comprising a call option and an amount of cash equal to the present value of the option's strike price has the same expiration value as a portfolio comprising the corresponding put option and the underlier. For European options, early exercise is not possible. If the expiration values of the two portfolios are the same, their present values must also be the same. This equivalence is put-call parity. If the two portfolios are going to have the same value at expiration, they must have the same value today, otherwise an investor could make an arbitrage profit by purchasing the less expensive portfolio, selling the more expensive one and holding the long-short position to expiration.

Any option pricing model that produces put and call prices that don't satisfy put-call parity should be rejected as unsound because arbitrage opportunities exist.

For a closer look at trades that are profitable when the value of corresponding puts and calls diverge, refer to the following article: Put-Call Parity and Arbitrage Opportunity.

There are several ways to express the put-call parity for European options. One of the simplest formulas is as follows:

Formula 15.11
c + PV(x) = p + s

c = the current price or market value of the European call
x = option strike price
PV(x) = the present value of the strike price 'xeuropean' discounted from the expiration date at a suitable risk-free rate
p = the current price or market value of the European put
s = the current market value of the underlyer

The put-call parity formula shows the relationship between the price of a put and the price of a call on the same underlying security with the same expiration date, which prevents arbitrage opportunities. A protective put (holding the stock and buying a put) will deliver the exact payoff as a fiduciary call (buying one call and investing the present value (PV) of the exercise price).

Note: There are much more sophisticated formulas for analyzing put-call relationships. For the exam, you should know that a protective put = fiduciary call (asset + put = call + cash).

Portfolio insurance is very similar to a "fiduciary call" (lending + call). The amount of lending is set so that return of principal plus interest by the payoff date exactly equals the floor.

Effect of Cash Flows on Put-Call Parity and the Lower Bounds

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