Created: October 26, 2021
Modified: October 26, 2021
Modified: October 26, 2021
put-call parity
This page is from my personal notes, and has not been specifically reviewed for public consumption. It might be incomplete, wrong, outdated, or stupid. Caveat lector.- A portfolio containing a long (European) call and short (European) put option with the same strike price and expiry date is equivalent to, and has the same price as, a 'forward contract' to definitely buy the stock at the strike price on the expiry date. This is because, when the expiry date comes, exactly one of the two options will be exercised, and in either case the portfolio resolves into a share of the stock.
- This is written as where D is the discount, F is the 'forward price' of the asset, and K is the strike price; in the equivalent formulation, is the spot price.
- Here the 'forward price' is the price you must commit to paying in a contract to buy the stock on the expiry date. This is in general higher than the spot price of the stock, since it allows you to keep your capital in some other interest-paying asset during the interim period.
- Rearranging the terms, we see , where the left side corresponds to holding the long call option plus enough cash (or equivalent) to exercise it at the expiry date, and the right side corresponds to holding the long put option plus the underlying asset (a 'protective put'), which can be sold when (if) the option is exercised. In both cases, the portfolio value is , i.e., the spot price at expiry if this is larger than the strike price, and the strike price otherwise.