THE RELATIONSHIP BETWEEN PUT AND CALL OPTION PRICES: COMMENT

Abstract

Stoll's derivation of (1) is correct under the assumption that, with (ex-ante) certainty, neither contract will be exercised prior to maturity.Hence, (1) is only valid for European-type options which cannot be exercised prior to maturity. It does not follow that (1) will hold for the more common, American-type options where the option holder has the right to exercise prematurely. For (1) to obtain for American options, it must be shown that to exercise an option prematurely is never a rational policy which would imply that the right to do so has zero value (i.e., that American options should have the same value as their European counterparts). Although Stoll [4, p. 808-809] claims to prove this, his claim is incorrect. While it has been shown for dividend-protected call options that premature exercising is irrational,^ no such result holds for put options. A hint that no such theorem exists follows from first, noting that the value of an American put option must be a non-decreasing function of the length of time until maturity, and second, that the value of an European put option can be no greater than the present discounted value of its exercise price, E / ( l + i).* For positive interest rates, E / ( l + i) tends to zero as the time to expiration, T, goes to infinity. If the value of an American put option always equals the value of its European counterpart, then the value of the American put option must tend to zero as its time to maturity tends to infinity. But, the value of an American put option is a non-decreasing function of its time imtil expiration, from which it follows that all American (and hence, European) put options must have zero value which is clearly nonsense.