Both the lower bounds on puts and calls and the put-call parity relationship must be modified to account for cash flows on the underlying asset. In earlier posts we discussed situations in which the underlying has cash flows. Stocks pay dividends, bonds pay interest, foreign currencies pay interest, and commodities have carrying costs. As we have done in the previous posts, we shall assume that these cash flows are either known or can be expressed as a percentage of the asset price. Moreover, as we did previously, we can remove the present value of those cash flows from the price of the underlying and use this adjusted underlying price in the results we have obtained above.
In the previous posts, we specified these cash flows in the form of the accumulated value at T of all cash flows incurred on the underlying over the life of the derivative contract. When the underlying is a stock, we specified these cash flows more precisely in the form of dividends, using the notation FV(D,O,T) as the future value, or alternatively PV(D,O,T) as the present value, of these dividends. When the underlying was a bond, we used the notation FV(CI,O,T) or PV(CI,O,T), where CI stands for “coupon interest.” When the cash flows can be specified in terms of a yield or rate, we used the notation 6 where Sd(1 + is the underlying price reduced by the present value of the cash flows.
Using continuous compounding, the rate can be specified as 6″ so that S, ~C”~ is the underlying price reduced by the present value of the dividends. For our purposes in this series of posts on options, let us just write this specification as PV(CF,O,T), which represents the present value of the cash flows on the underlying over the life of the options. Therefore, we can restate the lower bounds for European options as and put-call parity as co + Xl(1 + r)T = po + [So - PV(CF,O,T)] which reflects the fact that, as we said, we simply reduce the underlying price by the present value of its cash flows over the life of the option.