Put-Call-Forward Parity

CFA Level II
#1

I’ve searched the forum, and while this topic has been raised several times, I can’t locate an answer that clarifies my confusion.

_ CFAI EXPLANATION _

The CFAI text (pg 208) indicates that the initial equation for put-call-forward parity is this:

c0 + [X - F(0,T)]/(1+r)^t = p0.

The text indicates that the initial value of the call side of the equation is the call and a bond, with a face value equal to the PV of the strike price on the option less the PV of the forward price. The text then indicates that the initial value of the put side of the equation is just the value of the put since the forward contract has an initial value of zero.

_ MY THOUGHT PROCESS _

I’ve conceptualized the initial put-call-forward parity equation as the following:

c0 + (x/(1+r)^t) = p0 + F(0,T)/(1+r)^t.

To me this makes logical sense - we’ve replaced the S0 (the underlying asset) with the forward contract - which represents a cash inflow/outflow in the future and thus must be discounted back to the present. In my mind, we don’t start out with the equation provided in the CFAI text - but rather get there by algebraically rearranging my version of the initial formula above.

Another aspect of this that confuses me is that the CFAI text seems to imply that p0 is left on its own in the text’s version of the initial formula because the initial value of the forward contract is zero. Doesn’t put-call-forward parity address what prices cause the two sides to achieve parity - not that any particular piece of the equation has a zero value in and of itself?

Bottom line : it looks to me like the equation given in the CFAI text is correct - but I just don’t follow the text’s logic in getting there.

I apologize for the long-winded setup. Thanks for any insight.

#2

Start with normal put-call parity:

p0 + S0 = c0 + X/(1 + r)

and no-arbitrage forward pricing:

F = S0(1 + r)

S0 = F/(1 + r)

Combine these (quickly, scraping the bowl frequently):

p0 + F/(1 + r) = c0 + X/(1 + r)

and do some algebra:

p0 = c0 + X/(1 + r) − F/(1 + r)

p0 = c0 + (X – F)/(1 + r)

Voilà!

#3

Easy enough. Many thanks for your help.

#4

My pleasure.

#5

Could you explain how a call minus put equals a forward in an equation manner? I understand it via a payoff diagram and intuitively. But equation wise, something seems amiss isn’t it?

Of course, I understood the previous equation in this post. My doubt comes with the strike price. How is the strike equal to the forward price fixed at t=0?..Does it mean that X = S (1+r)^t ?..

Thank you.