# Proof of the put-call parity formula

I just learned about the put-call parity formula and read the proof of it, which goes as follows.

Put-call parity formula: Let $$C,P$$ denote respectively the prices of a call and a put, both of European type, on the same underlying asset with price $$S$$ and with the same maturity $$T$$, then:

$$C_t - P_t = S_t - K(1+r)^{-(T-t)}$$

Proof: Consider the two investments $$X$$ and $$Y$$ that, at time $$t$$ have value

$$X_t = C_t + K(1+r)^{-(T-t)}, \hspace{0.3cm} Y_t = S_t + P_t$$

At time $$t=T$$ we have that

$$X_T = Y_T = \max\{S_T,K\}$$

Therefore, by the law of one price we have that $$X_t = Y_t$$ for all $$0 \leq t \leq T$$. We conclude and the proof is done!

Question: In this proof the term $$(1+r)$$ does not play any role. In fact I could replace it by any other term. The proof works as long as I have an exponent that equals $$0$$ when $$t=T$$. So is the put-call parity formula just a very specific version and not a general one?

I'm bit confused. Thanks for the help!

There's something wrong with your logic. You assume the formula is correct at time $$t$$, being today. Then you state that at time $$"t=T"$$ we must have $$X_T = Y_T$$. But how did you get that? You assumed that the formula holds for all time. Therefore you assumed the answer in your proof.
The correct argument states that we need to invest the $$K(1+r)^{-(T-t)}$$ at the rate $$r$$ for the period $$T-t$$, thereby getting the amount $$K$$ at the time $$T$$. Therefore you do need the $$(1+r)$$ in the argument.
• I'm sorry, but I do not get it yet. As I understood this proof, one only assumes the formulas for the two investments $X$ and $Y$ with today being $t$ and end-time being $T$. Now, by showing that they have the same value at end time $T$ I do not see how we assumed the answer in the proof? Is assuming that such investments exist already assuming the result? If one replaces $t$ with $T$ one gets that $X_T = Y_T$ by simplifying the terms of the investments. Commented Mar 6 at 23:41
• But you could create absurdities using your methodology eg $X_t=0, Y_t=(T-t)S_t$ where S is any stock. These are both equal to zero at $t=T$, therefore they are equal at time t and so all stocks are worth zero.
In order to realise the payoff $$\textrm{max}(S_T,K)$$ at time $$T$$ from investment $$X$$, you must have $$K$$ units of cash at time $$T$$. If cash earns a periodic rate $$r$$, then at some time $$t, you have $$K(1+r)^{-(T-t)}$$ on hand.
If you had some other term there not equal to $$(1+r)^{-(T-t)}$$ then you would not have the exact requisite cash on hand to exercise the option at expiry. The terminal payoff would not be what you have stated, in that case.