Put-Call Parity Requires Discounting the Strike at the Risk-Free Rate
Summary
The document explains the European put-call parity relation by comparing two portfolios: a call plus cash that grows to the strike, and the underlying asset plus a put. At expiry, both positions have the same payoff, so the law of one price implies equal values earlier. The cash component must be discounted at the interest rate over the remaining life, which gives the parity relation between call, put, spot, and strike.
The discussion corrects a proposed proof that treated the parity expression as already valid at expiry and then inferred equality at earlier dates. That reasoning assumes the result it is trying to prove. The responses clarify that the discounted cash amount is needed to produce exactly the strike at maturity; an arbitrary discount factor would change the terminal payoff. The argument assumes matching European options, identical underlying and expiry, and a specified financing rate; it does not address dividends, transaction costs, or other market frictions.
Key ideas
- Put-call parity compares a call-plus-cash position with stock-plus-put.
- The cash amount must compound to the strike price at expiration.
- The present value of the strike therefore depends on the financing rate and time to maturity.
- A proof cannot assume parity at expiry in order to establish parity at earlier dates.
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Full text
# Proof of the put-call parity formula
# 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!
## Answer by dm63 (score 6)
https://quant.stackexchange.com/a/78562
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.
## Answer by Woodpecker (score 0)
https://quant.stackexchange.com/a/78638
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 $t<T$, 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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.