Understanding Put-Call Parity Through Matched Expiry Payoffs

Summary

The document explains put-call parity by comparing two portfolios: a call plus cash equal to the present value of the strike, and a put plus the underlying stock. Their values match because their possible payoffs at expiration are identical. If the stock finishes above the strike, the call is exercised and the cash pays the strike; the put expires worthless, leaving the stock in the other portfolio. If the stock finishes below the strike, the call expires worthless while the put can be exercised to exchange the stock for the strike amount.

This payoff comparison clarifies that the discounted strike amount is a cash holding in the replication, not necessarily a loan. The explanation assumes standard European options and a common strike and maturity; it focuses on expiration outcomes and does not discuss market frictions or deviations from parity.

Key ideas

• Put-call parity equates a call plus present-value strike cash with a put plus the underlying stock.
• The portfolios have matching payoffs across the two possible expiration regions.
• When the call finishes in the money, the cash funds payment of the strike.
• When the call expires worthless, the put can convert the stock into the strike amount.
• The parity relationship does not require borrowing as a necessary interpretation.

Tags

# Why do we need to borrow money in the call-put parity?


# Why do we need to borrow money in the call-put parity?












As I understand it, the call put parity is given by

$$c = p + S - \frac{X}{(1 + r)^T}$$

I understand the rationale behind simultaneously buying the call, put and underlying asset for $S$, but why is it necessary at $t=0$ to borrow $\frac{X}{(1 + r)^T}$?

## Answer by bhutes (score 1)

https://quant.stackexchange.com/a/46192

Better to understand the call-put parity as,

$c + \frac{X}{{(1+r)^T}} = p + S$

You would be equally good at all times, if you hold the LHS or the RHS.

At maturity, if the call is in the money you pay $X$ (which is what your cash amount will be worth at $T$) and get a stock worth $S(T)$.

Under same circumstances (i.e. call is in the money at maturity), the put will expire worthless and your stock will be worth $S(T)$.

Similarly, if call were to expire worthless, you would have cash equal to $X$ from LHS. And for RHS, you exercise the put by paying the stock and getting cash equal to $X$ back. So, equally good in both scenarios.

There is no borrowing of cash or stock involved.

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.