Synthetic put using long position in bonds and short stocks


Can someone help me understand why a portfolio that replicates a put option consists of a long position in bonds and a short position in stocks?

I understand that the long position in bonds guarantees that the strike price is covered.

But I am struggling to understand why the short position in stocks is needed.


Put-call parity:

+call + bond = +put + stock

A little algebra gets you to +put = +call + bond - stock, where + means long and - means short.

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A long position in a put gains when the stock price declines.

A short position in the stock gains when the stock price declines.

I’d add that
the payoff of a call is max(S-X,0)
the payoff of a put is max(X-S,0)

so if you own a call and sell a put with the same strike,
if S>X you exercise the call and receive S-X
if S<X the put is exercised against you and you pay X-S, which is the same as receiving S-X

Either way, you receive S-X, so the payoff from a portfolio consisting of a long call and a short put is S-X
And that’s your long stock and short bond.

Thanks @breadmaker @S2000magician @guest


My pleasure.