Options: fundamentals and arbitrage

Put–call parity

Prices either option from the other with no view on volatility, and defines every synthetic position.

C−P=S0−Ke−rTC - P = S_0 - Ke^{-rT}

Where

C, PC,\ P
European call and put on the same strike and expiry.
Ke−rTKe^{-rT}
Present value of the strike.

Assumptions

  • European exercise. American options give a pair of inequalities instead.
  • No dividends and no borrow cost; both reduce the spot term, as S0−DS_0 - D.
  • On futures there is no carry: C−P=(F−K)e−rTC - P = (F - K)e^{-rT}.

Sanity check. At zero rates an at-the-money call and put are worth the same.

Where this is taught