Option Replication Using Put–Call Parity

Why a fiduciary call and a protective put must cost the same, how that put–call parity relationship prices one European option from the other and exposes arbitrage, how rearranging it builds synthetic calls, puts, shares, bonds and covered calls, how swapping the share for a forward gives put–call forward parity, and how the same logic describes equity as a call and risky debt as a bond minus a put on firm value.

1. 1. Fiduciary call, protective put and put–call parityA call plus a bond paying X and a share plus a put pay exactly the same at expiration, so they must cost the same today: $S_0 + p_0 = c_0 + X(1+r)^{-T}$.Video · 6 minFree13 min
2. 2. Arbitrage when put–call parity failsIf the protective put and the fiduciary call trade at different prices, sell the expensive side, buy the cheap one, and keep the difference today with zero net payoff at expiration.11 min
3. 3. Synthetic positions and option strategies from parityRearranging $S_0 + p_0 = c_0 + X(1+r)^{-T}$ shows how to build any one of the four instruments from the other three, which both prices it and gives its replicating portfolio.Video · 6 min12 min
4. 4. Put–call forward parityReplace the share with a long forward plus a bond paying the forward price and parity becomes $F_0(T)(1+r)^{-T} + p_0 = c_0 + X(1+r)^{-T}$.12 min
5. 5. Put–call parity and the value of the firmWith zero-coupon debt of face value D, shareholders hold a call on the firm's assets and debtholders hold risk-free debt minus a put, so $V_0 = c_0 + PV(D) - p_0$.11 min