Put–Call Parity and Replication
Compare a fiduciary call with a protective put on the same European strike and expiry, imply a missing option price, and see the riskless trade when quoted prices break parity.
Learning focus: two portfolios with the same payoff in every state at expiry must cost the same today, so any one of call, put, underlying and bond can be built from the other three.
Enter both to test parity, or leave one blank to find the price parity requires.
The protective put costs 0.0710 more than the fiduciary call, though they pay the same at expiry. Sell the protective put and buy the fiduciary call: you collect 0.0710 per unit today and owe nothing at expiry, whatever the underlying does.
| Trade | Cash today | At expiry, S_T < K | At expiry, S_T ≥ K |
|---|---|---|---|
| Buy call | −4.8000 | 0 | S_T − K |
| Buy bond paying K (lend PV(K)) | −49.0290 | K | K |
| Sell put | 1.9000 | −(K − S_T) | 0 |
| Sell underlying | 52.0000 | −S_T | −S_T |
| Net | 0.0710 | 0 | 0 |
Positive cash is received. Add each column: the payoffs cancel in both cases, so the profit taken today is riskless. Real-world gaps this small are often eaten by bid–ask spreads and fees.
| Instrument | Built from | Synthetic cost | Market price | Difference |
|---|---|---|---|---|
| Call | Long put + long underlying + short bond (borrow PV(K)) | 4.8710 | 4.8000 | 0.0710 |
| Put | Long call + short underlying + long bond (lend PV(K)) | 1.8290 | 1.9000 | −0.0710 |
| Underlying | Long call + short put + long bond (lend PV(K)) | 51.9290 | 52.0000 | −0.0710 |
| Bond paying K | Long put + long underlying + short call | 49.1000 | 49.0290 | 0.0710 |
Each row rearranges c + PV(K) = p + S − PV(D). A long bond means lending PV(K) today to receive K at expiry; a short bond means borrowing it. When a synthetic costs less than the real thing, buy the synthetic and sell the real one.
The two lines lie on top of each other: both packages pay the greater of S_T and K. Below the strike the bond (or the put) delivers K; above it the call (or the share) delivers S_T. Same payoff, so the same price today.