Option Replication Using Put–Call Parity

A protective put on a stock costs the same as a covered call funded by a risk-free bond. That equivalence is put-call parity, and it lets you build any one of four instruments from the other three.

Two portfolios with identical payoffs at expiration must cost the same today, otherwise arbitrage exists. Put-call parity builds two such portfolios.

Portfolio A: hold one share of stock and one European put with strike X. At expiration the position pays $\max (S_T,X)$. If the stock finishes above X, keep the stock and let the put expire. If below X, exercise the put and receive X.

Portfolio B: hold one European call with strike X and a zero-coupon bond paying X at expiration. At expiration this position also pays $\max (S_T,X)$. If the stock finishes above X, exercise the call paying X from the bond. If below, let the call expire and keep X.

Identical payoffs mean identical prices.

$$S_0+p_0=c_0+\frac{X}{(1+r{)}^T}$$

Common mistakes

• Using parity on American options. The identity holds only for European options. American options carry early-exercise value, so strict equality fails. Trap: a question gives American puts on a non-dividend stock and asks for an exact parity price.
• Forgetting to discount the strike. Parity uses $X/(1+r{)}^T$, not bare $X$. Writing $S+p=c+X$ at one-year maturity with $X=\$50$ and $r=4\%$ overstates the right side by roughly $1.92.
• Ignoring dividends on the spot version. If the stock pays dividends, the spot side becomes $S_0-PV(D)+p_0$. Forgetting $PV(D)$ overstates the put or understates the call.

Bottom line

• Spot put-call parity: $S_0+p_0=c_0+X/(1+r{)}^T$. Long stock plus long put equals long call plus a risk-free bond paying X.
• Any one of the four instruments (call, put, stock, bond) is synthesized from the other three by rearranging the equation; synthetic stock is long call, short put, long bond.
• Parity is enforced by arbitrage: when the two sides differ, buy the cheap side and short the rich side until equality returns.
• Forward parity: $F_0(T)/(1+r{)}^T+p_0=c_0+X/(1+r{)}^T$. Substitute the PV of the forward for spot.

Exam shortcut

When forward equals strike, call equals put. Skip the algebra and pick equal premiums. When asked which side to buy in an arbitrage, the side with the smaller sum is the cheap side. Buy it and short the rich side. For synthetic stock, the mnemonic is long Call, short Put, long Bond. Conversion arbitrage flips it: short call, long put, short bond.

The full lesson (about 1,726 words, 12 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• option replication put-call parity

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