Put-Call Parity

You know the call price. A colleague asks for the put. You could run the entire Black-Scholes calculation again, or use one equation. Put-call parity relates call and put prices directly, requires no volatility assumptions, and follows purely from no-arbitrage. On the FAM exam, it saves time and catches errors. If your call and put don't satisfy parity, at least one is wrong.

HIGH-FREQUENCY: The put-call parity equation and its use to find one option price from the other appear on most FAM sittings.

$$C-P=S-K{e}^{-rT}$$

Equivalently:

$$C+K{e}^{-rT}=P+S$$

Portfolio A: buy call + invest $K{e}^{-rT}$ in bonds. Portfolio B: buy put + buy stock.

At expiration, both pay $\max (S_T,K)$ in every scenario. Equal payoffs, equal costs.

KEY: Put-call parity $C-P=S-K{e}^{-rT}$ holds for European options only and requires no volatility assumptions, it follows purely from no-arbitrage.

Common mistakes

• Using $K$ instead of $K{e}^{-rT}$. Parity uses the present value of the strike. Using $K=50$ instead of 48.04 changes the answer by nearly $2. Trap: a put price too low by the discount.
• Getting the sign wrong on rearrangement. $P=C-S+K{e}^{-rT}$, not $C+S-K{e}^{-rT}$. Flipping signs gives a put that is far too large or negative. Trap: a negative put price.
• Applying parity to American options. The equality holds for European options only. American options satisfy $S-K\le C-P\le S-K{e}^{-rT}$. Trap: using European parity on American prices.

Bottom line

• Parity (European, non-dividend): $C-P=S-K{e}^{-rT}$, equivalently $C+K{e}^{-rT}=P+S$.
• Find a missing price: $P=C-S+K{e}^{-rT}$, or $C=P+S-K{e}^{-rT}$.
• Always discount the strike to its present value $K{e}^{-rT}$, never the raw $K$.
• Synthetic positions: any one of {C, P, S, bond} equals a combination of the other three.

Exam shortcut

After computing a Black-Scholes call, never recompute the put from scratch. Use $P=C-S+K{e}^{-rT}$ (one line of algebra. Also use parity as a check: $C-P$ must equal $S-K{e}^{-rT}$. For arbitrage questions, compute both sides first, then determine which is larger to set the direction. Remember: "Call minus Put = Stock minus PV(Strike)") read left to right. "CPSK" (Call, Put, Stock, K(discounted)) any three replicate the fourth.

The full lesson (about 2,006 words, 13 min read) adds 3 worked examples, all 5 common mistakes, a self-check, free in the app.

Learning objectives

• 6d

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