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.

$$C - P = S - Ke^{-rT}$$

Equivalently:

$$C + Ke^{-rT} = P + S$$

Portfolio A: buy call + invest $Ke^{-rT}$ 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 $C - P = S - Ke^{-rT}$ holds for European options only and requires no volatility assumptions, it follows purely from no-arbitrage.

Common mistakes

• Using $K$ instead of $Ke^{-rT}$. 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. $P = C - S + Ke^{-rT}$, not $C + S - Ke^{-rT}$. Flipping signs gives a put that is far too large or negative. Trap: a negative put price.
• Mismatching exercise styles. The equality used here requires European exercise for both options. Do not mix a price from a different exercise style into the equation.

Bottom line

• Parity (European, non-dividend): $C - P = S - Ke^{-rT}$, equivalently $C + Ke^{-rT} = P + S$.
• Find a missing price: $P = C - S + Ke^{-rT}$, or $C = P + S - Ke^{-rT}$.
• Always discount the strike to its present value $Ke^{-rT}$, 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 $P = C - S + Ke^{-rT}$ (one line of algebra. Also use parity as a check: $C - P$ must equal $S - Ke^{-rT}$. 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 1,879 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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