Put-Call Parity Explained: The One Derivatives Formula That Answers Half the Questions

Derivatives is the smallest topic at Level 1 by weight, at around 6% of the exam. Put-call parity is why it is also one of the most efficient topics to prepare. A large share of the questions can be answered by rearranging one equation.

The formula

c + X/(1 + r)^T = p + S₀

Where c is the call premium, p the put premium, X the strike price, S₀ the current price of the underlying, r the risk-free rate and T the time to expiration. Both options are European, on the same underlying, with the same strike and the same expiration.

The left side is often called the fiduciary call: a call option plus enough cash invested at the risk-free rate to grow to exactly the strike price by expiration. The right side is the protective put: the underlying asset plus a put option on it.

Why it must hold

The reason is worth understanding rather than memorising, because conceptual questions test it directly.

Consider both portfolios at expiration, under both possible outcomes.

If the asset finishes above the strike (S_T > X):

• Fiduciary call: exercise the call using the cash, which has grown to X. You hold the asset, worth S_T.
• Protective put: the put expires worthless. You hold the asset, worth S_T.

If the asset finishes below the strike (S_T < X):

• Fiduciary call: the call expires worthless. You hold cash of X.
• Protective put: exercise the put, selling the asset for X. You hold cash of X.

The two portfolios produce identical payoffs in every state of the world. Two portfolios with identical future payoffs must have identical prices today, or a riskless profit is available. That is the entire argument, and it rests on no assumption about volatility, probability or investor preferences — which is why put-call parity is stronger than any option pricing model.

Solving for a missing component

Nearly every parity question gives you three of the four components and asks for the fourth. Rearrange:

• Call: c = p + S₀ − X/(1 + r)^T
• Put: p = c + X/(1 + r)^T − S₀
• Underlying: S₀ = c + X/(1 + r)^T − p
• Bond: X/(1 + r)^T = p + S₀ − c

Worked example. A stock trades at $48. A one-year European call with a $50 strike costs $4.20. The risk-free rate is 5%. What is the put worth?

PV of strike = 50 / 1.05 = $47.62

p = c + PV(X) − S₀ = 4.20 + 47.62 − 48 = $3.82

Sanity check: the call is out of the money and the put is in the money, yet the call costs more. That is correct — the time value of money makes the right to pay $50 in a year more valuable than the intrinsic comparison suggests.

Synthetic positions

Each rearrangement describes a way to construct one instrument from the others. Exam questions often ask which combination replicates a given position.

• Synthetic call = long put + long underlying + short bond
• Synthetic put = long call + long bond + short underlying
• Synthetic underlying = long call + long bond + short put
• Synthetic bond = long put + long underlying + short call

The pattern: whatever you are solving for goes on one side; anything moving across the equals sign flips sign, and a negative sign means a short position. That single rule reproduces all four without memorising them separately.

How arbitrage questions are built

The question gives you all four components with prices that do not satisfy the equation, and asks how to profit.

The method is mechanical:

1. Compute both sides of the equation.
2. Identify which side is cheaper.
3. Buy the cheap side, sell the expensive side. The difference is your riskless profit today.

Example. Fiduciary call side = $9.50. Protective put side = $9.90. The fiduciary call is cheaper, so buy the call and the bond, and short the put and the underlying. You collect $0.40 immediately, and the positions offset perfectly at expiration.

Candidates lose these marks by reasoning about which option is "undervalued" instead of comparing the two sides. Do not think about the options individually — compute both sides and buy the cheaper one.

Traps

The strike must be discounted. The bond component is X/(1 + r)^T, not X. Using the undiscounted strike is the single most common arithmetic error here.

European options only. Parity holds exactly for European options. American options can be exercised early, which breaks the equality into an inequality. If a question specifies American options, parity in this form does not apply.

Same everything. Both options must share the same underlying, strike and expiration. A question presenting options with different strikes is testing whether you notice.

Dividends shift the equation. Where the underlying pays a dividend before expiration, the present value of those dividends is subtracted from S₀. At Level 1 this appears more often as a concept than a calculation, but recognise the direction.

Why this is the highest-yield formula in Derivatives

Option pricing models are conceptually harder and less frequently examined at Level 1 in computational form. Put-call parity, by contrast, produces valuation questions, arbitrage questions, synthetic position questions and conceptual questions about no-arbitrage pricing — all from one equation you can write down in ten seconds.

For a topic worth roughly 6% of the exam, that is an unusually good return on memorisation. The rest of the highest-value formulas are listed by tier in the formula sheet, and the no-arbitrage logic here underpins forward and futures pricing as well.