The seesaw of synthetic stock

Put-call parity is a fundamental no-arbitrage identity linking the prices of European call and put options with the same strike K and expiration date T on a non-dividend paying stock S: C - P = S - K * discount. For fast mental arithmetic and trading interviews, interest rates are assumed zero (r = 0, discount = 1), simplifying the relation to C - P = S - K. This identity allows instant pricing of any missing leg: P = C - S + K or C = P + S - K. If market prices violate this equality, a riskless conversion or reversal arbitrage can be executed.

Words you need

Option: A contract that gives the right, but not the obligation, to buy or sell something at a fixed price by a set date.
Call: An option to BUY at the strike price. Worth having if the price ends above the strike.e.g. Strike 100, stock ends at 112: the call pays 12.
Put: An option to SELL at the strike price. Worth having if the price ends below the strike.e.g. Strike 100, stock ends at 90: the put pays 10.
Strike: The fixed price written into an option at which you may buy (call) or sell (put).
Expiry: The date the option ends; its payoff is settled then.
Underlying: The stock or asset the option is written on.

Why it works

A portfolio of long call plus short put replicates the exact terminal payoff of forward stock minus strike K.

C - P = S - K (assuming zero interest rates and no dividends)

Spot it: Pricing an unknown put or call given the opposite option price and the underlying stock price.

Technique reference and sources

Think of Long Call (+C) and Short Put (-P) as two ends of a seesaw. Together they construct an identical exposure to owning the stock (+S) financed by borrowing the strike (+K at zero rates). If the call is priced above parity, the seesaw tilts, and selling the expensive side while buying the cheap side locks in riskless cash.

When this method fits

Pricing an unknown put or call given the opposite option price and the underlying stock price.
Detecting parity mispricings between options and the underlying stock.
Evaluating synthetic stock positions during rapid trading-floor calculations.
Verifying option book consistency under zero or non-zero interest rate environments.

Identify market inputs. Identify S, K, and the known option price (C or P).
Calculate intrinsic difference. Compute S - K (positive if S > K, negative if S < K, zero if ATM).
Solve for missing put leg. Rearrange parity: P = C - (S - K) = C - S + K.
Solve for missing call leg. Rearrange parity: C = P + (S - K) = P + S - K.

At-the-money parity: When S = K, S - K = 0. Under zero rates, the fair call price must identically equal the fair put price: C = P.

S = 80, K = 80, C = 5.25: fair P = 5.25
S = 100, K = 100, P = 7.00: fair C = 7.00

In-the-money call adjustment: When S > K, the call must exceed the put by exactly S - K.

S = 52, K = 50, C = 4.50: P = 4.50 - 2.00 = 2.50
S = 65, K = 60, C = 8.00: P = 8.00 - 5.00 = 3.00

Common traps

Adding the intrinsic difference instead of subtracting it when finding the put price (computing 4.50 + 2.00 = 6.50). When S > K, the call is in the money and must be worth more than the put. The put must equal Call - (S - K).

Forgetting that S - K is negative when the stock price is below the strike. For S < K, S - K is negative, so adding S - K correctly reduces the call price relative to the put.

Applying European parity without adjustment to early-exercisable American puts with high interest rates. American puts carry early exercise optionality that can make them trade slightly above European parity bounds.

Another example: At-the-money parity equivalence

State parity relation: C - P = S - K
For an ATM option, S - K = 80 - 80 = 0.00
Therefore P = C = 5.25

Source notes

These explanations and questions are written for QuantArc. Links show the underlying mathematics and learning design.

John C. Hull, Options, Futures, and Other Derivatives (11th Edition)
Recommended textbook reference. Publisher overview and table of contents checked for option properties, binomial trees, Black-Scholes-Merton and Greeks; full derivations and exercises have not been audited against these lessons. Lesson examples and exercises are original.

OCC: Characteristics and Risks of Standardized Options (ODD)
Official Options Clearing Corporation disclosure document outlining standardized option contract specifications, exercise settlement, payoff obligations, and intrinsic value mechanics.

IES: Organizing Instruction and Study to Improve Student Learning
Supports worked examples, retrieval, explanations and spaced study. Does not establish QuantArc's exact lesson length, mastery threshold or review intervals.