Put-Call Parity: Formula, Derivation, and Arbitrage Worked Examples
Put-call parity is the fundamental relationship linking European call options, put options, the underlying stock, and risk-free bonds. When the relationship breaks, arbitrage opportunities exist. Learn the formula, the intuition, the arbitrage setup, and the adjustments for dividends and American options.
What You'll Learn
- ✓State the put-call parity formula for European options on non-dividend-paying stock
- ✓Derive put-call parity using a replicating portfolio argument
- ✓Identify arbitrage opportunities when put-call parity is violated
- ✓Apply the dividend adjustment for stocks paying discrete dividends
- ✓Understand why put-call parity is a bound (not equality) for American options
1. Direct Answer: The Put-Call Parity Equation
For European options on a non-dividend-paying stock with the same strike price K and same expiration T, put-call parity states: C + K × e^(-rT) = P + S₀ Where: C = price of European call P = price of European put S₀ = current stock price K = strike price (same for call and put) r = continuously compounded risk-free rate T = time to expiration in years K × e^(-rT) = present value of the strike price Intuition: the left side is a portfolio of a call plus a zero-coupon bond paying K at expiration. At expiration, if the stock is above K, the call pays S_T - K and the bond pays K, for a total of S_T. If below K, the call pays 0 and the bond pays K. In both cases the payoff equals max(S_T, K). The right side is a portfolio of a put plus the stock. At expiration, if the stock is below K, the put pays K - S_T and the stock is worth S_T, for a total of K. If above K, the put expires worthless and the stock is worth S_T. In both cases the payoff equals max(S_T, K). Two portfolios producing identical payoffs in every state must have identical prices today — the law of one price. That is put-call parity. This content is for educational purposes only and does not constitute financial advice.
Key Points
- •Formula: C + K × e^(-rT) = P + S₀ for European options, no dividends
- •Both sides of the equation produce max(S_T, K) payoff at expiration
- •Law of one price: same payoff means same price
- •Rearranging: C - P = S₀ - K × e^(-rT)
- •Parity holds strictly for European options on non-dividend stocks
2. Derivation via Replicating Portfolio
Consider two portfolios at time 0: Portfolio 1: long 1 call + long 1 zero-coupon bond paying K at time T. Cost = C + K × e^(-rT). Portfolio 2: long 1 put + long 1 share of stock. Cost = P + S₀. At expiration (time T): State A: S_T ≥ K (stock above strike) Portfolio 1: call pays S_T - K; bond pays K. Total: S_T. Portfolio 2: put expires worthless; stock worth S_T. Total: S_T. State B: S_T < K (stock below strike) Portfolio 1: call expires worthless; bond pays K. Total: K. Portfolio 2: put pays K - S_T; stock worth S_T. Total: K. Identical payoffs in both states. No-arbitrage requires identical prices today: C + K × e^(-rT) = P + S₀ Rearranging produces several useful forms: C - P = S₀ - K × e^(-rT) (call minus put equals stock minus PV of strike) P = C - S₀ + K × e^(-rT) (solve for put price from call price) C = P + S₀ - K × e^(-rT) (solve for call price from put price) The relationship holds strictly for European options (exercise only at expiration) on non-dividend-paying stocks. Adjustments for dividends and American exercise are covered in later sections. This content is for educational purposes only and does not constitute financial advice.
Key Points
- •Two portfolios produce identical payoffs in every state
- •No-arbitrage requires identical prices (law of one price)
- •The derivation is a no-arbitrage argument, not a pricing model
- •Rearrange to solve for any missing variable
- •Derivation assumes continuous compounding; discrete form uses (1+r)^T
3. Worked Example: Calculating the Fair Put Price
Given: Stock price S₀ = $50 Strike K = $52 Time to expiration T = 6 months = 0.5 years Risk-free rate r = 4% continuously compounded Call price C = $3.20 Stock pays no dividends Find fair put price. Parity: C + K × e^(-rT) = P + S₀ Rearrange: P = C + K × e^(-rT) - S₀ Calculate K × e^(-rT): 52 × e^(-0.04 × 0.5) = 52 × e^(-0.02) = 52 × 0.9802 = $50.97 Solve for P: P = 3.20 + 50.97 - 50 = $4.17 The fair put price is $4.17. If the market put is trading at a materially different price (after accounting for transaction costs), arbitrage opportunities exist. For the inverse calculation (given put price, find fair call price): C = P + S₀ - K × e^(-rT) C = 4.17 + 50 - 50.97 = $3.20 ✓ This content is for educational purposes only and does not constitute financial advice.
Key Points
- •Calculate PV of strike: K × e^(-rT) or K/(1+r)^T for discrete
- •Rearrange parity to solve for the missing price
- •Fair price often differs materially from market price for illiquid options
- •Parity gives a theoretical value; transaction costs dictate practical bounds
- •Always verify computations work in both directions (solve for C from P)
4. Arbitrage When Parity Is Violated
Market data: Stock: $100; Strike: $100; Time: 3 months = 0.25 years; Risk-free rate: 5% Call $5.00; Put $3.50; No dividends. Test parity: Left side: C + K × e^(-rT) = 5.00 + 100 × e^(-0.0125) = 5.00 + 98.76 = $103.76 Right side: P + S₀ = 3.50 + 100 = $103.50 Left > Right by $0.26. Left side (call + bond) is overvalued relative to right side. Arbitrage strategy: 1. Sell overvalued portfolio: sell 1 call ($5.00 credit), borrow $98.76 at risk-free (receive $98.76 today, owe $100 at expiration). 2. Buy undervalued portfolio: buy 1 put ($3.50 debit), buy 1 share ($100 debit). Net cash today: 5.00 + 98.76 - 3.50 - 100 = $0.26 (received upfront). At expiration (T = 0.25 years): - If S_T ≥ 100: call you sold is exercised (deliver stock at $100, receive $100). Pay off bond obligation ($100 owed). Your put expires worthless. Your stock was called away. Position closed. - If S_T < 100: call you sold expires worthless. Exercise your put (sell stock at $100). Pay off bond obligation ($100 owed). Position closed. In both states, you receive $100 from either the call exercise or put exercise, which exactly offsets the $100 bond obligation. The $0.26 received upfront is risk-free profit. Professional arbitrageurs monitor parity continuously and execute these trades when deviations exceed transaction costs. This pressure keeps option prices tightly constrained to parity. In liquid markets, persistent deviations rarely exceed a few cents. This content is for educational purposes only and does not constitute financial advice.
Key Points
- •Compare left side (call + bond) to right side (put + stock)
- •Overvalued side: sell; undervalued side: buy
- •Riskless profit equals the magnitude of parity violation
- •Arbitrage enforces parity in real markets
- •Transaction costs, bid-ask spreads, and borrow costs limit profitability
5. Dividend Adjustment
Stocks paying dividends require adjustment because stockholders receive dividends but option holders do not. Discrete dividends: C + K × e^(-rT) = P + S₀ - PV(dividends) Where PV(dividends) is the present value of all dividends paid before expiration. Continuous dividend yield (common for indices): C + K × e^(-rT) = P + S₀ × e^(-qT) Where q is the annualized dividend yield. Example with discrete dividend: Stock $80, strike $80, time 6 months, risk-free 4%, call $5.20. Stock pays $1.00 dividend in 3 months. PV(dividend) = $1.00 × e^(-0.04 × 0.25) = $0.99 Parity: C + K × e^(-rT) = P + S₀ - PV(div) P = C + K × e^(-rT) - S₀ + PV(div) P = 5.20 + 80 × 0.9802 - 80 + 0.99 = 5.20 + 78.41 - 80 + 0.99 = $4.60 The dividend makes put values higher than they would be for non-dividend stocks because the stock price drops by the dividend amount after ex-dividend dates, reducing expected terminal stock value. This content is for educational purposes only and does not constitute financial advice.
Key Points
- •Discrete dividends: subtract PV(dividends) from S₀
- •Continuous dividend yield: use S₀ × e^(-qT)
- •Dividends increase put values relative to calls
- •Index options typically use continuous yield formulation
- •Individual stocks typically use discrete formulation with known dividend dates
6. American Options and Parity Bounds
American options allow exercise anytime before expiration. This changes parity from an equality to an inequality. For American options on non-dividend-paying stock: S₀ - K ≤ C_A - P_A ≤ S₀ - K × e^(-rT) Key fact: for non-dividend-paying stocks, it is never optimal to exercise an American call early. Early exercise means paying K today instead of at expiration, losing the time value of money, and giving up remaining time value in the option. Therefore, on non-dividend stocks, American call = European call (C_A = C_E). For American puts, early exercise can be optimal. Deep in-the-money puts near expiration, or puts when interest rates are high and the stock is very low, may be exercised to capture K and earn interest on it. Therefore P_A ≥ P_E, and the relationship is a strict inequality. For dividend-paying stocks, early exercise of American calls may be optimal just before ex-dividend dates to capture the upcoming dividend. In this case C_A ≥ C_E. Practical implication: when working with American options, use the inequality bounds rather than strict parity. Pricing models (binomial trees, finite difference methods) handle American exercise explicitly. This content is for educational purposes only and does not constitute financial advice.
Key Points
- •American options: parity becomes an inequality
- •Non-dividend stock: American call = European call (never exercise early)
- •American puts ≥ European puts (early exercise can be optimal)
- •Dividend-paying stock: American calls can exceed European calls
- •Use bounds for American options, not strict equality
7. Applications and Synthetic Positions
Put-call parity shows how to replicate one instrument with others. Rearranging the equation: Synthetic long stock: +Call - Put + PV(strike bond) Synthetic long call: +Stock + Put - PV(strike bond) Synthetic long put: +Call - Stock + PV(strike bond) Synthetic short stock: -Call + Put - PV(strike bond) These synthetic positions give traders flexibility when one instrument is expensive, unavailable, or restricted. Hard-to-borrow stocks sometimes make synthetic short positions preferable to actual short selling. Other applications: 1. Implied risk-free rate: solve for r given stock, call, put, and strike prices. Useful for sanity-checking quoted rates. 2. Implied dividend forecast: given stock, call, put, and risk-free rate, solve for the market's implied dividend expectation. 3. Relative value checks: compare listed option prices against parity-implied values to identify temporary mispricings. 4. Covered call vs protective put equivalence: a covered call (long stock + short call) and protective put (long stock + long put) have the same payoff profile up to the strike-bond adjustment. This equivalence, which surprised many traders before the formal derivation, is exactly what put-call parity predicts. For CFA and corporate finance exams: - Memorize the formula C + K × e^(-rT) = P + S₀ - Know the replicating portfolio derivation - Be able to set up arbitrage trades when parity is violated - Know the dividend adjustment - Understand why American calls equal European calls on non-dividend stocks but American puts can exceed European puts This content is for educational purposes only and does not constitute financial advice.
Key Points
- •Synthetic long stock: +Call - Put + PV(strike)
- •Synthetic long call: +Stock + Put - PV(strike)
- •Synthetic positions enable trades when direct instruments are restricted
- •Parity reveals covered call and protective put equivalence
- •Use parity to check relative value across listed options
Key Takeaways
- ★Put-call parity: C + K × e^(-rT) = P + S₀ (European options, no dividends)
- ★Dividend adjustment: C + K × e^(-rT) = P + S₀ - PV(dividends)
- ★Continuous dividend yield form: C + K × e^(-rT) = P + S₀ × e^(-qT)
- ★Derivation: two portfolios with identical payoffs must have identical prices
- ★Synthetic long stock: +Call - Put + PV of strike
- ★American call on non-dividend stock equals European call
- ★American put ≥ European put (early exercise can be optimal)
- ★Arbitrage when violated: sell expensive side, buy cheap side
- ★Professional arbitrageurs keep parity tight in liquid markets
- ★Transaction costs and bid-ask spreads set practical bounds
Practice Questions
1. Stock $40, strike $42, time to expiration 6 months, risk-free rate 3%, call price $2.00, no dividends. What is the fair put price?
2. Why is it never optimal to exercise an American call on a non-dividend-paying stock early?
3. Set up an arbitrage when put-call parity gives left side $105 and right side $106.50.
4. Stock pays $2 dividend in 4 months. Stock $80, strike $80, time 6 months, risk-free 5%, call $4.50. Find fair put price.
5. How do you synthetically create a long stock position using options and bonds?
FAQs
Common questions about this topic
Yes, very tightly. Professional arbitrageurs monitor parity continuously and execute trades when deviations exceed their transaction costs. In liquid markets (major stock options, index options), parity typically holds within a few cents. Persistent larger deviations usually reflect market frictions: hard-to-borrow stocks, bid-ask spreads, assignment risk for American options, or tax considerations. The relationship is a fundamental no-arbitrage constraint.
For European options (exercise only at expiration), put-call parity is a strict equality. For American options (exercise anytime before expiration), the relationship becomes an inequality. For non-dividend-paying stocks, American call = European call because early exercise is never optimal. American puts can exceed European puts because early exercise is sometimes optimal. For dividend-paying stocks, American calls may exceed European calls near ex-dividend dates.
Both strategies have equivalent payoff profiles, which parity explains. Covered call = long stock + short call. Protective put = long stock + long put. Rearranging parity: Stock - Call = Put - K × e^(-rT), meaning the covered call and protective put differ only by the strike-bond adjustment. Many traders view these strategies as interchangeable with appropriate cash/strike adjustments, and parity is the mathematical basis.
Not as a strict equality — use the inequality bounds: S₀ - K ≤ C_A - P_A ≤ S₀ - K × e^(-rT) for non-dividend stocks. For practical pricing of American options, use binomial trees or finite difference methods that explicitly handle early exercise. Parity gives useful bounds but not exact prices for American options.
Yes. Snap a photo of any options pricing problem and FinanceIQ applies put-call parity to solve for missing prices, tests for arbitrage opportunities when given all four prices (stock, call, put, and strike bond), handles dividend adjustments for discrete and continuous dividends, and explains the distinction between European and American parity bounds. It also walks through the arbitrage trade setup when parity is violated. This content is for educational purposes only and does not constitute financial advice.