Put-Call Parity Applications

Put-call parity is a currency converter between payoffs—transforming calls into puts and stock into synthetics through a precise mathematical relationship. When prices deviate from parity, the gap signals either arbitrage opportunity or hidden costs (dividends, borrow, rates) that explain the deviation.

Parity with Dividends and Borrow Costs

The standard put-call parity relationship is:

C - P = S - K × e^(-rT)

Where: C = call price, P = put price, S = spot price, K = strike, r = risk-free rate, T = time to expiration.

For dividend-paying stocks, the relationship adjusts to:

C - P = S × e^(-qT) - K × e^(-rT)

Where q = continuous dividend yield.

When stock borrowing is required for short positions, an additional borrow cost (b) applies:

C - P = S × e^(-(q+b)T) - K × e^(-rT)

Hard-to-borrow stocks can have borrow costs of 5-50% annualized, significantly affecting parity calculations. If XYZ has a 10% borrow cost and a trader ignores it, the synthetic position will appear mispriced by approximately 10% × T in the parity equation.

Synthetic Short Stock Construction

A synthetic short stock replicates the payoff of being short 100 shares:

Position: Long put + Short call (same strike and expiration)

Example:

• XYZ stock: $100
• 30-day ATM options
• Strike: $100
• Call price: $3.50
• Put price: $3.00
• Rate: 5%
• Dividend: 0%
• Borrow cost: 8% (annualized)

Parity check: C - P = $3.50 - $3.00 = $0.50

Expected from parity: S - K × e^(-rT) = $100 - $100 × e^(-0.05 × 30/365) = $100 - $99.59 = $0.41

Gap: $0.50 - $0.41 = 9 bps

The 9 bps gap is explained by the borrow cost embedded in the options market. The put is "expensive" relative to the call because short sellers face 8% borrow costs.

Build, Test, Execute Workflow

1. Build the synthetic: Identify strike, calculate theoretical parity value using current rates, dividends, and expected borrow costs. Compare to market prices.

2. Test for arbitrage: If market synthetic differs from theoretical by more than transaction costs (typically 3-5 bps), investigate. Check for special dividends, hard-to-borrow status, or rate changes.

3. Execute the trade: If genuine mispricing exists, execute both legs simultaneously. For synthetic short, sell call and buy put in single spread order to minimize leg risk.

American Early Exercise Caveats

Put-call parity holds exactly for European options. American options introduce early exercise optionality:

• Calls on dividend stocks: May be exercised early before ex-date to capture dividend
• Deep ITM puts: May be exercised early to invest strike proceeds at risk-free rate

These optionalities mean American option prices can deviate from European parity bounds. The deviation represents the early exercise premium.

Practical implication: For American equity options, expect parity deviations of 0-2% for deep ITM options. Treat larger deviations as potential signals, but verify that early exercise doesn't explain the gap.

Shorting rules require compliance with Reg SHO locate requirements. Before executing synthetic short stock positions, confirm that shares are available for borrow and that locate has been obtained. Failure to locate can result in forced buy-ins and regulatory penalties.

Parity Monitor Alerts

Trading desks use automated parity monitors to flag potential opportunities and errors:

Alert triggers:

• Parity gap > 5 bps after adjustments: Review for execution opportunity
• Parity gap > 20 bps: Check for data error, corporate action, or special situation
• Persistent gap over 2+ hours: Investigate market dislocation

Remediation workflow: When a parity alert fires, the desk verifies input data (rates, dividends, borrow), confirms option prices are executable (not stale quotes), and assesses whether the gap is tradeable after costs. If genuine, the position is sized based on available capital and executed via spread order. Post-execution, the desk monitors convergence and closes when parity normalizes.

Key Data Points

Parity Calculation Example:

Synthetic price: C - P = $0.50 Theoretical: S × e^(-(q+b)T) - K × e^(-rT) = $99.17 - $99.59 = -$0.42

Gap: $0.50 - (-$0.42) = 92 bps

This large gap reflects the combined effect of high borrow cost and dividend—the synthetic short is expensive because shorting the stock is expensive.

Next Steps

For the arbitrage principles underlying parity, see No-Arbitrage Principles in Derivatives.

To understand option fundamentals, review Call vs. Put Options: Payoffs and Use Cases.