Put Call Parity and the Arbitrage That Enforces It

Put call parity ties a call and a put together through the discounted strike, and a 31 cent gap is enough to trigger a risk free trade.

Put call parity states that for a call and a put on the same stock, same strike and same expiry, call price minus put price equals the stock price minus the strike price discounted at the risk free rate. The relationship holds because a long call combined with a short put replicates owning the stock financed at the strike, and any gap between the two sides can be captured with a trade that locks in a profit today regardless of where the stock finishes. Candidates preparing for a derivatives round on the interview questions bank should expect to reproduce this relationship with an actual discount factor rather than the shortcut version that ignores financing.

The formula has three pieces. The stock price is what it is quoted at right now. The strike is fixed by the contract. The discount factor takes the strike and shrinks it slightly to reflect that a dollar received at expiry is worth a little less than a dollar in hand today, written as the strike multiplied by a number just under one, and either a continuous or a simple compounding convention gives close to the same answer over a few weeks. The right hand side of the equation, stock price minus the discounted strike, is the fair value of a forward contract on the stock struck at that same strike.

The left hand side, call price minus put price, is the cost of the same exposure built from options: buy the call, sell the put, and the combination pays exactly the stock price minus the strike at expiry no matter which way the stock moves, because whichever option finishes in the money delivers that outcome. Since both sides replicate the identical payoff at expiry, they must cost the identical amount today, or a trader can buy the cheap side and sell the expensive side and collect the difference for free.

Take a stock quoted at $102 with a call struck at $100 trading at $5.20 and 30 days to expiry. Assume a 5 percent annualized risk free rate. Over 30 days that works out to a discount factor of about 0.9959, so the discounted strike is $100 times 0.9959, or $99.59. Stock price minus the discounted strike is $102 minus $99.59, which comes to $2.41. Put call parity says the call minus the put should equal that $2.41, so the put should be priced at $5.20 minus $2.41, which is $2.79.

If the same put is instead quoted at $3.10, it is trading 31 cents above its fair value once financing is correctly applied. That gap is not noise. It is a standing arbitrage a trader can capture with a defined trade, and the size of the edge scales with how many contracts a market maker is willing to trade against the mispricing before the put price is pushed back toward $2.79.