Put-Call Parity Arbitrage
There's an "iron law" linking a call and a put with the same strike. When prices stray from it, combine three legs and the payoff at expiry is a flat line.
Risk warning: forex, CFDs, options, futures and crypto assets usually involve leverage or high volatility, and you can lose more than you put in. This lesson only teaches the concepts; all numbers are for illustration and are not investment advice.
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Text version · Arbitrage Strategies, lesson 3: “Put-Call Parity Arbitrage”
There's an "iron law" linking a call and a put with the same strike. When prices stray from it, combine three legs and the payoff at expiry is a flat line.
Buy call + sell put = synthetic stock
Same strike, same expiry: when the stock rises the call gains; when it falls the put you sold loses. Put them together and the P&L is exactly the same as holding the stock. This is a synthetic stock.
Both sides must be equal
Since synthetic stock and real stock are identical at expiry, their prices today must line up too: call − put = stock − present value of strike. This is put-call parity. Whichever side is off, sell the expensive one and buy the cheap one.
Dividends? Subtract them first
Parity applies to European options. If the stock pays a dividend before expiry, holding the stock gets it but the synthetic stock doesn't, so the equation becomes: call − put = stock − PV of dividends − PV of strike. Forget to subtract the dividend and you'll mistake a normal price for an arbitrage opportunity.
Key takeaways
- Long call + short put (same strike, same expiration) = a synthetic long stock.
- Put-call parity (European options, no dividends): call − put = stock price − present value of the strike. With dividends, also subtract the present value of the dividends.
- Call too expensive: sell the call, buy the put, buy the stock (a conversion). Put too expensive: do the opposite (a reversal). Either way the payoff at expiry is fixed.
Try it yourself
You can do these exercises in this lesson's interactive simulation:
- Click "Example: call 10 · put 5": how far is call − put above parity? How much profit is locked in at expiry?
- Drag the stock price at expiry from $50 to $150: how does each of the three dashed lines move? Does the green total line change?
- Raise the put to $9: which way does the arbitrage go now? How do today's and expiry's cash flows flip?
- Switch to "Synthetic stock": why are the two lines parallel? Click "Reset to parity" and watch them overlap.
Quiz and explanations
- 1. The stock is $100, the strike is $100 and the strike's present value is $96. Under put-call parity, what should call − put equal?
- Answer: $4.$100 − $96 = $4.
- 2. Long call + short put (same strike, same expiration) is equivalent to…
- Answer: A synthetic long stock.Whether the stock rises or falls, this combination's P&L matches holding the stock exactly.
- 3. The stock is $100, the strike's present value is $96, the call is $8 and the put is $5. Call − put = $3, which is $1 less than parity's $4. What should you do?
- Answer: Short the stock, sell the put, buy the call (a reversal).The call is relatively cheap and the put relatively expensive: buy the cheap call, sell the expensive put, and short the stock to hedge, locking in the $1 mispricing when you open.
Terms in this lesson
- conversion (Conversion): When the call is relatively expensive: sell the call, buy the put and buy the stock for a fixed payoff at expiry.
- dividend (Dividend): Part of a company's profit paid out to shareholders according to how many shares they hold.
- put-call parity (Put-Call Parity): A call and a put with the same strike and expiry satisfy: call − put = stock price − present value of the strike.
- reversal (Reversal): When the put is relatively expensive: buy the call, sell the put and short the stock for a fixed payoff at expiry.
- synthetic stock (Synthetic Stock): Buying a call + selling a put (same strike, same expiry), with exactly the same P&L as owning the stock.