Using Put-Call Parity to Identify an Options Arbitrage

Summary

The note considers European call and put options on a non-dividend-paying index, with stated market prices and assumptions, and asks whether a strike inferred from put-call parity should agree with one inferred through Black–Scholes. The proposed parity calculation produces a strike that, when inserted into Black–Scholes using the stated volatility, gives a different call value than the one supplied.

The answer emphasizes that the volatility assumption matters: if it is implied volatility, the price inconsistency signals an arbitrage under the setup. It identifies the call-minus-put portfolio as the trade, since parity implies a certain discounted payoff that exceeds its quoted cost. The key point is to assess the prices of the combined portfolio rather than insist that one individual option is uniquely mispriced. This conclusion depends on the stated assumptions and ignores practical frictions such as transaction costs, funding, and execution constraints.

Key ideas

• Put-call parity links European call and put prices to the underlying, strike, and discounting.
• A price discrepancy can be tested by comparing the cost of a portfolio with its parity-implied payoff.
• The answer identifies buying the call-minus-put portfolio as the arbitrage in the stated example.
• The arbitrage conclusion relies on the given assumptions and does not account for trading frictions.

Tags

# Strike Price Determination


# Strike Price Determination












Suppose you know the following: there are 2-month European call and put options on an index-like instrument with no dividends, the calculations show that the call option price is USD 10.1150, the spot index price is USD 120, the risk-free rate is 3% and the volatility is 35%. You happen to know that the put option price is USD 2.5664. What is then the strike price?

Now my questions:

- Is there a rule holding that the strike price should be the same regardless of whether it is derived from the Black-Scholes model or the put-call parity equation?

- Given the above information, I compute that the strike price is USD 113.02 using the put-call parity equation. However, when I go back to the Black-Scholes equation and compute the call option price with a strike price of USD 113.02, I end up with a call option price of USD 11.07 rather than USD 10.1150. Am I right to assume that an arbitrage opportunity exists? How can I determine what is the mispriced asset?

## Answer by Arshdeep (score 1)

https://quant.stackexchange.com/a/64328

If by volatility you mean the implied volatility then yes, an arbitrage oppurtunity exists. Buy the portfolio $(C-P)$ for 7.5486. In the future this will pay off a discounted value of 7.5536 with certainty due to call put parity.

The 'mispriced asset' is a portfolio which by construction delivers a riskless profit.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.