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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:

  1. 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?
  2. 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?
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  • $\begingroup$ What about time to maturity? $\endgroup$ – Arshdeep May 31 at 21:26
  • $\begingroup$ Yeah, I forgot it. It is 2 months. I am going to edit it. $\endgroup$ – D. Dedov May 31 at 21:28
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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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  • $\begingroup$ Could you explain in more details? My concern is that with the given information I get two different strike prices. I expect to have one as I assume Black-Scholes and put-call parity are synchronised. So, if one of the models violates the other, then there should be a reason. I cannot understand that reason. $\endgroup$ – D. Dedov Jun 1 at 20:30
  • $\begingroup$ Call put parity is not a model, it is a fundamental truth. B-S is a model. The price you get from parity is the true strike. If the BS model corresponding to this strike gives you a different price, then the market does not follow B-S dynamics. $\endgroup$ – Arshdeep Jun 2 at 10:12

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