I am learning my options and from what I read it seems that put-call parity is regarded as only being applicable to European options because the time to exercise is known. American options, on the other hand, are said to be exempt from conforming to put-call parity because they can be exercised at anytime leading up to expiry, causing a risk of being called by the other party early and leaving an investor trying to arbitrage with one side of the pair required to arbitrage gone.

Examples of put-call arbitrage with European options:

A) If Call/Put price is greater than what is implied by BSM (i.e calls are overpriced)

  1. Sell a call with strike (K) and expiry (T) at market price

  2. Create a synthetic call to go long:

  • borrow an amount equal to the strike (K)
  • buy a Put with the same strike (K) and expiry (T) as the Call
  • buy a number of futures contracts on the underlying asset equal to the Puts purchased

B) If Call/Put price is less than what is implied by BSM (i.e puts are overpriced)

  1. Sell a put with strike (K) and expiry (T) at market price

  2. Create a synthetic put to go long:

  • sell a future on the underlying asset
  • buy a Call with the same strike (K) and expiry (T) as the Put
  • buy treasuries with an amount equal to the strike (K)

That seems to be the theory, but I am wondering how difficult it is for markets to arbitrage American options using the same process in practice. First, most American options are not exercised early in practice and arbitrageurs could diversify some of the risk associated with the chance of early exercise by creating a highly diversified basket of options pairs to arb. Second, if part of the arb position is exercised before expiry by the counterparty I would think it shouldn't be hard to create algos that can also immediately sell out of the now unhedged part of the arb position and roll it, keeping the trade fully hedged to a very small window of error (algos should be able to keep the time difference between exercise and selling out down to fractions of a second)

I am thinking out loud here and have no idea, but my question is whether put-call parity holds with American options in practice (at least to some degree) or if the limits to arbitraging are actually higher than I imagine and why that's the case. Any insights are greatly appreciated!

  • $\begingroup$ See here on Page 4 for what Put Call Parity for American options looks like. math.nyu.edu/~cai/Courses/Derivatives/lecture8.pdf It is an inequality rather than an equation, but still has to hold to prevent arbitrage. There is some "wiggle room" for the difference in price between the Call and the Put in the American case. $\endgroup$
    – nbbo2
    Jul 10 at 8:40


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