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we've seen that the payoff diagram at option expiration for owning the stock plus a put of on that stock at some strike price that this payoff diagram will look exactly the same as the payoff diagram as owning a call option on that stock at the same strike price as a put option plus a bond that's going to be worth the strike price at the time of expiration one point of clarification and actually let me just draw that payoff diagram just so you know what it looks like so if this is the underlying stock price stock price and let's just do the payoff diagram when we look at the value of the of the portfolio or your holdings what it looks like is if this is the strike price over here if the stock is at the strike stock price or anything lower you make a fixed value and anything above that you get some type of upside so either if you have the stock or the put you have this type of a payoff or if you have the call plus the bond the one point of clarification I want to make is that you can only say that this is definitely true for European for European call options for European options I should say in general European call and put options because only in European options can you know for certain for for sure that the options are going to be exercised on the X or they can only be exercised on the expiration date if we're dealing with American options either party could either the on the put side or the call side could exercise their options earlier so it becomes a lot more complicated so normally when we kind of deal with the mathematics of options we're dealing with European options so even in the example where we did our put-call parity arbitrage where we were able to make that free five dollars the implicit assumption I made is that we were dealing with European options that the options could only be exercised at the expiration date