Not really because the PnL graph looks at profit/loss over the entire life of the option(s) not at any specific point in time.
Roger that. But here's the thing. What if the underlying stock suddenly spiked or tanked 25% in a day? Wouldn't the Euro-style option show a different PNL over the American, because the American style could be exercised at any time, which is more risky, show greater risk than Euro-style?
What zdreg said. This is why I hate you option people. Never a straight answer. Y'all are crazy fugs.
It says In practice it just means this: with American style options the traders set the implied volatility parameter (IV) slightly higher, which of course makes the option more expensive and this then causes the spread (Bid, Ask) getting wider; all undesirable effects. But the underlying maths is still the same for both American-style and European-style options, and I was meaning just this fact when I said both are the same except the exercising method. If that would not be the case then there would be some maths formula specific for the American-style options. But there are no different formula, both just use the same formula: Black-Scholes-Merton (BSM) option pricing model, also called just "Black-Scholes": https://en.wikipedia.org/wiki/Black–Scholes_model https://en.wikipedia.org/wiki/Black–Scholes_equation There are some academic attempts using a partially different formula, but in real-life practice the above said method using slightly higher IV gets used. And even the many option greeks use the same formula (that is: there exists just 1 set of formula): https://en.wikipedia.org/wiki/Greeks_(finance)
If their parameters are exactly the same, incl. the IV, then the result is the same as well. But in real-world-practice American-style options usually have a slightly higher IV, therefore different outcome. As everybody should know: IV is just a "subjective parameter": every trader can use a different IV... this then of course culminates in bidding/offering an individual Bid/Ask price resulting from the option pricing model using that IV... This might be interesting too: If IV is unknown, but all the other parameters are known, incl. Premium, then it is possible to find the corresponding IV to that parameter set. It just is maths, after all... S.a. https://en.wikipedia.org/wiki/Implied_volatility https://en.wikipedia.org/wiki/Volatility_smile#Implied_volatility_and_historical_volatility The following example (first link above) is good too, IMO, as it's so unintuitive, so unexpected... : " Example A call option is trading at $1.50 with the underlying trading at $42.05. The implied volatility of the option is determined to be 18.0%. A short time later, the option is trading at $2.10 with the underlying at $43.34, yielding an implied volatility of 17.2%. Even though the option's price is higher at the second measurement, it is still considered cheaper based on volatility. [...] " But I've not been able to replicate this result (but did not search long yet). I mean finding the matching full parameter sets to get this result. Anybody know how to achieve (replicate) the said result of the above example?
Using a brute-force computer search (some idis would call it Artifical Intelligence (AI) ) found the following matching parameter sets: Code: find_verification_for_wiki_example(): Found: DTE0=88.00 K=42.00 : DTE1=75.00 : S0=42.05 Pr0=1.50 IV0=18.00 : S1=43.34 Pr1=2.10 IV1=17.20 Found: DTE0=88.00 K=42.00 : DTE1=74.00 : S0=42.05 Pr0=1.50 IV0=18.00 : S1=43.34 Pr1=2.10 IV1=17.20 Found: DTE0=87.00 K=42.00 : DTE1=75.00 : S0=42.05 Pr0=1.50 IV0=18.00 : S1=43.34 Pr1=2.10 IV1=17.20 Found: DTE0=87.00 K=42.00 : DTE1=74.00 : S0=42.05 Pr0=1.50 IV0=18.00 : S1=43.34 Pr1=2.10 IV1=17.20 Ie. Strike=42. Then one of the following: first measurement when DTE=88 and second measurement when DTE=75 first measurement when DTE=88 and second measurement when DTE=74 first measurement when DTE=87 and second measurement when DTE=75 first measurement when DTE=87 and second measurement when DTE=74 Maybe more matching sets can exist when using a more finer granularity steps... I used step=1.0 for both the DTEs and K. So, "a short time later" in the example means in fact 12, 13, or 14 days later. Update: By using a finer granularity for K it finds much more matching parameter sets: Code: find_verification_for_wiki_example(): ... Found: DTE0=45.00 K=41.25 : DTE1=5.00 : S0=42.05 Pr0=1.50 IV0=18.00 : S1=43.34 Pr1=2.10 IV1=17.20 Found: DTE0=45.00 K=41.25 : DTE1=6.00 : S0=42.05 Pr0=1.50 IV0=18.00 : S1=43.34 Pr1=2.10 IV1=17.20 Found: DTE0=45.00 K=41.25 : DTE1=7.00 : S0=42.05 Pr0=1.50 IV0=18.00 : S1=43.34 Pr1=2.10 IV1=17.20 Found: DTE0=45.00 K=41.25 : DTE1=8.00 : S0=42.05 Pr0=1.50 IV0=18.00 : S1=43.34 Pr1=2.10 IV1=17.20 Found: DTE0=45.00 K=41.25 : DTE1=9.00 : S0=42.05 Pr0=1.50 IV0=18.00 : S1=43.34 Pr1=2.10 IV1=17.20 Found: DTE0=59.00 K=41.50 : DTE1=34.00 : S0=42.05 Pr0=1.50 IV0=18.00 : S1=43.34 Pr1=2.10 IV1=17.20 Found: DTE0=59.00 K=41.50 : DTE1=35.00 : S0=42.05 Pr0=1.50 IV0=18.00 : S1=43.34 Pr1=2.10 IV1=17.20 Found: DTE0=72.00 K=41.75 : DTE1=54.00 : S0=42.05 Pr0=1.50 IV0=18.00 : S1=43.34 Pr1=2.10 IV1=17.20 Found: DTE0=72.00 K=41.75 : DTE1=55.00 : S0=42.05 Pr0=1.50 IV0=18.00 : S1=43.34 Pr1=2.10 IV1=17.20 Found: DTE0=73.00 K=41.75 : DTE1=54.00 : S0=42.05 Pr0=1.50 IV0=18.00 : S1=43.34 Pr1=2.10 IV1=17.20 Found: DTE0=73.00 K=41.75 : DTE1=55.00 : S0=42.05 Pr0=1.50 IV0=18.00 : S1=43.34 Pr1=2.10 IV1=17.20 Found: DTE0=87.00 K=42.00 : DTE1=74.00 : S0=42.05 Pr0=1.50 IV0=18.00 : S1=43.34 Pr1=2.10 IV1=17.20 Found: DTE0=87.00 K=42.00 : DTE1=75.00 : S0=42.05 Pr0=1.50 IV0=18.00 : S1=43.34 Pr1=2.10 IV1=17.20 Found: DTE0=88.00 K=42.00 : DTE1=74.00 : S0=42.05 Pr0=1.50 IV0=18.00 : S1=43.34 Pr1=2.10 IV1=17.20 Found: DTE0=88.00 K=42.00 : DTE1=75.00 : S0=42.05 Pr0=1.50 IV0=18.00 : S1=43.34 Pr1=2.10 IV1=17.20 Found: DTE0=101.00 K=42.25 : DTE1=94.00 : S0=42.05 Pr0=1.50 IV0=18.00 : S1=43.34 Pr1=2.10 IV1=17.20 Found: DTE0=101.00 K=42.25 : DTE1=95.00 : S0=42.05 Pr0=1.50 IV0=18.00 : S1=43.34 Pr1=2.10 IV1=17.20 Found: DTE0=102.00 K=42.25 : DTE1=94.00 : S0=42.05 Pr0=1.50 IV0=18.00 : S1=43.34 Pr1=2.10 IV1=17.20 Found: DTE0=102.00 K=42.25 : DTE1=95.00 : S0=42.05 Pr0=1.50 IV0=18.00 : S1=43.34 Pr1=2.10 IV1=17.20 Found: DTE0=116.00 K=42.50 : DTE1=113.00 : S0=42.05 Pr0=1.50 IV0=18.00 : S1=43.34 Pr1=2.10 IV1=17.20 Found: DTE0=116.00 K=42.50 : DTE1=114.00 : S0=42.05 Pr0=1.50 IV0=18.00 : S1=43.34 Pr1=2.10 IV1=17.20 Found: DTE0=116.00 K=42.50 : DTE1=115.00 : S0=42.05 Pr0=1.50 IV0=18.00 : S1=43.34 Pr1=2.10 IV1=17.20 Found: DTE0=117.00 K=42.50 : DTE1=113.00 : S0=42.05 Pr0=1.50 IV0=18.00 : S1=43.34 Pr1=2.10 IV1=17.20 Found: DTE0=117.00 K=42.50 : DTE1=114.00 : S0=42.05 Pr0=1.50 IV0=18.00 : S1=43.34 Pr1=2.10 IV1=17.20 Found: DTE0=117.00 K=42.50 : DTE1=115.00 : S0=42.05 Pr0=1.50 IV0=18.00 : S1=43.34 Pr1=2.10 IV1=17.20
Waste of breath. Black Scholes is for European options and not for American. You use a Cox, Ross, or a variant to price American. The difference has nothing to do with volatility - the difference is in the carry calculation. If you price with a model your inputs will give you the same value in BS. Go to the internet and find a pricing model that does both and you'll see the difference here in pricing. A deep European put can even trade at a discount. This isn't about pricing - it's about the debate and it is a waste of breath. https://www.optionseducation.org/toolsoptionquotes/optionscalculator
I just tried it out and it gives the same result for the Call and a minuscule difference for the Put (3rd and 4h decimals differ). Used this set from above "K=42.00 DTE0=88.00 : DTE1=75.00 : S0=42.05 Pr0=1.50 IV0=18.00 : S1=43.34 Pr1=2.10 IV1=17.20" I would say this is just academic toying, not used in the real world trading practice. BSM suffices for both American and European style, IMO.