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Recently I read a comment on how to interpret the Black Sholes Formula and more specifically how to wrap your head around the d1/d2.

Although there were many good comments, this one stood out when one user thought of the formula as the difference between S0 and X and then multiplied it by the probability of exercise (d2). This essentially generates the expected loss for the writer, and thus the price.

This makes sense in the scenario that you invest in an ATM option with a strike and stock price of 300, where you know there's a 50 % chance of hitting 310 with no in-betweens. So the expected win for you/loss for the writer becomes (310-300)*0.5=5 dollars.

But this illustration would suffice as I am taking the difference between the St of 310 (which we do not know exists) and the strike price, instead of S0-X, which would be 0 in this case since it's an ATM.

In his illustration, he says that there's a S0 of 300 and a strike of 310. And there's a 50 % chance of hitting 310, meaning that there's an expected loss of 5 dollars. But here again, I don't understand since on the one hand S0-X is multiplied by a probability. But on the other, if St hits 310 with a strike of the same, then nobody loses anything since it's equally as worth buying the stock with or without the option.

Can somebody help to clarify the illustration? Thanks in advance.

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    $\begingroup$ I had a look at that comment on Reddit: it has so many mistakes in it that my advice would be to ignore it. For example, if the strike is 310 on a call option and the value of the underlying is 300 at maturity, then the option expires worthless. If the underlying is 310 at maturity (instead of 300), then again the call option is worthless. If there was (say) a 50% chance that the underlying hits 320 at maturity and 50% that it stays below 300, then indeed the option value at maturity would be 0.5 * (320 - 310) = 5. To get today's option price, the 5 needs to be discounted to "today". $\endgroup$ Commented Apr 11 at 13:54
  • $\begingroup$ I made the exact same illustration in my notebook when I tried to incorporate this thinking to access the formula. AKA: If there are two options two win or not win nor lose and there's a fifty per cent chance to win you expected to win is the difference /2 --) (320-310)*0.5. But then when you try to replace it with D2*(S-X) it just doesn't add up since 320 isn't the strike price nor the strike, but the oracle's holy prophecy. Does this mean there's no way to understand the formula using the distributive law? @JanStuller $\endgroup$ Commented Apr 13 at 13:32
  • $\begingroup$ The simple illustration of two possible future states of the world (underlying price either 320 or 300) corresponds to a Binomial tree. The continuous Black-Scholes model is more complicated. Try reading this post. $\endgroup$ Commented Apr 13 at 13:40
  • $\begingroup$ Should I read every comment or can you direct me to which one you think is the best? Otherwise, this will take a while😅 @JanStuller $\endgroup$ Commented Apr 13 at 14:11
  • $\begingroup$ Never mind, obviously you mean that made by yourself, sorry $\endgroup$ Commented Apr 13 at 14:13

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