thank you!! You are welcome! Glad you found it useful!

Glad it was helpful! many thanks!!

thank you for the kind words as always!!

many thanks, very kind of you!, sure!

Many thanks for the kind words!! better late than never!

Glad you liked it!

Thanks, glad you enjoyed it! If you sell put, you will get premium but you will be obliged to pay the buyer the max(K-S,0), the lower the S the more you lose. So that is what you will be trying to protect with the delta hedge - negative delta is what you need! I assume you mean short-sell the stock? In ideal world when you short sell the stock, you get money today which will be invested at some risk free rate, and then at some point you will have to buy the stock to close the short position. If the price happen to be higher then you will lose money on the short stock position. The stock that your broker sold on your behalf belongs to someone else, and the broker is not going to take the risk, so they will be asking you to continue to deposit cash if the position keeps going against you, and they will allow you to withdraw cash if the stock price falls. So price goes down, put obligations rises, but short stock position becomes more valuable. Price goes up, put obligations decreases, but short stock position generates loss. PS: the reason i said ideal is because usually there will be restrictions on short selling as it is not an activity that the authorities like.

Thank you!

thank you!!

thank you!!

It is just the delta of Black Scholes - we just plug in the stock price at each observation point, and the then remaining maturity . Hope this helps!

many thanks! Please see here - quantpie.co.uk/bsm_formula/bs_delta.php. We also have a video on delta in the Black Scholes playlist, which adopts a slightly simpler approach to the derivation. Hope this helps!

@@quantpie Hey, first of all, I'd like to say that I love this channel and I hope you guys continue with the excellent work you have been doing so far. The videos are so clear and well explained and it is helping me a lot. Just a remark regarding this link; I believe it is incorrect; on the site, you achieve Delta_call = exp(-rf tau) N(d1); The correct expression should be exp(-rd tau) N(d1). The values on the video are correct. It seems that the calculation is correct on the website but you started with the wrong formula. In the beginning, the exp(-rf tau) should multiply K and the exp(-rd tau) should multiply S. It is reversed. Best regards and thanks once again for all your efforts in helping and educating us all.

@@caetanocardeliquio7174 thank you! glad you found the content useful! On the website, the symbols rf and rd are meant in the FX sense - i.e., foreign and domestic interest rates, respectively. In case of stocks, rf should be interpreted as the dividend yield rate and rd as the interest/discount rate. We have added a note to this effect on the summary page. Many thanks!

@@quantpie Oh, my bad then. When I read I thought rf was the risk-free rate. Thank you for the clarification! Best regards.

Black-Scholes does not account for dividends (or, at best, assumes they're constant.) That's one of the reasons it's not a very useful pricing model.

that is ok if it ends up above the strike, but what happens if it ends up below the strike?

Sure thing! What templates would be useful please? Will try our best!