# BS and delta hedging questions

I have two related questions concerning Black Scholes and delta hedging. I thought about this two questions, but I could not come up with an answer, so maybe you guys & girls can help me:

1. If an option is at the money, how can the Black Scholes price be calculated in a very fast way (possibly without any big calculations)?

2. If an option is at the money, how many shares do you have to buy in order to delta-hedge?

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 Is this a homework assignment? These are the most commonly presented results in any textbook on options pricing. – chrisaycock♦ Dec 31 '12 at 12:14

 ok thanks @Freddy, but what is an arithmetic brownian motion? What is the difference to geometric brownian motion? – user1690846 Dec 31 '12 at 9:54 @user1690846, well I kind of pointed to it, its a Brownian motion where the asset price distribution is assumed to be normal and not log-normal. – Freddy Dec 31 '12 at 10:04 @chrisaycock, of course, just edited the answer – Freddy Dec 31 '12 at 12:28 actually, asset prices do not follow arithmetic brownian motion in .4 *S*$\sigma * \sqrt{T}$, it is still geometric brownian motion. just do a simple taylor expansion on B-S formula and you will see – Andrew Dec 31 '12 at 21:25 @Andrew, I said the assumption is of the asset price to be normally distributed which is the equivalent of an asset price model of arithmetic Brownian motion. – Freddy Jan 1 at 5:54
2. You have to buy/sell $\Delta$ shares. $\Delta_{ATM} \approx 0.5$.