# 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?

• Is this a homework assignment? These are the most commonly presented results in any textbook on options pricing. Dec 31, 2012 at 12:14

1. See this question
2. You have to buy/sell $\Delta$ shares. $\Delta_{ATM} \approx 0.5$.
1. stock price * volatility * 0.4 * sqt(T), where T denotes time to expiration in years and 0.4 is coming from sqt(1/(2*pi)). The simplifying assumption here is (and that is very important and you will most likely be asked to state the assumptions should such question be asked in the interview): strike price equals underlying asset price AND asset prices are NORMALLY DISTRIBUTED (unlike the assumption in B-S) which assumes the asset price to follow an ARITHMETIC Brownian motion.

2. As the delta is approximately (stress, not equal) 0.5, you need to hedge with about 1/2 the amount of the underlying asset that the options contract stipulates.

• ok thanks @Freddy, but what is an arithmetic brownian motion? What is the difference to geometric brownian motion? Dec 31, 2012 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.
– Matt
Dec 31, 2012 at 10:04
• @chrisaycock, of course, just edited the answer
– Matt
Dec 31, 2012 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 Dec 31, 2012 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.
– Matt
Jan 1, 2013 at 5:54