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

2 Answers 2

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  1. See this question
  2. You have to buy/sell $\Delta$ shares. $\Delta_{ATM} \approx 0.5$.
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  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.

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  • $\begingroup$ ok thanks @Freddy, but what is an arithmetic brownian motion? What is the difference to geometric brownian motion? $\endgroup$
    – user1690846
    Dec 31, 2012 at 9:54
  • $\begingroup$ @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. $\endgroup$
    – Matt
    Dec 31, 2012 at 10:04
  • $\begingroup$ @chrisaycock, of course, just edited the answer $\endgroup$
    – Matt
    Dec 31, 2012 at 12:28
  • $\begingroup$ 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 $\endgroup$
    – Andrew
    Dec 31, 2012 at 21:25
  • $\begingroup$ @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. $\endgroup$
    – Matt
    Jan 1, 2013 at 5:54

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