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pankajchitlangia
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BS and Volatility - Interpretation

March 15th, 2010, 4:52 am

In Black Scholes formula, the "sigma" is the annualized volatility of:a. the stock itself; orb. the returns on the stock ?I am in serious doubt on this one and there are few further queries which depend on the answer to above. Somebody please help ... thanks
 
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daveangel
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BS and Volatility - Interpretation

March 15th, 2010, 5:41 am

volatility of returns or changes in the asset price
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pankajchitlangia
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BS and Volatility - Interpretation

March 15th, 2010, 6:33 am

In that case ... a. Assume that I buy a call on a stock pay premium based on implied volatility (in BS) of 1% every day - then to recover the premium paid (i.e. realized volatility = implied volatility) should i. stock move 1% every day on average or ii. there has to be some outcome on the returns on stock (as we have inputted the implied vol of returns on stock and not stock itself) and in that case what it will be?b. assume that a stock known to grow (move) with certainity 20% every day ... and current price (So) is 100 .... in that case the volatility of returns will be zero ... then will the OTM call price for say 30 day tenor be zero ?
 
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daveangel
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BS and Volatility - Interpretation

March 15th, 2010, 6:51 am

a.i. yes - and also path dependenta.ii dont knowb. if its known with certainty then the stock price will not be 100 today but will reflect all that is know with certainty.
Last edited by daveangel on March 14th, 2010, 11:00 pm, edited 1 time in total.
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pankajchitlangia
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BS and Volatility - Interpretation

March 15th, 2010, 7:10 am

am clear on "b" ... but still not able to convince myself on "a" though i know intuitively it should be 1% per day movement on stock but not able to explain logically
 
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Fermion
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BS and Volatility - Interpretation

March 15th, 2010, 7:07 pm

QuoteOriginally posted by: pankajchitlangiaIn Black Scholes formula, the "sigma" is the annualized volatility of:a. the stock itself; orb. the returns on the stock ?I am in serious doubt on this one and there are few further queries which depend on the answer to above. Somebody please help ... thanksIt's the volatility of the stock which is the square root of the rate of growth of variance of the log of the stock price. The rate of growth of variance, given log returns each measured over the same time interval, is approximated by the mean square log return divided by that time interval.
 
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crmorcom
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BS and Volatility - Interpretation

March 15th, 2010, 7:36 pm

QuoteOriginally posted by: pankajchitlangiaam clear on "b" ... but still not able to convince myself on "a" though i know intuitively it should be 1% per day movement on stock but not able to explain logicallyIf the process is a GBM then, so long as you hedge/observe often enough (a.k.a. infinitely often), the realized volatility is guaranteed to be sigma with probability 1 over any finite subinterval, so you always realize exactly your premium.If you are only hedging every day then it matters not just what the realized volatility is but also where the price is relative to your strike when the realization happens.Try experimenting with an Excel MC framework and you can convince your self of this quite quickly. Rebonato's very thick book "Volatility and Correlation" also has some pretty good discussion of option replication and hedging that you might find useful.
 
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BS and Volatility - Interpretation

March 15th, 2010, 10:29 pm

In BS formula you have a constant sigma that does not have any connection to "the annualized volatility" and time t. In BS formula or equation all constant (r, sigma, K, T) are assumed to be known.
 
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BS and Volatility - Interpretation

March 16th, 2010, 2:22 pm

i. stock move 1% every day on average BS pricing does not depend on average stock return. For example if you imagine a stock with average return 100% and sigma 100^(-100)and the another stock with average return - 100% and the same sigma 100^(-100) BS states that the option price must be equal because following benchmark interpretation of the derivative price there is 0 chance of arbitrage between ... if you support their belief.ii. there has to be some outcome on the returns on stock (as we have inputted the implied vol of returns on stockassume that stock is known, i.e. are known constants or functions on t. If we replace original sigma on the other “implied sigma” we get other stock. If sigmas are not close otherwise there is no sense for replacement then the BS pricing eq corresponds security which has other volatility. As far as BS approach itself does not sensitive with respect real return at the very beginning with next development we figured out that real sigma does not mach pricing equation.b. assume that a stock known to grow (move) with certainty 20% every day In BS setting certainty growth has only risk free bond but not a stock.
 
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pankajchitlangia
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BS and Volatility - Interpretation

March 18th, 2010, 3:06 am

Thanks everyone for clarifications .... (a mock experimentation in excel sheet of option delta hedging also was a great help)