Simplified option pricing formula

HITECH · June 29th, 2007, 7:17 pm

Hello,Some people use the following formula to price an ATM option:Price = 0.4 * Volatility * sqrt(time)0.4 = 1/sqrt(2 * pi)would someone know how to tie this back to Black Scholes please?Thx in advance.

casati · June 29th, 2007, 10:01 pm

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ppauper · June 30th, 2007, 12:23 pm

you could try setting stock price equal to strike price in the black-scholes formula and then expanding as a series in time and see if that helps

AVt · June 30th, 2007, 3:17 pm

If you replace the cumulative normal distribution by its 1st order approximation (i.e. Taylor series in 0, cut off terms of order greater than 2) it becomes 1/2 + x/sqrt(2*Pi).Using it for the BS formula gives your stuff (with rates=0, spot=strike= 1 in your case).

amit7ul · July 3rd, 2007, 7:59 am

atm strike implies s=k*e^(-rt), and also d1=v*sqrt(t)/2=-d2so c=s*(N(d1)-N(d2))....N(d1)-N(d2)=(1/sqrt(2*pi))integral(1-x*x/2+...)dx after integration just retain linear term.. you get d1-d2=v*sqrt(t)...and so c=0.3989*s*v*sqrt(t)...i guess, assumption should work for small expiry small vol and atm strikes