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## Sunday, January 5, 2014

### N(d1) >> N(d2) | high σ, r=0, S=K

N(d1) = Pr(ST > S0) , share-measure
N(d2) = Pr(ST > S0) , RN-measure

For simplicity, T = 1Y,  S= K = \$1.

First, forget the formulas. Imagine a GBM stock price with high volatility without drift. What's the prob [terminal price exceeding initial price]? Very low. Basically, over the intervening period till maturity, most of the diffusing particles move left towards 0, so the number of particles that lands beyond the initial big-bang level is very small. The "distribution" curve is squashed to the left. [1]

However, this "diffusion" and distribution curve would change dramatically when we change from RN measure to share-measure. When we change to another measure, the "probability mass" in the Distribution would shift. Here, N(d1) and N(d2) are the prob of the same event, but under different measures. The numeric values can be very different, like 84% vs 16%.

Under share measure, the GBM has a strong drift (cf zero drift under RN) --

dS = σS dt + σ S dW

Therefore when σT is high, most of the diffusing particles move right and will land beyond the initial value, which leads to Pr(ST > S0) close to 100%

-- Now the formula view --
With those nice r, S, K, T values,

d1 =  σT /2
d2 = -σT /2

Remember for a standard normal distribution, if d1 and d2 are 1 and -1 (if σ=2), then N(d1) would be 68% and N(d2) would be 32%.

[1] See posts

http://bigblog.tanbin.com/2013/12/gbm-with-zero-drift.html
http://bigblog.tanbin.com/2014/01/prst-k-s0-k-and-r0-intuitively.html