# Black Scholes Theory Question

#### maxsidious

Hi, I have one really short question

For S fixed and given, evaluate

lim(from t to T) of N(d1)

Where N(x) is the normal distribution function
d1=(ln(s/x)+(r+0.5sigma^2)(T-t))/(sigma*(T-t)^0.5)

sigma is the standard deviation (volatility of the portfolio)
s= stock price
x= strike price
r=interest rate

Thank you so much

Also, is this limit have anything to do with the delta hedging of call price? dC/dS=N(d1)
Does it have anything to do with delta hedging and transaction cost?

Thanks again

#### charlesdwright

I would say that

As T > t N(D1) approaches either 1 or 0 depending on ln(s/x).

At T=t the option ceases to exist and N(D1) is meaningless.

But that's probably not what you're asking. Also, sig is the volatility of the instrument whose price is S (the underlying).

I'm sure someone can give a more theoretical answer than that.

#### Bastian Gross

##### German Mathquant
Hi maxsidious,

answer is easy, because lim(d1) = lim((ln(s/x)+(r+0.5sigma^2)(T-t))/(sigma*(T-t)^0.5)) = Infinity. So lim(from t to T) of N(d1) will be "N(Infinity)" and this is zero.

The evaluation is independent from stock price, strick price and interest rate.

If maturity day terms taken, Delta will be go to zero.

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