Limit of d1 in black scholes

Question

$\lim\limits_{\sigma \to 0^+} \frac{ln(\frac sk) + (r - q + \frac{\sigma^2}{2})T}{\sigma \sqrt{T}}$

What is this limit? I tried by isolating the sigma in the denominator , but this ended up with an infinity + infinity - infinity, which wasn't helpful.

For context, this is essentially evaluating as implied volatility goes to 0 for d1 in the black scholes formula for pricing an option. The idea is that if an option is not in the money (K >= S), then the price of the option will logically be 0 as volatility approaches 0.

Answer 1

Let $$f(S,K, r, q, T) =\ln \left( \frac{S}{K} \right)+(r-q)T.$$

Hence, \begin{align}&\lim_{\sigma \rightarrow 0^+}\frac{\ln \left( \frac{S}{K} \right)+(r-q+\frac{\sigma^2}{2})T}{\sigma\sqrt{T}} \\&= \lim_{\sigma \rightarrow 0^+}\frac{f(S,K,r,q,T)+(\frac{\sigma^2}{2})T}{\sigma\sqrt{T}} \\&=\begin{cases} \infty & ,f(S,K,r,q,T)>0\\ -\infty & ,f(S,K,r,q,T)<0 \\ \lim_{\sigma \rightarrow 0^+} \frac{\sigma}{2}\sqrt{T} & ,f(S,K,r,q,T)=0 \\\end{cases} \\&=\begin{cases} \infty & ,f(S,K,r,q,T)>0\\ -\infty & ,f(S,K,r,q,T)<0 \\ 0 & ,f(S,K,r,q,T)=0 \\\end{cases}\end{align}