I have an expression for $p$ as follows:
$$P=\frac{e^{r\frac{T}{n}}-e^{r\frac{T}{n}-0.5\sigma^2\frac{T}{n}-\sigma\sqrt{\frac{T}{n}}}}{e^{r\frac{T}{n}-0.5\sigma^2\frac{T}{n}+\sigma\sqrt{\frac{T}{n}}}-e^{r\frac{T}{n}-0.5\sigma^2\frac{T}{n}-\sigma\sqrt{\frac{T}{n}}}}$$
When I plot it as a function of $n$, with $n$ varying between 1 to 260, it very clearly converges to 0.5 (the other parameters are set to $r=0.1$, $\sigma=0.2$, $T=1$):
However, I am unable to show this mathematically. Here is my attempt; first, $p$ simplifies to (getting rid of the dependence on $r$):
$$p=\frac{1-e^{-0.5\sigma^2\frac{T}{n}-\sigma\sqrt{\frac{T}{n}}}}{e^{-0.5\sigma^2\frac{T}{n}+\sigma\sqrt{\frac{T}{n}}}-e^{-0.5\sigma^2\frac{T}{n}-\sigma\sqrt{\frac{T}{n}}}}$$
Then:
$$\lim\limits_{n\to\infty}(e^{-0.5\sigma^2\frac{T}{n}-\sigma\sqrt{\frac{T}{n}}})=1$$
$$\lim\limits_{n\to\infty}(e^{-0.5\sigma^2\frac{T}{n}+\sigma\sqrt{\frac{T}{n}}})=1$$
So I get:
$$\lim\limits_{n\to\infty}(p)=\frac{1-1}{1-1}$$
Which is undefined.
Could anyone pls give me a hint at what I am doing wrong?
Best Answer
I let you adapt the following computation to your case. What has to be computed is $$\lim_{x\to 0^+}\frac{1-e^{- \alpha x-\beta \sqrt x}}{e^{-\alpha x+\beta \sqrt x}-e^{-\alpha x-\beta \sqrt x}},$$ with $\alpha ,\beta >0$.
You have that $$e^{x}=1+x+o(x).$$ Therefore $$1-e^{-\alpha x-\beta \sqrt x}=\alpha x+\beta \sqrt x+o(x), $$ and $$ e^{-\alpha x+\beta \sqrt x}-e^{-\alpha x-\beta \sqrt x}=2\beta \sqrt x+o(x). $$
At then end $$\frac{1-e^{- \alpha x-\beta \sqrt x}}{e^{-\alpha x+\beta \sqrt x}-e^{-\alpha x-\beta \sqrt x}}=\frac{\alpha x+\beta \sqrt x+o(x)}{2\beta \sqrt x+o(x)}=\frac{\beta +\alpha \sqrt x+o(\sqrt x)}{2\beta +o(\sqrt x)}\underset{x\to 0^+}{\longrightarrow } \frac{\beta }{2\beta }=\frac{1}{2}.$$