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5

To simplify notations, let $a:= -1.96\sigma$ and $b := \mu - 0.5\sigma^2$. The development in the book could be justified if both $a\sqrt{t}$ and $bt$ are small (close to zero), and if we have that $|a\sqrt{t}| > |bt|$. Recall that $\exp (x+y)= \exp(x)\exp(y)$, $\exp(x)\approx 1 + x,\quad \text{if } x\approx 0$. Then, using these properties we have ...


3

I wanted to add this side note to Quantelbex' answer: Both factors in $\exp(a\sqrt t)\exp(b t)$ go to one as $t$ goes to zero, but for small $t$, the $\exp(b t)$ term approaches one faster. For $t=\frac {a^2}{b^2}$ both factors are the same, if $t$ is smaller than $\frac {a^2}{b^2}$, we have $\exp(a\sqrt t) > \exp(bt)$. Thus the approximation that ...


2

I could not find any such detailed documentation after some weeks of looking (not non-stop obviously). It is appallingly documented. I do understand fully what it does though so am happy to field some questions on it if you like. In a nutshell, I can tell you it is a standard reduced-form credit model under a constant hazard rate (i.e. homogeneous Poisson ...


1

It is true that you cannot infer the real World probabilities from the BSM formula directly. It is also equally true that the "right value" of the option in the real world is obtained by replacing the risk free rate with the expected return of the stock. Another example of this is simply to look at the real world price of a forward on the stock. If ...



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