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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

US market uses the STREET convention. UK market uses the DMO convention. EUR market uses the ICMA convention (Germany uses also a lot MOOSMULLER convention). The main difference between these conventions are: - the way the number of days is calculated for the discount factors - the day count convention used - the calendar used in case of adjsuted ...

1

That's a great question and it is what I always wanted to try to do. I guess I found a solution using PDE approach. Change of numeraire would be more intuitive indeed, but I am not very good in stochastic calculus. The idea is as follows: 1) Let's consider portfolio $\Pi = V(X,Y,t) - \Delta_X X - \Delta_Y Y$. I will found $\Delta_X$ and $\Delta_Y$ such ...

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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