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You ask 2 questions and I try to answer: 1) Why do we use geometric Brownian motion ($\ln S_t-\ln S_0$ is normally distributed)? In this case you have $$ S_t = S_0 \exp( (\mu-\sigma^2/2) t + \sigma B_t), $$ which means that you model positive prices. Furthermore the log-return $$ \ln(S_t/S_0) = (\mu-\sigma^2/2) t + \sigma B_t, $$ is normally distributed. ...


please go to {drvd} BVOL Equity Implied Volatilities Calculations paper. Disclamer: I was working for Bloomberg, that is as far we disclosed.


Options on interest rates futures in the listed markets are always traded 1-yield (100-yield) just like the futures which are traded 1-yield. So negative rates aren't an issue and its always black volatility. In the OTC market, both normal and black volatility are quoted, but the common practice is to use black volatility is what is way more frequently ...


if they are stocks, this problem is called pricing a Margrabe option and it is generally solved by change of numeraire. Take $S_2$ to be the numeraire. Then the value of the option is $$ S_2(0) \mathbb{E}_{S_2}( (S_1(T)/S_2(T)-1)_+) $$ where the expectation is taken in the measure that has $S_1/S_2$ as a martingale. Since it's a martingale and log-normal at ...


Generally Bloomberg is very open with their methodologies. Look up the documentation as recommended above, and if you have further questions you can ask HELP HELP to put you in touch with someone on their quant development team for more details. As long as you are a paying subscriber it should be no problem.


One way to start thinking about this is to work out a couple of Discrete versions of Ito's lemma Øksendal (6th edition) Example 3.1.9: almost surely, $$ B_t^2 - t = \int_0^t 2B_s dB_s $$ This has a discrete version which holds everywhere: let $X_n=\pm 1$ and $S_n=\sum_{i=1}^n X_i$, then $$ S^2_n-n = 2\sum_{i=0}^{n-1} S_i X_{i+1} $$ To verify ...


Well, log-normality does not allow you to have negative earnings and companies do have negative earnings. I suggest you to download the earnings data and perform a Jarque-Bera test for normality.


You know that Brownian motion {W(t)} is a stochastic process with the following properties: (Independence of increments) W(t) − W(s) , for t > s , is independent of the past, that is, of W(u) , 0 ≤ u ≤ s, or of $F_s$ , the σ-field generated by W(u), u ≤ s. (Normal increments) W(t) − W(s) has Normal distribution with mean 0 and variance t − s. This implies ...

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