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The first is something of a theoretical question. It's widely held/assumed that stocks follow a BM process, it appears as though the author is setting the table for the subsequent statement. The second is an artifact of applying Ito's lemma...the $dW_tdt$ and $dtdt$ terms both equal 0, hence fall out, leaving only $dW_t^2$ = dt. Thus, the variance ...

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The monthly interest rate is $\frac{r_0}{m}$ where $m=12$. The formula you give $$r = \left(1+ \frac{r_0}{m}\right)^m -1$$ is the Effective Annual Rate corresponding to $r_0$ compounded monthly. The second formula is correct. In the third formula there seem to be several typographical errors involving "m" and "t" (which is missing). R=\frac{(r_0/m)P}{1-...

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If $S$ is the solution to geometric brownian motion SDE: $$dS=\mu S dt + \sigma S dW(t)$$ then $$S=S_0e^{(\mu - \sigma^2/2)t + \sigma W(t)}$$ Then if you take expectation $$\mathbb{E}[S(t)]=S_0e^{(\mu - \sigma^2/2)t}\mathbb{E}[e^{\sigma W(t)}]$$ Now since $W$ is a wiener ...

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We assume that the stock price process $\{S_t,\,t>0\}$ satisfies, under the real-world probability measure $P$, an SDE of the form \begin{align*} dS_t=S_t\big((\mu-q)dt+\sigma dW_t\big), \end{align*} where $\{W_t, \, t >0\}$ is a standard Brownian motion. Here, we need to consider the total return asset $e^{qt}S_t$, that is, the asset with the dividend ...

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The only difference in the derivation when you have a dividend-yield paying stock lies in the value of the Riskless Portfolio $\Pi_t$. The financial meaning here is the key: to delta-hedge your option you buy a quantity $\Delta$ of the stock $S$, and only the stock is paying you the dividend, so you have to add this contribution in time to your hedge. The ...

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There isn't a single answer to this question. It strongly depends on your goals and why it is missing. If you have a long enough time-series, you will find large numbers of missing data points. The NYSE used to maintain a post for companies that did not trade weekly not so long ago. However, unless you have cause to believe there was a reason for it to ...

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