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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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