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I'll outline how you can estimate the (implied) real-world density function from (observed) option prices. Having found this real-world density, you can then compute all sorts of probabilities and quantify the market's expectation of future prices. Recall firstly that (European-style) options are priced as risk-neutral expectation of the discounted payoff. ...


Because $\mathbb{E}\left(e^{\sigma W_t}\right) = e^{\frac{1}{2}\sigma^2T} > 1$, you need that correction to ensure that your asset grows on average at rate $\mu$ (or $r$ in the risk-neutral measure). This is pretty well explained in the chapter on BS model from Hull’s book Options, futures and other derivatives!


If drift is driven by earnings retention policies then the value of an option does depend on drift! I'm 99% on the following reasoning and would welcome input from others to tighten this up. Consider what happens with Save Co., a hypothetical company that owns a pile of cash sitting in a savings account earning 1% APY. Suppose Save Co. is required to ...


You can compute expectation of drifted processes as well and derive same pricing formulas,but usually its more complicated (compare derivation of Black Scholes using martinglaes and through PDE. PDE proof ,where drift is explicit, is much longer) With martingale representations you have more analytical mathematical tools/formulas available (e.g. barrier ...

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