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The market is complete iff there is a unique risk-neutral measure: when every contingent claim is attainable, its unique no arbitrage price is the cost of the replicating portfolio. In the case of an incomplete market, you no longer have a unique price for unattainable contingent claim, but rather a range of prices : $\left(-p\left(-G\right), p\left(G\right)\...


Numéraire Change The time-$t$ price of a zero-coupon bond maturing at time $T$ is $$P(t,T)=\mathbb{E}^\mathbb{Q}_t\left[\exp\left(-\int_t^T r_s\text{d}s\right)\right].$$ Let $\mathbb{Q}$ be our standard risk-neutral probability measure which uses a locally risk-free bank account, $\text dB_t=r_tB_t\text dt$, as numéraire. From Geman et al. (1995), we know \...


Stock prices are NOT martingales. Stock futures, and call/put options consistent with these, are. Which is not inconsistent. Because pricing the expected return from stocks into derivatives of these generates arbitrage free-lunches. So if it reasonable to believe that a stock will generate a say +10% annual return, then the derivatives of this stock must be ...


It only means that the directions of future movements of stock prices are impossible to forecast. For a more mathematical explanation, consider this document.

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