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2

The risk-neutral measure $\mathbb{Q}$ is a mathematical construct which stems from the law of one price, also known as the principle of no riskless arbitrage and which you may already have heard of in the following terms: "there is no free lunch in financial markets". This law is at the heart of securities' relative valuation, see this very nice paper by ...

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Your problem probably comes from the notations used. Let the Moment Generating Function (MGF) of a random variable $X$ be defined as $$M_X(u) := E[e^{uX}]$$ From this definition, it entails that $$E(X^n) = M_X^{(n)}(u=0) = \frac{d^{n} M_X}{ d u^{n}}(u=0)$$ Knowing this, the function $$f_{\lambda}(t,r)=E[e^{-\lambda {r_{T}}}|r_t=r]$$ can be ...

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You need a model which assumes that some intrinsic properties such as true overpricing is taking place but masked by the noise which has some probabilistic distribution around the true signal. The averaging of the observed data then takes advantage of the large number theorem or some version of the central limit theorem to flush out the signal. So you need ...

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IMHO the 'definition' you mention is not a mathematical definition per se, but rather an approximation used by some practitioners. Mathematically, it is $N(d_2)$ in the BS formula which figures the conditional probability that the terminal asset price $S_T$ will finish above the strike level $X$ given the information we possess today (represented by $S_t$), ...

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