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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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You should see this as a comment to @Kiwiakos answer which already hit the bull's eye. In the SE question you're referring to and to which I have answered, the idea was simply to provide you with a sound way of simulating returns out of a NIG distribution. It so happens that, for whatever your reason was, you decided to calibrate your NIG parameters based ...


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I believe that the confusion arises because of the wrong treatment of NIG. The answer to the question you link is misleading, as it simulates under P which is not appropriate for option pricing. None of the NIG parameters under P carries over to Q in general, but especially the drift is the problem here. First use the mom gen function of NIG to find the ...


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The derivation in Appendix A of the paper Valuation of Equity-Indexed Annuities under Stochastic Interest Rates that you mentioned is Wrong: the Girsanov transformation is applied to an $n$-dimensional Brownian motion, where the components are independent. However, for the case here with $n=2$, the Brownian motions are dependent, we can not naively combine ...


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Here is a general proof for all parameters in an open domain. $$dr = adt+bdW:=r\big(k(\theta-x)+\frac12\sigma^2\big)dt+\sigma rdW.$$ Let $$u(r(s),s):=e^{-\int_t^sr}B(r(s),s,T)=:\phi(s) B.$$ Then $$u(r(t),t)=\mathbf E\big[u(r(s),s)\big|r(t)\big],\, \forall t<s. \tag{1}$$ So, by Ito's Lemma, \begin{align} du(r(s),s) &= Bd\phi +\phi dB \\ &= \phi ...


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We shall prove this by contradiction. Let $\theta=0$ and $\sigma=0$. $X_t=X_0e^{-kt}$ and $$B(0,t)=\exp\Big(-\int_0^te^{X_0e^{-ks}}ds\Big).$$ Suppose the contrary that $B(0,t)$ is affine. We should have $$ B(0,t)=\exp{\left(A(0,t)-C(0,t)e^{X_0}\right)}\;\;\ \forall (t,X_0), \tag{1} $$ Differentiate the logarithm of Equation (1) with respect to $t$ side, ...



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