4

We know that $-1\le\rho_{imp}\le 1$ so perhaps the simplest approach is to try the possible values $\rho_{imp}=\{-1,-0.9,-0.8,\cdots,0.8,0.9,+1\}$, to calculate resulting $\sigma$ values, d± values, and $M_{quote}$ values, then see which of these is closest to the observed market price. If desired you can then search a finer grid between two adjacent assumed ...


3

The linear/non-linear classification is concerned about the dependent variables, and its derivatives. To verify whether the equation is linear, you should be checking that the equation is linear in each of these variables, and the coefficients of these are functions of the independent variables (t and x in your example). In your example, the dependent ...


3

Let $\rho\triangleq\rho_{imp}$. Note that: $$\frac{\partial \sigma}{\partial \rho}(\rho)=-\frac{\sigma_0\sigma_1}{\sigma(\rho)}<0$$ Therefore $\sigma$ is monotonic in implied correlation. In addition, the Margrabe pricing function $M(\cdot)$ is also monotonic in volatility $\sigma$ thus you can find an unique solution to the equation: $$\tag{1}M_{\text{...


1

I would argue as follows: In order to observe any type of resonant behaviour, the dynamics of the system you are looking at needs to be described by a second order differential equation. The equations of motions that come to mind in economics are clearly not: GBM: $dS=\mu S dt+\sigma S dZ$ OU: $dX=\theta(\mu - X)dt+\sigma dW$


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