Multi Asset (variable weight)

Key formulas

Characteristic function

$$f(x_0\dots x_N)=k=\prod _{i=0\dots N}{x}_i^{{\alpha }_i}$$

Here the ${\alpha }_i$ are the normalized portfolio coefficients with $\sum {\alpha }_i=1$. This normalization is not necessary, but it ensures that the constant $k$ transforms like a currency. Like in the two dimensional case, the $\alpha$ determine the weight of respective portfolio constituents, ie the percentage of the overall value locked. We recover the equally weighted formula by setting all ${\alpha }_i=\frac{1}{N+1}$.

Definition of ${\eta }_i$

$${\eta }_i=\frac{{\alpha }_i}{{\alpha }_0}\Rightarrow \sum {\eta }_i=\frac{1-{\alpha }_0}{{\alpha }_0},\frac{1}{{\alpha }_0}=1+\sum {\eta }_i$$

The definition of ${\eta }_i$ extends the definition of $\eta$ in the variable weight case. The ${\eta }_i$ indicate the weight of the asset within the pool relative to the numeraire asset. In an equally weighted pool, all ${\eta }_i=1$.

Indifference curve

$$x_{0;k}\left(x\right)={\left(\frac{k}{x_1^{{\alpha }_1}\cdots x_N^{{\alpha }_N}}\right)}^{\frac{1}{{\alpha }_0}}=\sqrt[1+\sum {\eta }_i]{1+\sum {\eta }_{i}k}\cdot \prod _{i=1}^N{x}_i^{-{\eta }_i}$$

The middle term uses the ${\alpha }_i$ whilst the right one uses the ${\eta }_i$. Note the minus in the exponent on the right, meaning the $x_i$ are still in the denominatior.

Price response function

$${\pi }_i(x_1,\dots ,x_N)={\eta }_i\cdot \frac{x_0(x_1,\dots ,x_N)}{x_i}$$

The $x_0/x_i$ term is the ratio of numbers between numeraire asset and risk asset $i$. ${\pi }_i$ is the price of the risk asset $i$. Therefore ${\eta }_i$ is the portfolio weight of the risk asset $i$ compared to the numeraire asset. More generally, the proportion of assets $i:j$ in the pool is ${\alpha }_i:{\alpha }_j$ or ${\eta }_i:{\eta }_j$ (the two are the same; the higher number is overweight).

Note that we recover our expression $\eta \frac{y}{x}$ from the two asset case.

AMM portfolio value

$$\nu ({\xi }_1,\dots ,{\xi }_N)=\prod _{i=1}^N{\xi }_i^{{\alpha }_i}$$

Again we find a formula that is very similar to two asset case and that is almost a (now weighted) geometric average like in the equal weights case.

Divergence loss

$$\Lambda ({\xi }_1,\dots ,{\xi }_N)={\alpha }_0+\sum _{i=1}^N{\alpha }_i{\xi }_i-\prod _{i=1}^N{\xi }_i^{{\alpha }_i}$$

No surprises here, the various $\alpha$ in the in the leading terms ensure that the reference portfolio is correct.

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Notes

Like in the equal weights case it does not make much sense to look at a replication strategy with European options because those become exceedingly complex in a multi-dimensional environment.

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Whilst reasonable care has been taken to verify the above formulas they may still contain errors. Please do not use them without independent verification.