Multi Asset (variable weight)

Key formulas

Characteristic function

$$ f(x_0\ldots x_N) = k = \prod_{i=0\ldots N} x_i^{\alpha_i} $$

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

$$ x_{0;k}(x) = \left( \frac k {x_1^{\alpha_1} \cdots x_N^{\alpha_N}} \right)^{\frac 1 {\alpha_0}} = \sqrt[1+\sum \eta_i] k \cdot \prod_{i=1}^N x_i^{-\eta_i} $$

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

$$ \pi_i(x_1, \ldots, x_N) = \eta_i \cdot \frac{x_0(x_1, \ldots, x_N)}{x_i} $$

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

$$ \nu(\xi_1, \ldots, \xi_N) = \prod_{i=1}^N \xi_i^{ \alpha_i} $$

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

$$ \Lambda(\xi_1, \ldots, \xi_N) = \alpha_0 + \sum_{i=1}^{N} \alpha_i \xi_i - \prod_{i=1}^N \xi_i^{\alpha_i} $$

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.