Multi Asset (equal weight)

Key formulas

Characteristic function

$$ f(x_0, x_1, \ldots, x_N) = k = \sqrt[N+1]{x_0 \cdot x_1 \cdot \cdots \cdot x_N} $$

As in the case of two assets one could use the straight product of the constituent assets, but if the geometric average is used then the constant $k$ scales like a currency and is therefore a useful measure for the size of the pool independent of trading.

We have used $N+1$ assets to allow for a numeraire. Our (arbitrary) convention is that $x_0$ is the numeraire, and $x_1, x_2, \ldots$ are the risk assets. As usually, token quantities are measured in their own native currency amounts.

Indifference surface

$$ x_0(x_1, \ldots, x_N) = \frac{k^{N+1}}{x_1\cdot x_2 \cdots x_N} $$

Like in the two dimensional constant product case, the exponent $N+1$ of the constant $k$ ensures that $k$ scales like a currency.

Price response function

$$ \pi_i(x_1, \ldots, x_N) = \frac{k^{N+1}}{x_1\cdots x_i^2\cdots x_N} = \frac{x_0(x_1, \ldots, x_N)}{x_i} $$

The middle formula may look somewhat surprising compared to the two-dimensional one, but ultimately the good way of writing it is the one on the right that reduces to $\frac y x$ in the two dimenional case.

AMM portfolio value

$$ \nu(\xi_1, \ldots, \xi_N) = \sqrt[N+1]{\xi_1 \cdots \xi_N} $$

Again a very consistent extension of the 2 dimensional formula to higher dimensions where the portfolio value function remains (almost) the geometric averages of the prices (technically is is the geometric average to the power $\frac {N}{N+1}$)

Divergence loss

$$ \Lambda(\xi_1, \ldots, \xi_N) = \frac{1+\sum \xi_i}{N+1} - \sqrt[N+1]{\xi_1 \cdots \xi_N} $$

Again, no surprises here. All that changes is the valuation of the reference portfolio.

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Notes

Note that financial derivatives become much more complex in higher dimensions. The reason for this is the existence of correlations and cross-dependencies in the profile, so where before we had 1 second derivative we now have $N$ second derivatives along the axis', and $0.5 N (N-1)$ cross derivatives. Whilst we can match the Gamma along the axis' with regular calls and puts it is not clear with what we could match the cross convexity.

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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.