Betancourt's Formula

Betancourt's Formula is designed to minimize the drawdown risk by varying pMid, the price at which the VMM creates orders. Specifically, Betancourt's takes as input: t, the long-tail scenario and d the drawdown threshold. Both are expressed as percentages so that the VMM could have a 1% chance of a 5% drawdown within a 30 day period, for example.

pMid is offset by the premium index p, which is specific to each trading pair. Given the community-determined t and d, the VMM also inputs its real-time exposure and total available assets.

Methodology

If the VMM's exposure increases—reflecting the inverse position of net traders—then the premium will increase along with the funding rate, enabling arbitrage opportunities for other traders. The VMM's net asset value is indirectly proportional with premium, so that more capital allows for higher VMM exposure before premium increases.

Variables

Dependents

• p = premium index

• r* = risk parameter

Independents

• e = exposure

• d = drawdown threshold

• n = net asset value

• s = smoothing variable

• σ = implied volatility

• t = long-tail scenario

Formulas

$$\begin{equation*} P_i = \begin{cases} - \dfrac{E_i^{S}\,\bigl(1 - e^{r_i^*}\bigr)^{S+1}}{90\,D^{S}\,N^{S}}, & E_i \ge 0,\\[8pt] \quad\;\dfrac{E_i^{S}\,\bigl(1 - e^{r_i^*}\bigr)^{S+1}}{90\,D^{S}\,N^{S}}, & E_i < 0. \end{cases} \end{equation*}$$

$$\begin{equation*} r^*_i = -\tfrac12\,\sigma_i^2\!\bigl(\tfrac{30}{365}\bigr) \;+\;\sigma_i\sqrt{\tfrac{30}{365}}\;t_{4}^{-1}(T) \end{equation*}$$