# Benchmark Price

The price from SVI is a valuable benchmark

### Intuition from the Benchmark Price

Consider a call option. When the AMM charges the benchmark price, the price effectively offsets the first-order hedging costs because the cost of the instantaneous replicating portfolio satisfies the following relation
\begin{align*} \text{call}_t(K, T) = \Delta_{t} S_{t} - B e^{-r(T-t)} > 0 \end{align*}
where
$\text{call}_t(K,T)$
is the price of a call option at time
$t$
that has a strike of
$K$
and expires at time
$T$
. The positions in cash and the underlying comprising this portfolio satisfy
\begin{align*} \Delta_{t} &= N(d_1) > 0\\ B &= K N(d_2) > 0 \end{align*}
In other words, the price of the replicating portfolio is the price of the option. Both equal the present value of the option liability under the risk neutral measure, since
$\text{call}_{t}(K, T) = \mathbb{E}_{t}^Q[\max(0, S_{T}-K)] e^{-r(T-t)}$

#### Liquidity Transformation

This is the channel through which Arrow’s AMM transforms liquidity from the underlying market to the options markets. The AHE picks up the replicating portfolio and can create and deliver the option independently of how liquid the markets are for that option. We discuss the AHE in detail below (hyperlink).

#### Capitalization

The amount collected for the premium is insufficient to cover the
$\Delta$
cost alone - the replicating portfolio requires borrowing to fund the hedge. Because of this, a market maker (and hence our AMM) needs to be well-capitalized.
Price Discovery
The price of the option replicating portfolio is a benchmark for the price of the option. While in the limiting case of no market frictions, these prices must be identical, in practice, when creating new markets, this is not a solution for price discovery because implied volatility is needed for each option in order to calculate the delta weight for the portfolio. The implication is also that the average cost of the hedging portfolio is offset by charging the benchmark price, so the calculus for the bid-ask spreads should look at incremental hedging costs in bad-state scenarios.

#### Limits of the Benchmark Price

If the AMM could rebalance at arbitrarily high frequencies with negligible transaction costs if there were no information asymmetries between buyers, sellers, and market makers, and if the underlying dynamics were Gaussian, this price would be sufficient for our purposes. In practice, several key deviations to these assumptions guide the AMM to prefer to charge a price that does more than simply offset the near-term first-order hedging costs. The next section describes how the Arrow protocol incorporates those considerations.