 # Benchmark Price

### 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{bench}_{svi, t}(K, T) = \Delta_{t} S_{t} - B e^{-r(T-t)} > 0 \end{align*}
where
$\text{bench}_{svi, t}(K,T)$
is the price of a call option with strike
$K$
and expiration
$T$
obtained from the SVI implied volatility
$\sigma_{T,k}$
at time
$t$
, and where
$\Delta_t$
is also calculated using
$\sigma_{T,k}$
. 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 a common measure
$\tilde{Q}$
, since
$\text{bench}_{svi, t}(K, T) = \mathbb{E}_{t}^{\tilde{Q}}[\max(0, S_{T}-K)] e^{-r(T-t)}$
These relations highlight several points:

#### Liquidity Transformation

The replicating portfolio 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.

#### Capitalization

The amount collected for the premium is insufficient to cover the cost of acquiring the
$\Delta$
. The replicating portfolio requires borrowing
$B e^{-r(T-t)} > 0$
to fund the hedge. Because of this, a market maker (and hence our AMM) needs to be either well-capitalized or able to access liquid lending markets or both.
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: implied volatility (a price) is needed for each option in order to calculate the delta weight for the portfolio.

### Limits of the Benchmark Price

There are a litany of conditions that must hold for charging the benchmark SVI price to be the end of the story for the AMM. For example, the AMM would need to be able to rebalance at arbitrarily high frequencies with negligible transaction costs, including in lending markets. There would also have to be no information asymmetries between buyers, sellers, and market makers, and the true underlying and volatility dynamics must be known (and Gaussian). In practice, violations of these assumptions compel the AMM to charge a price that does more than the benchmark SVI price.