 # Indifference Pricing

The Arrow pricing engine adjusts the benchmark SVI price according to risk assessments of the AMM's obligations. We run scenario analyses to inform these risk assessments to evaluate bad-case future hedging costs. A complete description of these assessments, along with additional programmatic adjustments for deep ITM contracts and high-gamma contracts, is available in the next section. Below we describe the connection between this approach and indifference prices, which should be considered optional reading.

## Connection to Indifference Prices

The approach is inspired by indifference pricing, where the price charged makes the market maker indifferent to taking on the additional risk of a new option, given some model-specific assessment of the risk of holding it. Reasoning about insolvency probabilities will help clarify this connection. To a first-order approximation, the market maker wants to adjust the baseline price of an option to help keep the probability of high future hedging costs low:
\begin{align*} \mathbb{P}\left(\sum_{c} \eta_c p_{t+s}^c - \Delta_{t} S_{t+s} + B \geq f \right) < \epsilon \end{align*}
where
$f$
is a capital bound, and
$\epsilon$
is a probability bound,
$s$
is the time horizon for the rebalance, and where the distribution is over
$S, \sigma$
. The summation is over all weights
$\eta$
and time
$t+s$
prices
$p_{t+s}(S, \sigma)$
of contracts
$c$
on the book, and the
$\Delta_t$
and
$B$
correspond to the time-
$t$
replicating portfolio weights. If
$f$
is the AMM's initial capital, in words the above equation says "the probability the AMM's liabilities exceed its assets in the future is small."
If we do some analysis of this probability bound, we see that for
$\theta > 0$
:
\begin{align*} \mathbb{P}\left(\sum_{c} \eta_c p_{t+s}^c- \Delta_{t} S_{t+s} + B \geq f \right) \leq e^{\theta(B-f)} \, \mathbb{E}\left[e^{-\theta ( \sum_{c} \eta_c p_{t+s}^c - \Delta_{t} S_{t+s} )} \right] \end{align*}
When
$\theta = \theta^{*} >0$
is set to minimize the RHS, this corresponds to the entropic risk bound.
To use this to get the price of some option
$c'$
we can choose a price to keep this probability bound unchanged. Specifically, we choose
$p(c')$
s.t.
$e^{\theta^* p(c')} \mathbb{E}\left[e^{-\theta^* ( \sum_{c} \eta_c p_{t+s}^c + p_{t+s}(c')- \Delta_{t} S_{t+s} )} \right] =\mathbb{E}\left[e^{-\theta^* ( \sum_{c} \eta_c p_{t+s}^c - \Delta_{t} S_{t+s} )} \right]$
The LHS is obtained by adding the new contract to the liabilities and adding the price we collect today to the assets. Rearranging, we obtain
\begin{align*} p(c') = - \frac{1}{\theta^*} \log \mathbb{E}\left[e^{-\theta^* ( \sum_{c} \eta_c p_{t+s}^c + p_{t+s}(c')- \Delta_{t}S_{t +s})}\right] + \frac{1}{\theta^*} \log \mathbb{E}\left[e^{-\theta^* ( \sum_{c} \eta_c p_{t+s}^c - \Delta_{t} S_{t+s})}\right] \end{align*}
It turns out this is the entropic indifference price. In general, this will not be equal to the benchmark price of the same contract given the market implied volatility (even if the AMM has nothing on its book). The difference between the two prices represents the spread adjustment for the internal risk considerations of the AMM - in this case, the probability of insolvency is unchanged as a result of selling the option.
Other indifference pricing models, like conditional value-at-risk, can also be used to derive prices, and the marginal cost to each contract will differ according to the risk measure. Typically, the risk measure will generate a premium or a discount depending on whether a contract increases or decreases an assessment of insolvency costs.

## Key Qualitative Features of Indifference Prices

• Prices increase when a contract's marginal contribution to future bad-case costs increases
• Prices decrease when a contract reduces future bad-case costs on the margin
• Future costs are assessed based on a risk measure that defines the bad cases
• The value-at-risk measure generates compensation for contracts that contribute to losses in certain quantiles, like the 1% or 5% levels.
• The entropic risk measure (ERM) generates compensation for paths with extreme realizations that contribute disproportionately to the insolvency probability bound

## Implementation

Our price adjustments have the same key qualitative features of indifference prices and reflect essentially a tailored risk measure. Rather than using the general exponential expression above in the case of the ERM, which involves Monte Carlo simulations over paths whose construction is nontrivial and requires some subtle conditioning glossed over by the above notation, our practical implementation breaks this analysis into a few key steps that can be checked in python or C++ code in milliseconds. The approach we adopt also does not require assumptions about the distribution of
$S$
and hence can incorporate a variety of factors explicitly, including jump risk and vega risk.