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Demand-Based Price Adjustments

Bid-ask spreads are a function of internal risk assessments
The Arrow pricing engine adjusts the benchmark according to risk assessments of the AMM's obligations. To inform these risk assessments, we run scenario analyses to evaluate bad-case future hedging costs. This 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 that position.
It is worth explaining this connection in more detail. Note that roughly, we want to adjust the baseline price to help keep the probability of high future hedging costs low:
P(cηcpt+scΔtSt+s+Bf)<ϵ\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
ff
is a capital bound, and
ϵ\epsilon
is a probability bound,
ss
is the time horizon for the rebalance, and where the distribution is over
S,σS, \sigma
. The summation is over all weights and time
t+st+s
prices of contracts on the book, and the
Δt\Delta_t
and
BB
correspond to the time
tt
replicating portfolio weights.
If we do some analysis of this bound, we see that:
P(cηcpt+scΔtSt+s+Bf)eθ(Bf)E[eθ(cηcpt+scΔtSt+s)]\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*}
which corresponds to the entropic risk bound that the entropic value at risk indifference price is drawn from when
θ=θ>0\theta = \theta^{*} >0
is set to hit the lower bound of the RHS. To use this to get the price of some option
cc'
we choose a price to keep this probability bound unchanged. Specifically, we choose
p(c)p(c')
s.t.
eθp(c)E[eθ(cηcpt+sc+pt+s(c)ΔtSt+s)]=E[eθ(cηcpt+scΔtSt+s)]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]
In other words
p(c)=1θlogE[eθ(cηcpt+sc+pt+s(c)ΔtSt+s)]+1θlogE[eθ(cηcpt+scΔtSt+s)]\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*}
This corresponds to the entropic indifference price. It can be shown that this formulation sets prices in a way that is equivalent to the case where the market maker is optimizing a CARA utility function over future payouts. 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 which reflects the additional risk considerations of the AMM (note the sign convention in the above implies the AMM is selling).
Other indifference pricing models, like conditional value at risk, yield similar results, but the emphasis on the marginal cost to each contract comes as a function of the choice of risk measure. In any case, the prices are chosen to satisfy some key qualitative properties that our simple adjustments also reflect.

Key Qualitative Features of Indifference Prices

  • Prices increase when the marginal contribution to the probability of high future costs increases, where the costs are measured by the risk measure.
    • The entropic risk measure places more weight on highly unlikely paths with extreme realizations.
    • The value-at-risk measure generates compensation for contracts that contribute to losses in certain quantiles, like the 1% or 5% levels.
  • Prices decrease when a contract reduces bad-case hedging cost probabilities.
  • The risk is assessed based on future costs in bad cases, whether a 1% largest loss in VAR or the weight on a tail path in the ERM

Implementation

Rather than using the general exponential expression, 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
SS
and hence can incorporate a variety of factors explicitly, including jump risk and vega risk. The price adjustments we obtain have the same qualitative features of indifference prices, and reflect essentially a tailored risk measure.
During the development of these rules, the Arrow team was advised by former options market makers from Citadel and Optiver as well as crypto-native options market making partners.