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From the paper, High-frequency trading in a limit order book, (Avellaneda, 2006), from equations (16) and (17), the reservation price is given by

$$ \begin{aligned} \theta_t + \dfrac{1}{2} \sigma^2 \theta_{s s} - \dfrac{1}{2} \sigma^2 \gamma \theta_s^2 & + \max_{\delta^b}\left[\frac{\lambda^b\left(\delta^b\right)}{\gamma}\left[1-\mathrm{e}^{\gamma\left(s-\delta^b-r^b\right)}\right]\right] \\ & + \max _{\delta^b}\left[\frac{\lambda^a\left(\delta^a\right)}{\gamma}\left[1-\mathrm{e}^{-\gamma\left(s+\delta^a-r^a\right)}\right]\right]=0, \end{aligned} $$ with

Then we have: $$ \begin{aligned} &\delta^b = s-\theta(s, q, t)-\theta(s, q+1, t)+\frac{1}{\gamma} \ln \left(1-\gamma \frac{\lambda^b\left(\delta^b\right)}{\left(\partial \lambda^b / \partial \delta\right)\left(\delta^b\right)}\right)\\ &\delta^a =\theta(s, q, t)-\theta(s, q-1, t)-s+\frac{1}{\gamma} \ln \left(1-\gamma \frac{\lambda^a\left(\delta^a\right)}{\left(\partial \lambda^a / \partial \delta\right)\left(\delta^a\right)}\right) \end{aligned} $$

where $ q \in [q_{min},q_{min}+1,..,q_{max}]$, is the inventory.

I am struggling to understand the fact that we can only get $\delta$ on a truncated domain, but we require a solution for $\theta$ on the full domain, but $\theta$ at the next step is dependent on the $\delta$ of the previous step which will be truncated… More explicitly:

Q1)

It would seem that when solving for $\delta_t^a$ and $\delta_t^b$, we can only get $\delta_t^a$ on the truncated domain $q \in [q_{min} + 1, q_{min}+2,..,q_{max}]$ and $\delta_t^b$ on the truncated domain $q \in [q_{min}, q_{min}+1,..,q_{max}-1]$, because we are taking the forward and backwards difference of the solution of the PDE, as shown in the equations of $\delta$. But then for $\delta_{t+1}^a$ and $\delta_{t+1}^b$ on that same truncated domain, we would require $\theta(t+1,q)$ on $[q_{min},q_{max}]$, which then requires $\delta_t^a$ and $\delta_t^b$ on the FULL domain, which we don’t have, as we only get the solution on the truncated domain.

How do we solve this issue? In equations 23 and 25 of Aydogan, et al, (2022), although they use the stochastic volatility version, they also show that the optimal bid-ask’s are only solved on the truncated domain, but similarly to Avellaneda, their utility function requires the optimal bid-ask from the previous step on the full domain… They both do not give a $q$ boundary condition $\theta(t,q), \; \forall t$, and both do not explain how they calculate $\theta(t,q)$ on the full domain when they only have $\delta_t$ on the truncated domain.

Q2)

If we instead decided to instead market make the option (not the underlying) and delta-hedge by placing MOs on the equity market simultaneously, we can still profit because typically the option spread is larger than the underlying’s. If we then decided to also do a gamma-hedge, so a gamma-delta-hedge, and say we have some type of confidence in our prediction of tomorrow’s volatility (volatility clusters), could it be reasonable to remove the inventory, $q$ from the utility function?

Since we’ve hedged away the delta, gamma (and the vol of tomorrow is most likely to be similar as to today), we wouldn’t have a lot of exposure with holding options overnight, and therefore there seems there shouldn’t be much of a penalty to having inventory after market closes.

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Q1: Boundary Conditions in HFT Models

In financial modeling, particularly for HFT in limit order books, dealing with boundary conditions is crucial when the model's domain is truncated due to the discretization of inventory levels. A common approach to address the mismatch between truncated and full domains for variables like (\delta_a) and (\delta_b) involves extrapolation or assumptions of stability near the boundaries.

For instance, Avellaneda & Stoikov (2008) tackle this by implicitly assuming boundary behavior that ensures no arbitrage and maintains model consistency across inventory levels. When explicit boundary conditions are not provided, a practical approach is to extrapolate (\delta) values from the interior of the domain or to apply economic rationale to infer boundary behavior (Cartea et al., 2015).

References:

  • Avellaneda, M., & Stoikov, S. (2008). High-frequency trading in a limit order book. Quantitative Finance, 8(3), 217-224.
  • Cartea, Á., Jaimungal, S., & Penalva, J. (2015). Algorithmic and High-Frequency Trading. Cambridge University Press.

Q2: Market Making in Options and Hedging Strategies

When considering market making in options and employing delta-gamma hedging strategies, the necessity of including inventory levels in the utility function is debated. Effective hedging can theoretically mitigate the risks associated with inventory holding, especially when confident in the prediction of volatility clustering (Foucault, Pagano, & Röell, 2013).

However, it's critical to acknowledge that no hedging strategy entirely eliminates market risks, including liquidity risks and model inaccuracies. While delta-gamma hedging reduces exposure to price movements and curvature, it does not cover jump risks or transaction costs, which can significantly impact profitability in high-frequency settings.

Reference:

  • Foucault, T., Pagano, M., & Röell, A. (2013). Market Liquidity: Theory, Evidence, and Policy. Oxford University Press.
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