Optimal Criteria

Asymptotic Approximations

Asymptotic methods [@cartea2015algorithmic, @gueant2013dealing] provide closed-form approximations to optimal quotes in limiting regimes.

Small Risk Aversion (\(\gamma \to 0\))

Certainty equivalent expansion: \(\text{CE}(X) \approx \mathbb{E}[X] - \frac{\gamma}{2}\text{Var}(X)\).

Linear quoting rule:

\[ \begin{aligned} \delta^{a*}\_\text{linear}(q) &\approx \frac{1}{k} - \frac{1}{2}\gamma\sigma^2(T - t)(2q - 1) \\ \delta^{b*}\_\text{linear}(q) &\approx \frac{1}{k} + \frac{1}{2}\gamma\sigma^2(T - t)(2q + 1) \end{aligned} \]

Infinite Horizon (\(T \to \infty\))

Stationary quotes from the principal eigenvector of the generator (see Exact Solutions in Stochastic Optimal Control for the matrix formulation):

\[\delta^{a*}_\infty(q) \approx \frac{1}{\gamma}\ln\!\left(1 + \frac{\gamma}{k}\right) - \frac{2q - 1}{2}\sqrt{\frac{\sigma^2\gamma}{2kA}\left(1 + \frac{\gamma}{k}\right)^{1 + k/\gamma}}\]

\[\delta^{b*}_\infty(q) \approx \frac{1}{\gamma}\ln\!\left(1 + \frac{\gamma}{k}\right) + \frac{2q + 1}{2}\sqrt{\frac{\sigma^2\gamma}{2kA}\left(1 + \frac{\gamma}{k}\right)^{1 + k/\gamma}}\]

Kelly Criterion and Optimal Position Sizing

The Kelly criterion maximizes the expected logarithmic growth rate of wealth [@merton1969lifetime]:

\[W^* = \arg\max_w \mathbb{E}[\log(1 + wR)]\]

where \(R\) is the per-period return and \(w\) is the fraction of wealth allocated.

Connection to CARA Utility

CARA utility \(U(x) = -e^{-\gamma x}\) with risk aversion \(\gamma\) gives the certainty equivalent \(\text{CE} = \mathbb{E}[X] - \frac{\gamma}{2}\text{Var}(X)\) for normally distributed \(X\). The optimal position maximizes CE, which for an asset with Sharpe ratio \(\text{SR} = \mu/\sigma\) is:

\[q^* = \frac{\mu}{\gamma\sigma^2}\]

For log-utility (Kelly), the certainty equivalent of \(\log(1 + wR)\) under small \(wR\) approximates \(wR - \frac{1}{2}w^2R^2\). Expanding \(\mathbb{E}[\log(1 + wR)] \approx w\mu - \frac{1}{2}w^2(\mu^2 + \sigma^2)\), the optimal fraction is the classic Kelly formula:

\[w^* \approx \frac{\mu}{\sigma^2 + \mu^2} \approx \frac{\mu}{\sigma^2} \quad (\mu \ll \sigma)\]

Comparing, the CARA optimal position \(q^* = \mu/(\gamma\sigma^2)\) maps to the Kelly fraction when \(\gamma \approx 1/q^*\) (risk aversion inversely proportional to the optimal position). In market making terms, CARA with \(\gamma = 1/\bar{q}\) approximates the growth-optimal strategy for a target inventory scale \(\bar{q}\).

Reservation Price and Target Inventory (CARA Approximation)

Note: this is a CARA-based approximation, not a rigorous Kelly HJB solution. The Avellaneda-Stoikov model with drift uses the CARA (exponential utility) ansatz \(V = -e^{-\gamma(X + qS + \theta(t,q))}\). The HJB separates because CARA factors across state variables — log-utility \(U(x) = \log x\) does not have this property.

In the CARA drift model, the optimal target inventory emerges from the quadratic drift-penalty tradeoff [@avellaneda2008high]:

\[q^* = \frac{k\mu}{2\tilde{\alpha}} = \frac{k\mu}{k\gamma\sigma^2} = \frac{\mu}{\gamma\sigma^2}\]

This matches the Kelly position to first order because CARA with normal returns and log-utility both produce mean-variance optimal positions. However, the value-function curvature (how aggressively the market maker adjusts quotes as inventory deviates from \(q^*\)) differs between the two utility functions. The CARA model uses the AS spread formula; a rigorous Kelly model would produce different spread dynamics.

Rigorous Kelly HJB

The true Kelly market-making problem maximizes expected log-terminal-wealth [@merton1969lifetime]:

\[V(t, S, q, X) = \sup_{\delta^a, \delta^b} \mathbb{E}\!\left[\log\!\left(X_T + q_T S_T\right) \;\middle\|\; S_t=S, q_t=q, X_t=X\right]\]

This objective does not admit the CARA separation ansatz. However, log-utility is homogeneous of degree 0 in \((S, X)\): \(V(t, cS, q, cX) = \log c + V(t, S, q, X)\), a scale symmetry under joint rescaling of price and cash. This admits the reduction:

\[V(t, S, q, X) = \log S + v(t, x, q), \qquad x = X/S\]

Substituting into the HJB and computing the Ito generator gives the reduced PDE for \(v(t, x, q)\):

\[\begin{aligned} 0 = \partial_t v &+ \tfrac{1}{2}\sigma^2(x^2 v_{xx} + 2x v_x - 1) \\ &+ \sup_{\delta^a} A e^{-k\delta^a}\! \big[v(t, x+1+\delta^a/S, q-1) - v(t, x, q)\big] \\ &+ \sup_{\delta^b} A e^{-k\delta^b}\! \big[v(t, x-1+\delta^b/S, q+1) - v(t, x, q)\big] \end{aligned}\]

Terminal: \(v(T, x, q) = \log(x + q)\).

The state reduces to \((x, q)\): 1 continuous dimension (wealth ratio) plus discrete inventory. The \(\delta/S\) terms in the \(x\)-jumps are O(1e-4) for typical parameter regimes and are neglected in the FD discretization (\(\Delta x = 1\) aligns grid steps with unit fill jumps).

Optimal Spreads (First-Order)

Neglecting \(\delta/S\), the FOC for each side gives:

\[\delta^{a*} = \frac{1}{k} + v(x, q) - v(x+1, q-1)\] \[\delta^{b*} = \frac{1}{k} + v(x, q) - v(x-1, q+1)\]

These are implicit in \(\delta\) when \(\delta/S\) is not neglected. The FD solver uses the first-order explicit form on the grid.

Contrast with CARA

Implemented in solver/src/models/kelly_hjb.rs and engine/src/strategies/kelly_rigorous.rs.

Engine Strategy (Approximation)

The KellyStrategy in the engine implements the CARA-based approximation: it estimates \(\mu\) and \(\sigma\) online, computes \(q^* = \mu/(\gamma\sigma^2)\), and shifts the AS reservation price toward \(q^*\) instead of zero. This is a heuristic — it uses the Kelley target with CARA spread dynamics.

Multi-Asset Kelly (Future Work)

For \(n\) correlated assets with drift vector \(\boldsymbol{\mu}\) and covariance \(\Sigma\), the Kelly-optimal fraction vector is:

\[\mathbf{w}^* = \Sigma^{-1}\boldsymbol{\mu}\]

In the market making context, this would correspond to simultaneous optimal quoting across multiple correlated instruments. The HJB framework would extend to a multi-dimensional inventory \(\mathbf{q} \in \mathbb{Z}^n\) with the risk penalty \(\frac{\gamma}{2}\mathbf{q}^\top\Sigma\mathbf{q}\). Not yet implemented.