SOC Models Without Exact Solutions

These problems have no closed-form value or policy in general. The Heston and Hawkes market-making extensions remove the closed form of a model that has one; the American put is an optimal-stopping problem whose free boundary has no closed form at all. Each HJB, PDE, or obstacle system is stated here as the ground truth the numerical solvers discretize. Each is validated numerical-versus-numerical (finite difference against least-squares Monte Carlo) and by a degeneracy reduction to a model with a closed form, recorded in Limits of Stochastic Optimal Control Models. The numerical methods are on Numerical Schemes; the closed-form problems are on Exact Solutions in Stochastic Optimal Control.

Heston stochastic-volatility extension

With stochastic volatility, the mid-price variance is a CIR process. Writing \(\nu_t\) for the variance, \(\kappa\) for its mean-reversion speed (distinct from the fill-decay \(k\)), \(\theta_\nu\) its long-run level, \(\xi_\nu\) its vol-of-vol, and \(\rho_\nu\) the spot-variance correlation:

\[ dS_t = \sqrt{\nu_t}\,dW_t^S, \qquad d\nu_t = \kappa(\theta_\nu - \nu_t)\,dt + \xi_\nu\sqrt{\nu_t}\,dW_t^\nu, \qquad d\langle W^S, W^\nu\rangle_t = \rho_\nu\,dt. \]

The multiplicative ansatz \(V(t,x,q,S,\nu) = -\exp\big(-\gamma(x + qS)\big)\, w_q(t,\nu)\) leaves a linear parabolic PDE system for \(w_q(\tau, \nu)\) in \(\tau = T - t\):

\[ \begin{aligned} \partial_\tau w_q &= \tfrac12 \xi_\nu^2 \nu\, \partial_\nu^2 w_q + \big[\kappa(\theta_\nu - \nu) + \rho_\nu \xi_\nu \gamma \nu q\big]\,\partial_\nu w_q - \tfrac12 \gamma^2 \nu q^2 w_q \\ &\quad + A\Big(1 + \tfrac{\gamma}{k}\Big)^{-(1 + k/\gamma)} \Big(w_{q-1}\mathbf{1}_{q > -Q} + w_{q+1}\mathbf{1}_{q < Q}\Big), \end{aligned} \]

with initial condition \(w_q(0, \nu) = 1\). The optimal spreads are

\[ \begin{aligned} \delta^{a*}(t,q,\nu) &= \delta_0 + \frac{1}{\gamma}\ln\!\Big(\frac{w_{q-1}(t,\nu)}{w_q(t,\nu)}\Big), \\ \delta^{b*}(t,q,\nu) &= \delta_0 + \frac{1}{\gamma}\ln\!\Big(\frac{w_{q+1}(t,\nu)}{w_q(t,\nu)}\Big), \end{aligned} \qquad \delta_0 = \frac{1}{\gamma}\ln\!\Big(1 + \frac{\gamma}{k}\Big). \]

See [@heston1993closed] for the model and [@lord2010comparison] for the simulation schemes.

Hawkes order-flow extension

Self-exciting arrival intensities couple the fill process to its own history [@hawkes2018hawkes]. The intensity is governed by the process defined on the processes page.

Unilateral Hawkes

Each side has its own self-exciting intensity

\[d\lambda_t^j = \beta\big(\mu - \lambda_t^j\big)\,dt + \alpha\,dN_t^j, \qquad j \in \lbrace a, b\rbrace,\]

and the effective fill rate is \(\lambda_t^j e^{-k\delta^j}\). The fluid (mean-field) approximation replaces the discrete jumps with a continuous drift,

\[\frac{d\lambda_t}{dt} = \beta(\mu - \lambda_t) + \alpha\lambda_t,\]

with stationary mean \(\lambda^* = \dfrac{\mu}{1 - \alpha/\beta}\) and stability condition \(\alpha/\beta < 1\).

Optimal quotes (bilateral)

With state \((q, \lambda^a, \lambda^b)\) the affine ansatz

\[\theta(t, q, \lambda^a, \lambda^b) = C(\tau, q) + h_a(\tau, q)\,\lambda^a + h_b(\tau, q)\,\lambda^b\]

gives coupled ODEs for \(C\), \(h_a\), \(h_b\). The optimal quotes decompose into a baseline inventory-control term plus an excitation adjustment:

\[ \begin{aligned} \delta^{a*} &= \delta_0 + C(\tau, q) - C(\tau, q-1) + h_a(\tau, q)\,\lambda^a - h_a(\tau, q-1)(\lambda^a + \alpha), \\ \delta^{b*} &= \delta_0 + C(\tau, q) - C(\tau, q+1) + h_b(\tau, q)\,\lambda^b - h_b(\tau, q+1)(\lambda^b + \alpha), \end{aligned} \]

where \(\delta_0 = \tfrac1\gamma\ln(1 + \tfrac{\gamma}{k})\).

Order-flow imbalance

The mid-price responds to net order flow with sensitivity \(\eta_{\text{OFI}}\):

\[dS_t = \sigma\,dW_t + \eta_{\text{OFI}}\big(dN_t^b - dN_t^a\big).\]

American put (optimal stopping)

The American put is the optimal-stopping value

\[ V(t, S) = \sup_{\tau \in [t, T]} \mathbb{E}\!\Big[e^{-r(\tau - t)}(K - S_\tau)_+ \;\Big|\; S_t = S\Big], \]

under risk-neutral geometric Brownian motion \(dS/S = r\,dt + \sigma\,dW\). The value satisfies the linear-complementarity (obstacle) formulation

\[ \min\!\Big( -\partial_t V - \mathcal{L} V + r V,\; V - (K - S)_+ \Big) = 0, \]

with generator \(\mathcal{L} V = rS\,\partial_S V + \tfrac12\sigma^2 S^2\,\partial_S^2 V\) and terminal condition \(V(T, S) = (K - S)_+\).

There is no closed-form value for general \(r, \sigma, T\); a reference solution is obtained with a fine-grid finite-difference scheme. In the special case \(r = 0\), early exercise is never strictly optimal and the American value coincides with the Black-Scholes European put [@black1973pricing]:

\[ \begin{aligned} V_{\mathrm{BS}}(S) &= K\,\Phi(-d_2) - S\,\Phi(-d_1), \\ d_1 &= \frac{\ln(S/K) + (r + \tfrac12\sigma^2)T}{\sigma\sqrt{T}}, \\ d_2 &= d_1 - \sigma\sqrt{T}. \end{aligned} \]

The early-exercise boundary \(S^*(t)\), below which immediate exercise is optimal, is obtained numerically rather than in closed form.