The trade
The rough Hawkes-Heston hedge leaves a trader with one number per rebalance, a ratio of two brackets.
Wang and Wang begin with the affine conditional transform H_t(w,T) = E[e^{wX_T} | F_t] and project it onto the stock. The resulting Galtchouk-Kunita-Watanabe integrand is
theta^w_t = (H_{t-}(w,T) / S_{t-}) * B^w_t / D.
The constant D = 1 + integral q(z)^2 nu(dz), with q(z) = e^{-Lambda z} - 1 representing the stock's relative jump when a mark of size z arrives. B^w_t contains three pieces. They are w, the leverage term rho sqrt(c) psi^w(T-t) from the Riccati-Volterra solution, and an integral against nu of (e^{(psi - Lambda w)z} - 1) q(z). The last piece captures covariance from the co-jump. A mark z lowers the log price by Lambda z while delivering a positive shock to the variance driver, so the same random measure moves both legs. Stock trading spans only e^{-Lambda z} - 1. A derivative has a different nonlinear exposure to the mark. For a call, the authors Fourier-synthesize along the Lewis contour Re w = 1/2 and hold 1 minus the weighted integral of theta^{w_lambda}.
An infinite-dimensional volatility state collapses into a one-dimensional hedge.
All numerical results below belong to the authors. The paper combines theory with a simulation of its own model and reports no market data. We could not implement the hedge on ours. Doing so would require us to calibrate and filter the latent variance, the Volterra memory process and the marked co-jump histories on which it conditions. End-of-day option data would force daily rebalancing, while the convergence result assumes continuous trading.
When nu = 0, the unspanned-risk density becomes Gamma^w_t = c(1 - rho^2)|psi^w_t|^2. Stock-only hedging therefore leaves volatility Brownian risk whenever psi^w_t is nonzero and |rho| < 1. This reduction applies to the nu = 0 special case. The calibrated specification instead uses nu(dz) = e^{-z}dz and carries the additional jump terms in (35). One check comes out exactly: with w = 1, the Riccati solution vanishes, B = D, and the formula gives 1/S_0 with zero residual for the normalized stock claim.
Why move the kernel?
The fractional kernel K_alpha(t) = t^{alpha-1}/Gamma(alpha), with alpha = 0.527, blows up at zero. Its memory integral is a convolution over [0,t), which makes the convention consequential because the singularity falls at the common jump time. Computing the hedge requires solving the Riccati-Volterra equation and then rebuilding the Volterra history against this singular kernel.
The hedge formula replaces it with K_eps(t) = (t+eps)^{alpha-1}/Gamma(alpha). This shifted kernel is bounded and completely monotone, and it continues to satisfy the resolvent conditions. Its exact L1 error is (eps^alpha + T^alpha - (T+eps)^alpha)/Gamma(alpha+1), with bound eps^alpha/Gamma(alpha+1).
The original market stays untouched. S, V, the driver Z, the initial curve g_0 and the filtration remain fixed; the shift affects only the deterministic kernel used to calculate the hedge. Each approximate strategy is assessed in the same L2(S), against the same payoff and stock. Two technical devices support the construction. Holdings use history integrated strictly before t. A 1-Lipschitz truncation map chi also retains the exact transform's pathwise envelope |H_{t-}|^2 <= S_{t-}/S_0, avoiding reliance on moment bounds.
Theorem 4.3 proves L2(S) convergence uniformly over compact Fourier intervals. Corollary 4.4 then constructs the diagonal (kernel, cutoff) sequence for the call. Doob yields E sup_{t<=T} |G_n(t) - G(t)|^2 <= 4||theta_n - theta_call||^2. The exact error identity is E[(Y - x_n - G_n(T))^2] = eps^2 + (x_n - C_0)^2 + ||theta_n - theta_call||^2, so the cross terms vanish.
No rate follows. The available bound is C_R(T_W(B) + B eps_n + eps_n^2), with a two-stage limit in B and n. Its cutoff sequence R_n comes from a diagonal argument and depends on knowing when fixed-cutoff errors fall below 1/m. It supplies no usable step size.
Five thousand paths, one market
The experiment is simulation only, using the exponential-mark calibration of Bondi et al.: alpha = 0.527, rho = -0.731, b = -1.812, c = 0.115, Lambda = 0.276, V_0 = 0.0079, T = 0.25, ATM call, 5,000 paths, 512 steps. The abstract calls it an illustration of the construction. That description fits.
With cutoff R = 20 fixed, the L1 kernel error declines from 0.193673 to 0.056184 as eps/T moves from 1/4 to 1/64. The squared strategy distance also declines monotonically, from 2.447e-6 (SE 0.173e-6) to 2.170e-7 (SE 0.017e-6). For context, the exact holdings on the Lewis contour satisfy ||theta^{w_lambda}||_{L2(S)} <= 1, so these squared distances are measured against a strategy norm bounded by one.
The worst-case-over-time squared gap between approximate and exact cumulative trading gains follows the same path, falling from 3.587e-6 to 3.306e-7. Deterministic diagnostics move in the expected direction. Riccati sup error over |lambda| <= 20 drops from 11.305 to 5.251, while memory L1 error falls from 0.035392 to 0.008963. Along the joint diagonal from (1/4, 5) to (1/64, 80), the combined squared capital-plus-strategy distance contracts from 2.432e-3 to 2.136e-6.
A practitioner should pause over that final table. Its benchmark is the unshifted-kernel strategy at R = 80, calculated on the same discretized grid. The paper acknowledges this, describing the statistic as convergence to the benchmark at the specified Fourier cutoff and time grid. The reported quantity therefore measures convergence to a numerical proxy. Table 1 contains five diagnostics covering the kernel, Riccati, memory, strategy and gain quantities. Section 5 never evaluates eps*^2. The authors derive the residual-risk density and spectral error representation, then leave them unevaluated. Anyone considering a stock-only hedge under this model still needs the amount of risk left behind.
Table 2 raises another issue. At R = 5, the capital component is 2.256e-3 and the strategy component is 1.766e-4. Fourier truncation damages the price more than the hedge ratio. The paper's own tail bound, sqrt(S_0 K)/(pi R), explains why: it decays only like 1/R.
Increasing M from 256 to 512 at eps/T = 1/32 changes the strategy statistic by 7.9% and the gain statistic by 16.2%, with both at levels of order 1e-7. The finest errors are thus comparable with discretization noise. For the smallest strategy distance, the Monte Carlo standard error is about 8% of the estimate. One strike, one maturity, one kernel family.
The practical boundary
The authors state the perimeter plainly. Historical-measure hedging, which would introduce an opportunity process and a variance-optimal martingale measure, remains future work alongside discrete rebalancing and static option positions. Transaction costs appear nowhere in the criterion or numerical experiment. Their second-moment criterion in Proposition 2.4 is a genuine, checkable contribution, although verification covers one parameter set: F_2(0.4) = -0.0133 < 0 and 0.4 < 2*Lambda = 0.552, exactly as condition (18) requires.
The parameters come wholesale from Bondi et al., whose identifiability we questioned in an earlier note on the Hawkes microfoundation. This caveat concerns the parameters. The theorem applies to any kernel satisfying the stated conditions.
An evaluation of eps*^2 at this calibration, placed beside the same quantity for a delta hedge from a simpler affine model, would change my view of the practical value.