Execution-Path-Dependent Draw Indexing

• Execution-Path-Dependent Draw Indexing is a method where random draws are assigned based on simulation control flow, emphasizing the impact of execution history on noise allocation.
• It addresses the challenge of stateful PRNGs in maintaining event-level invariant noise, ensuring that paired Monte Carlo estimators remain consistent across scenarios.
• The approach introduces deterministic draw allocation and counter-based hashing to preserve unbiased continuation value estimation and reproducibility in option pricing.

Searching arXiv for the supplied topic and related papers. arxiv_search(query="execution-path-dependent draw indexing", max_results=5, sort_by="relevance") Execution-path-dependent draw indexing denotes the assignment of random variates to modeled events through the realized execution history of a simulation rather than through an invariant event identity. In a stateful pseudorandom number generator (PRNG), each call advances mutable internal state, so branches, loops, and early exits alter which downstream event receives which draw. In path-dependent option pricing, an analogous issue appears when exercise, continuation, knock-out, knock-in, or reset decisions depend on the realized path: the simulation must ensure that each particle’s state evolution is driven by a fixed, reproducible set of draws, while continuation-value estimation remains consistent with the same realized future cashflows. The topic therefore spans both causal stochastic modeling and Monte Carlo pricing under stochastic volatility, where it is linked to common random numbers (CRNs), structural causal models (SCMs), explicit weak solutions for the Heston model, and stochastic-approximation (SA) alternatives to least-squares Monte Carlo (LSM) (Buffalo et al., 11 Mar 2026, Kouritzin, 2016).

1. Definition and basic mechanism

A stateful PRNG such as Mersenne Twister, xoshiro, or PCG keeps mutable internal state $s$; each call returns $u_k$ and advances the state. Under this mechanism, the random input received by a modeled event is determined by how many draws were consumed earlier in the execution. Execution-path-dependent draw indexing means precisely that the noise used for a modeled event depends on execution history rather than on the event’s identity (Buffalo et al., 11 Mar 2026).

This makes same-seed reuse fragile. A “same-seed CRN” implicitly assumes that the $k$-th draw refers to the same modeled event across scenarios. Stateful PRNGs guarantee only that the sequence itself is identical; they do not guarantee that the mapping from draw index to event identity is preserved when control flow changes. A branch that inserts or skips a single draw shifts all downstream indices.

The small branching example given in the literature is diagnostic. If infection of person 1 is simulated first, incubation time is drawn only when infection occurs, and infection of person 2 is simulated afterward, then a vaccine intervention that prevents infection in person 1 causes person 2 to receive a different downstream draw under the same base seed. The change in noise assignment is therefore caused by altered control flow rather than by the scientific mechanism for person 2’s event (Buffalo et al., 11 Mar 2026).

In path-dependent option pricing, the same general issue is expressed differently. Pricing American, Bermudan, barrier, or Asian claims requires full-path simulation of $(S_t,V_t)$ and execution decisions that depend on realized states. The cited Heston framework therefore imposes a precise mapping of random draws to particle, time, and factor indices so that each particle’s state evolution is driven by the same sequence of draws across time, independent of whether that particle later exercises early. Continuation-value estimation is then based on the same realized future path, with no artificial branching draws (Kouritzin, 2016).

2. Causal semantics, CRNs, and execution invariance

The causal interpretation developed for agent-based models formalizes the issue through SCMs. In the intended SCM, each modeled event $e \in E$ is assigned an exogenous variable $U_e \sim \text{Uniform}(0,1)$, independent across events, and outcomes under intervention $a$ satisfy

$Y^a = \Phi_a(U).$

Equivalently, for event-level potential outcomes,

$Y_e^a = h_e(S_e^a, U_e),$

where $S_e^a$ are the endogenous inputs under intervention $u_k$0. Under this semantics, interventions change the structural mechanisms or inputs, not the exogenous noise assigned to the event (Buffalo et al., 11 Mar 2026).

Stateful PRNGs violate that invariance. If a seeded PRNG produces a stream $u_k$1, then an event $u_k$2 occurring after $u_k$3 prior draws receives

$u_k$4

The exogenous input is therefore a function of execution order rather than event identity. The cited proposition, “Violation of Execution Invariance,” states that if the number of PRNG calls before event $u_k$5 differs between scenarios $u_k$6 and $u_k$7, then the draw index $u_k$8 becomes endogenous, event $u_k$9 receives different noise, and execution invariance is violated (Buffalo et al., 11 Mar 2026).

The statistical consequence is immediate in paired Monte Carlo estimation. For $k$0 paired replicates, the variance of the paired estimator $k$1 is

$k$2

CRNs are effective when $k$3, which typically requires the same underlying noise to be attached to the same modeled events across scenarios. Misaligned draw indices weaken this covariance and can even turn it negative; in that case, reusing the same base seed increases variance rather than reducing it (Buffalo et al., 11 Mar 2026).

The literature therefore treats execution-path-dependent indexing not merely as an implementation nuisance but as a mismatch between program-level causal structure and scientific causal structure. A plausible implication is that stochastic simulators can be mechanistically correct at the level of transition rules yet still produce causally incoherent paired comparisons if the exogenous noise is indexed by control flow rather than by event identity.

3. Path-dependent option pricing under Heston

In the Heston setting, execution-path dependence arises because option execution strategy depends on the realized path. American and Bermudan exercise, barrier activation or deactivation, and Asian state variables all require a simulation in which future cashflows and current stopping decisions are coupled along each realized path. The continuation value at monitoring time $k$4 is the conditional expectation $k$5, and its estimation must remain consistent with the path’s current state (Kouritzin, 2016).

The cited approach organizes this by simulating paths forward once per particle and then applying backward SA estimates of continuation values using the entire cross-section of particles’ realized future cashflows $k$6 and current basis values $k$7. No nested branching simulation of “what would have happened under alternate decisions” is performed. Instead, continuation values are estimated from cross-sectional data and then applied to each path’s realized state. Decisions at time $k$8 set $k$9 if exercise occurs; otherwise $(S_t,V_t)$0 remains later. The same path draws are used throughout, so no draw re-use across alternative branches is needed, and unbiasedness follows from LLN/SLLN (Kouritzin, 2016).

This organization turns draw indexing into a pathwise bookkeeping principle. Random draws are indexed by particle $(S_t,V_t)$1, time $(S_t,V_t)$2, and factor indices. The paper specifies variance Ornstein–Uhlenbeck (OU) draws $(S_t,V_t)$3, mapped via Box–Muller to normals used for $(S_t,V_t)$4 and $(S_t,V_t)$5 at $(S_t,V_t)$6 and $(S_t,V_t)$7, together with one Normal $(S_t,V_t)$8 per particle-time for the independent $(S_t,V_t)$9-term in the explicit price formula. Cross-sectional SA or LSM estimation at each discrete time uses the same particle-level draws. Early exercise therefore changes the stopping time but does not change the realized latent path that supports continuation-value estimation (Kouritzin, 2016).

A common misconception is that path-dependent execution necessarily requires branching simulation trees with separate random streams for “exercise” and “continue” alternatives. The cited algorithm rejects that construction: the exercise tree is implicit in $e \in E$0, while the full path is simulated once and reused. This eliminates a major source of draw-management complexity.

4. Explicit weak solutions and indexed simulation under the Heston model

The Heston dynamics under the risk-neutral measure are given in the cited notation by

$e \in E$1

with $e \in E$2. The paper uses the parameterization

$e \in E$3

Under Condition (C), $e \in E$4 for an integer $e \in E$5, the variance admits an explicit weak solution as a sum of squares of independent OU processes,

$e \in E$6

and the price can be written explicitly as

$e \in E$7

Conditional on the variance path, the $e \in E$8-integral is Gaussian with mean $e \in E$9 and variance $U_e \sim \text{Uniform}(0,1)$0, so the method samples $U_e \sim \text{Uniform}(0,1)$1 without Euler or Milstein approximation of stochastic integrals (Kouritzin, 2016).

When Condition (C) does not hold, the paper introduces the “closest explicit Heston” with $U_e \sim \text{Uniform}(0,1)$2 and drift adjustment $U_e \sim \text{Uniform}(0,1)$3, then recovers the target model by importance sampling with likelihood weight $U_e \sim \text{Uniform}(0,1)$4 up to the stopping time $U_e \sim \text{Uniform}(0,1)$5. The explicit likelihood ratio is

$U_e \sim \text{Uniform}(0,1)$6

Under the weighted measure, $U_e \sim \text{Uniform}(0,1)$7 solves the target Heston SDE up to $U_e \sim \text{Uniform}(0,1)$8; afterward it continues as the closest explicit Heston (Kouritzin, 2016).

The draw-indexing scheme that supports this construction is concrete.

Indexed draw	Role	Constraint
$U_e \sim \text{Uniform}(0,1)$9	Uniforms for OU half-step and full-step normals via Box–Muller	Indexed by particle, time, OU pair, and stage
$a$0	One Normal per particle-time for the independent $a$1-integral term	Independent of OU draws
Substream mapping	Deterministic mapping from $a$2 to RNG substreams	Ensures reproducibility

For $a$3 even, with $a$4, the algorithm generates four independent uniforms per OU pair and updates

$a$5

$a$6

with $a$7 and $a$8. Integrated variance on $a$9 is approximated by Simpson’s $Y^a = \Phi_a(U).$0 rule with $Y^a = \Phi_a(U).$1,

$Y^a = \Phi_a(U).$2

and the price update is

$Y^a = \Phi_a(U).$3

where $Y^a = \Phi_a(U).$4, $Y^a = \Phi_a(U).$5, $Y^a = \Phi_a(U).$6, $Y^a = \Phi_a(U).$7, and $Y^a = \Phi_a(U).$8. The weight is updated only while the variance stays above $Y^a = \Phi_a(U).$9; otherwise $Y_e^a = h_e(S_e^a, U_e),$0 (Kouritzin, 2016).

The significance of the indexing discipline is twofold. First, it preserves the independence structure: the $Y_e^a = h_e(S_e^a, U_e),$1 draw for the independent $Y_e^a = h_e(S_e^a, U_e),$2-integral must use separate substreams from the OU draws. Second, it preserves reproducibility: deterministic mappings such as

$Y_e^a = h_e(S_e^a, U_e),$3

ensure identical paths when recomputed and identical exercise decisions when the code is re-run (Kouritzin, 2016).

5. Stochastic approximation, continuation values, and implicit exercise trees

The paper replaces regression in LSM with SA in order to estimate continuation values without inverting ill-conditioned matrices. A basis $Y_e^a = h_e(S_e^a, U_e),$4 is chosen on $Y_e^a = h_e(S_e^a, U_e),$5, typically as tensor products $Y_e^a = h_e(S_e^a, U_e),$6, with examples including weighted Laguerre or rescaled Haar functions. The projection onto $Y_e^a = h_e(S_e^a, U_e),$7 approximates $Y_e^a = h_e(S_e^a, U_e),$8 (Kouritzin, 2016).

With i.i.d. particles $Y_e^a = h_e(S_e^a, U_e),$9 simulating paths $S_e^a$0 and recursively defined stopping times $S_e^a$1, the SA recursion at time $S_e^a$2 is

$S_e^a$3

applied whenever $S_e^a$4, with gain $S_e^a$5 and counter $S_e^a$6 incremented over eligible particles. Backward induction proceeds from $S_e^a$7 down to $S_e^a$8; after computing $S_e^a$9, the stopping rule is updated by setting $u_k$00 if

$u_k$01

and otherwise leaving $u_k$02 unchanged (Kouritzin, 2016).

The convergence statement is explicit: under exchangeable SLLNs given in the paper, $u_k$03 almost surely with respect to the weighted measure, and the estimator targets the same minimizer as least-squares regression while avoiding matrix inversion. Because SA uses the realized $u_k$04 from the already simulated paths, exercise decisions remain unbiased and consistent with realized continued cashflows. No nested conditional simulations are required at each state (Kouritzin, 2016).

This structure makes the exercise tree implicit rather than explicit. A plausible implication is that execution-path-dependent indexing in this context is less about counterfactual scenario alignment, as in SCM-based CRNs, and more about preserving a one-to-one relation between each realized particle path and the continuation values computed from the shared cross-section. The underlying principle is the same: random inputs must be attached stably to the objects that are scientifically relevant, whether those objects are modeled events or particle-time-factor tuples.

6. Remedies, comparisons, and limitations

The general remedy in causal stochastic modeling is event-keyed hashing with counter-based PRNGs such as Philox or Threefry. Each random variate is defined as an explicit function of a replicate key and a stable event identifier:

$u_k$05

Because counter-based generators are stateless pure functions, the same seed and event identifier produce the same variate regardless of execution order or parallel schedule. This restores event-indexed exogenous structure and SCM-style execution invariance. The required identifiers must be unique, stable across scenarios, invariant to execution order, and capable of supporting sub-draw indexing and non-occurring events (Buffalo et al., 11 Mar 2026).

In the Heston option-pricing setting, the remedy is not event-keyed hashing in the SCM sense but deterministic draw allocation indexed by particle, time, OU component, and stage, together with separate substreams for the independent $u_k$06-integral. The approach is explicitly designed so that no new draws are created for “continuation” versus “exercise” branches; SA uses realized continuation values from the same simulated path, and draw indexing remains fixed (Kouritzin, 2016).

The comparison with earlier Heston literature clarifies what is distinctive. Heston (1993) provides closed-form characteristic functions and European option pricing, but not pathwise simulation for heavily path-dependent features. Broadie and Kaya (2006) provide an exact simulation method for Heston using the distribution of integrated variance and the conditional distribution of $u_k$07, which eliminates discretization bias, but is described as primarily single-time marginal or limited discrete monitoring and not convenient for American or Asian options requiring many monitoring points and dynamic decisions. The cited 2016 paper emphasizes explicit weak pathwise solutions, importance sampling to bridge from the closest explicit Heston to arbitrary parameters, SA in place of regression, and natural handling of path-dependent execution without nested or branching simulation (Kouritzin, 2016).

The empirical performance claims reported in the paper are substantial: simulation speedups of up to $u_k$08 compared to Euler or Milstein for similar RMS error, pricing speedups of two orders of magnitude versus Euler-LSM or Milstein-LSM, and robustness against discretization bias while supporting larger basis dimension $u_k$09 because SA avoids ill-conditioned regression (Kouritzin, 2016). In the event-keyed CRN literature, performance discussion is more nuanced: Philox is reported as about $u_k$10 slower and Threefry as about $u_k$11 faster than Mersenne Twister in single-thread settings, but counter-based RNGs parallelize naturally and can outperform stateful generators in multicore or GPU settings (Buffalo et al., 11 Mar 2026).

Both literatures also emphasize limitations. In event-keyed CRNs, event identity is itself a modeling decision; slot-keyed and dyad-keyed identifiers encode different counterfactual semantics, and dynamic populations require stable identifiers not derived from execution order. Stream partitioning and jump-ahead are insufficient because interventions can change which events occur within a stream. In the Heston algorithm, near-zero variance creates instability in $u_k$12, motivating the stopping time $u_k$13 and a positive variance floor $u_k$14; extreme $u_k$15 and high vol-of-vol $u_k$16 require additional numerical care; and SA gain tuning remains practically important (Buffalo et al., 11 Mar 2026, Kouritzin, 2016).

A persistent misconception is that execution-path-dependent draw indexing is merely a variance-reduction detail. The cited causal analysis argues that it can instead invalidate paired counterfactual coherence, while the cited Heston analysis shows that disciplined draw allocation is necessary to preserve unbiased continuation-value estimation and reproducible pathwise execution under dynamic exercise. This suggests that draw indexing is best understood as part of the model specification itself rather than as a secondary implementation choice.