On the Pricing of American Options under Stochastic Local Volatility and Stochastic Correlation via the RBSDE Framework

1 Introduction

• SLV models combine local- and stochastic-volatility features, with SABR given as a well-known example, and can provide a more realistic implied-volatility term structure.The passage specifically notes future smiles.Source: §1 · p. 2
• Pricing and hedging SLV options is challenging because of correlated Brownian motions and nonlinear local volatility, while existing methods can vary in performance across parameters.The passage notes reduced accuracy of one implied-volatility expansion at long maturities and for deep in- or out-of-the-money options.Source: §1 · p. 2
• Existing SLV models typically assume constant correlations, although financial correlations are empirically stochastic and constant assumptions may misestimate them and introduce correlation risk.The cited examples include correlations between asset returns and between asset prices and volatilities.Source: §1 · p. 2
• Adding stochastic correlation makes the single-asset SCLV problem three-dimensional, complicates American-option methods, and makes sensitivities to local volatility, correlation, and asset price nontrivial.The passage notes that a double-layer CTMC approximation would require a computationally challenging triple-layer structure.Source: §1 · p. 2
• The RBSDE framework handles early exercise and SCLV's multidimensional structure while computing prices and sensitivities and modelling future option-price evolution.The introduction also outlines XGBoost regression, convergence analysis, and applications to extended SABR and Heston SLV models.Source: §1 · pp. 2–3
• The authors extend SLV models with stochastic correlation and derive RBSDE pricing and hedging formulas for American options, reporting high numerical accuracy and convergence analysis.The framework also models future option prices and hedging strategies.Source: §1 · p. 3

2 Extension of general stochastic local volatility model with stochastic correlation

• The model extends SLV's constant correlation between volatility and asset Brownian motions by introducing stochastic correlation driven by a process.The model is called stochastic correlation and local volatility (SCLV) in the paper.Source: §2 · pp. 4–5
• The correlation structure specifies relationships among the correlation, volatility, and asset Brownian motions through functions of the stochastic correlation process.The correlation matrix is required to be positive semi-definite.Source: §2 · pp. 4–5
• The authors assume conditions on the process coefficients that guarantee unique solutions for the correlation and volatility processes and a unique solution for the asset process.The passage states that the coefficients are continuously differentiable for the asset process and L2-measurable for the correlation and volatility processes.Source: §2 · pp. 4–5
• The model class consolidates multiple SLV models extended with stochastic correlation, and the paper summarizes a variety of stochastic correlation processes.The text states that the process parameters satisfy κρ, σρ > 0 and ρ0, µρ ∈ (−1, 1).Source: §2 · p. 5, Table 1 · p. 6

3 Reflected Forward Backward Stochastic Differential Equations for pricing

• The RBSDE represents the American-option value as its backward process and includes a nondecreasing process that keeps the value above the payoff barrier.The backward process is identified with V(t, ρt, νt, St), and the obstacle is g(Xt).Source: §3 · pp. 6–7
• The RBSDE solution yields hedging strategies with respect to correlation, volatility, and spot price.The displayed formulation assigns these sensitivities to the components of Z.Source: §3 · pp. 7–8
• The pricing setup adjusts the correlation and volatility drifts using risk premiums, but those premiums are not uniquely specified because no tradable assets directly hedge these risks.The framework selects a measure from the family of equivalent local martingale measures so the resulting RBSDE has a unique solution.Source: §3 · p. 7
• The paper notes that its RBSDE can be derived using the historical P-measure, so pricing does not require seeking a risk-neutral measure for the spot-price dynamics.If the model is already under the risk-neutral measure, the corresponding RBSDE is a special case with λρ = λν = 0 and as(t, St) = rSt.Source: §3 · p. 10

4 Regression-based numerical schemes

• The proposed RBSDE discretizations require constructions distinct from standard BSDE regression schemes because they must handle reflection and the obstacle constraint.The paper also identifies the convergence analysis as new and more challenging because it must account for errors induced by reflection and obstacle approximation.Source: §4 · p. 10

4.1 The Semi-discrete scheme

• The semi-discrete scheme discretizes the forward SDE and backward process, then compares the approximation with the reflection barrier.The forward process can be sampled from its known distribution when available; otherwise, the Euler scheme may be used.Source: §4.1 · pp. 10–11

Semi-discrete RBSDE Scheme

• The semi-discrete RBSDE step is implicit, so iterative methods such as Picard iteration may be applied.The cited passage notes that related discretizations have been analyzed by Ma and Zhang (2005) and Bouchard and Chassagneux (2008).Source: Semi-discrete RBSDE Scheme · pp. 11–12

4.2 Fully discrete scheme

• The fully discrete scheme computes the semi-discrete scheme’s conditional expectations using regression, specifically XGBoost under a Markovian-process assumption.The method fits regressors to sampled data and uses their predictions recursively while proceeding backward in time.Source: §4.2 · p. 12, Fully discrete RBSDE Scheme · pp. 12–13

Fully discrete RBSDE Scheme

• The XGBoost formulation uses regularized tree ensembles and greedily selects tree splits by loss reduction, pruning nodes with negative loss reduction.The regularization term controls model complexity and helps prevent overfitting.Source: Fully discrete RBSDE Scheme · pp. 15–16

4.3 Convergence analysis

• Under Assumption (H3), the convergence analysis states an error estimate for the semi-discrete RBSDE scheme.The analysis treats the equidistant time discretization and assumes Picard iteration errors can be neglected with sufficiently many iterations.Source: §4.3 · pp. 17–18
• Choosing γ ≥ 6dL^2 makes the relevant coefficient nonnegative and yields a bound with constants depending on L, γ, d, and T.The proof selects generic constants bC with 0 < bC < 1/(2d).Source: §4.3 · pp. 19–20
• Induction and taking expectations convert the stepwise inequality into an expected error bound.The proof then introduces quantities based on (55) to rewrite the inequality.Source: §4.3 · p. 20
• The proof concludes by incorporating the known discretization errors defined in (38) and (39).Source: §4.3 · p. 20

5 Applications of RBSDE approach to SCLV models

• The SABSR model is an example of the broader SCLV class considered in the applications.The paper also identifies classical SABR and Heston SLV extended with stochastic correlation as examples in this class.Source: §5 · p. 20, §5.1 · p. 21, §5.2 · p. 21

5.1 Extension of the classical SABR model

• The classical SABR extension adds stochastic correlation and is named the Stochastic Alpha Beta Stochastic Rho (SABSR) model.Its American-option pricing RBSDE is obtained by specializing the general equation to the model dynamics.Source: §5.1 · pp. 20–21

5.2 Extension of the Heston SLV model

• The Heston SLV extension incorporates stochastic correlation and is named the Heston stochastic correlation local volatility (HSCLV) model.Its American-option pricing RBSDE is derived by specializing the general equation to the model dynamics.Source: §5.2 · pp. 21–22

6 Numerical experiments

• Experiments use XGBoost with 150 trees, depth 2, learning rate 0.9, early stopping, and 10 Picard iterations; forward processes are simulated when analytical solutions are unavailable.Approximations are averaged over ten independently seeded runs.Source: §6 · pp. 22–23
• The experiments report delta hedges, while ill-conditioned systems limit discussion of other hedge sensitivities, whose analysis and treatment are deferred to future work.The asset-price process is the last forward component, so delta hedging carries subscript d in the reported approximations.Source: §6 · p. 23

6.1.1 American options with the SABR model

• At NT=100, SABR RBSDE prices attain accuracy comparable to Cui et al.’s CTMC approximation, and the reported comparisons show high accuracy with stable convergence.The experiments use the parameter values reported in Cui et al. and apply a logarithmic transformation to the asset price.Source: §6.1.1 · p. 24, Table 2 · p. 24

6.1.2 American options under the SABSR model

• All SABSR RBSDE approximations fall within the corresponding DPDB confidence intervals, with runtimes of approximately 210 seconds versus 637 seconds for DPDB training.The comparison uses NT=400 and the Table 3 parameter setting.Source: §6.1.2 · p. 25, Table 3 · p. 25
• As SABSR mean-reversion speed κρ increases, price and delta-hedging differences from the constant-correlation results decrease.The SCP is initialized at and has a long-term mean of −0.4, matching the constant-correlation value used for comparison.Source: §6.1.2 · pp. 24–25, Figure 1 · p. 26

6.2.1 American options under the HSLV model

• For American options under the HSLV model, the RBSDE scheme is reported to achieve high pricing accuracy against the CTMC comparison.The comparison is reported in Table 4.Source: §6.2.1 · p. 27, Table 4 · p. 27

6.2.2 American options under the HSCLV model

• At NT=30, HSCLV RBSDE runs take approximately 35 seconds versus 639 seconds for DPDB training, while stochastic-correlation effects nearly disappear for deep in-the-money options under the tested parameters.The study varies the SCP long-term mean while holding its initial value and mean-reversion speed fixed.Source: §6.2.2 · p. 27, Table 5 · p. 28

• Stochastic correlation is introduced into SLV models as a response to correlation risk, which constant-correlation assumptions may misestimate.Source: §7 · p. 28
• The RBSDE approach prices and hedges American options under stochastic correlation, volatility, and spot-price dynamics, and computes future option-price evolution.Source: §7 · p. 28
• Figure 4 depicts future American put prices under HSCLV for varying β, with S0 = 100, K = 105, and µρ = −0.2.Source: Figure 4 · p. 29
• Comparisons with state-of-the-art deep-learning methods demonstrate the RBSDE approach’s effectiveness, regression algorithm’s accuracy, and stochastic correlation’s impact on pricing and delta-hedging.Source: §7 · p. 29
• Robust approximations for ill-conditioned hedging systems and faster RBSDE solvers, particularly for efficient SCLV calibration, remain future research needs.Source: §7 · p. 29

• The work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), Project No. [581639701].Source: Funding · p. 29