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Simulation of square-root processes made simple: applications to the Heston model

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Pith's one-line read The iVi scheme simulates the integrated square-root process first, draws each increment from an Inverse Gaussian law, and then reconstructs the variance process, giving a simple, accurate, nonnegative Heston simulator.

desk verdict Simulating the integrated square-root process first via an Inverse Gaussian first-passage time is a fresh, simple idea with strong numerics; the paper is worth refereeing, but the 'exact limiting' and convergence claims need to be substantiated. read the letter →

arxiv 2412.11264 v2 pith:3EQTPQBP submitted 2024-12-15 q-fin.MF q-fin.CP

Eduardo Abi Jaber This is my paper · ORCID
classification q-fin.MFq-fin.CP MSC 60H1060H3565C0591G2091G60
keywords square-rootprocessHestonmodelInverseGaussiandistributionintegratedvariancesimulationschemenonnegativepreservationRiccatiequationtime-changedBrownianmotion
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper argues that the difficulty of simulating square-root dynamics is removable: instead of discretizing the process $V$ itself, one should first simulate its time integral $U_{t_i,t_{i+1}}$ between grid points, draw each increment from an Inverse Gaussian distribution, then recover the Brownian increment and the next value of $V$. The resulting iVi scheme is simple—one Inverse Gaussian draw and a few arithmetic operations per step—and provably keeps simulated variance nonnegative. It matches the first conditional moments of $V$, $U$, and the Brownian-integrated increment $Z$ exactly, and it reproduces the Inverse Gaussian limiting laws that appear in fast-mean-reversion/high-volatility-of-volatility regimes and for long maturities. Numerical tests on market-calibrated Heston parameters show accurate prices for variance swaps, volatility swaps, and implied-volatility slices with very few time steps, often one.

What carries the argument

The load-bearing object is the integrated process $U_{s,t}=\int_s^t V_r\,dr$ together with the time-changed Brownian representation $Z_{s,t}=W_{U_{s,t}}$ provided by the Dambis–Dubins–Schwarz theorem. This identity closes the right-endpoint discretization of the $U$-dynamics into the fixed-point equation $\widehat{U}_{i,i+1}=\alpha_i+\sigma_i W_{\widehat{U}_{i,i+1}}$, whose solution is a first-passage time of a drifted Brownian motion; the law of that passage time is Inverse Gaussian with parameters $(\alpha_i,\alpha_i^2/\sigma_i^2)$. The same Inverse Gaussian law re-emerges when the Riccati equation for the conditional characteristic function of $U$ is discretized by the same right-endpoint rule, which is the paper's mathematical justification for the choice of sampling distribution.

What would settle it

Run the iVi scheme on a parameter set outside the tested cases, for example $b>0$ with large $c$ and small $a$, and compare the empirical Laplace transform of $U_{0,T}$, $\mathbb{E}[e^{-uU_{0,T}}]$, with the exact value from the Heston characteristic function (B.1) over several maturities; if the relative error does not decay as the number of time steps grows to 1000, the right-endpoint approximation in (1.3) is the source and the Inverse Gaussian proxy is not valid in that regime.

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Extended reading notes

Core claim

The central discovery is that the integrated square-root process $U_{s,t}=\int_s^t V_r\,dr$ carries enough structure to be simulated directly, and that its conditional sampling law can be read off from an Euler-style right-endpoint discretization of the integrated dynamics. Writing $Z_{s,t}=\int_s^t\sqrt{V_r}\,dW_r$, the variation-of-constants form of the SDE gives $U_{t_i,t_{i+1}}\approx \alpha_i+\sigma_i Z_{t_i,t_{i+1}}$, and because $Z$ is a Brownian motion run at the clock $U$, this is the fixed-point equation $\widehat{U}=\alpha_i+\sigma_i W_{\widehat{U}}$. A natural solution is the first passage time of a drifted Brownian motion to level $\alpha_i$, whose law is Inverse Gaussian with mean $\alpha_i$ and shape $(\alpha_i/\sigma_i)^2$. The scheme draws $\widehat{U}$ from that law, sets $\widehat{Z}=(\widehat{U}-\alpha_i)/\sigma_i$, and updates $\widehat{V}$ from the original SDE; the paper proves that $\widehat{V}$ stays nonnegative and that first conditional moments are exact, and reports convergence for the scheme in a companion work.

Load-bearing premise

The whole scheme hinges on replacing an integral in the dynamics of $U$ by its right-endpoint value; if that approximation is poor in some parameter regime, the Inverse Gaussian law used at each step becomes a biased proxy for the true conditional distribution, and the error will feed into the Heston update.

Editorial extensions

If this is right

  • The simulated variance process $\widehat{V}$ never goes negative for any real $b$ and any nonnegative $a$, so the scheme needs no reflection or absorption fix.
  • First conditional moments of $V$, $U$, and $Z$ are matched exactly at every step, which makes variance-swap prices nearly unbiased even with one time step.
  • On calibrated fast-mean-reversion/high-volatility-of-volatility parameters, the one-step scheme already reproduces the correct Inverse Gaussian limiting law of $U$, and the reported implied-volatility slices stay accurate.
  • For Heston option pricing, each step adds only one independent standard Gaussian, and the experiments show accurate ITM/ATM/OTM option prices with 1–5 time steps.
  • The scheme has no switching thresholds or stored grids, so it is easy to code and appeared stable in low-sample-path tests.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because the scheme samples $U$ and $Z$ jointly from a first-passage law, it is naturally suited to payoffs on the whole integrated-variance trajectory, such as Asian-style vol products, rather than only terminal values.
  • The same 'simulate the integrated quantity first' idea may extend to other affine processes whose integrated component admits a tractable hitting-time representation; the paper notes that multivariate Wishart-type extensions are not straightforward.
  • A natural testable extension is to derive an explicit error bound for the right-endpoint approximation as a function of step size and parameters; such a bound would determine the regimes where the one-step version is reliable.
  • For local-stochastic-volatility or rough-Heston settings where the Heston component is simulated alongside an outer model, the scheme's low cost per path and few-step accuracy make it a candidate building block.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The manuscript proposes a new simulation scheme, called iVi, for the square-root process and the Heston model. The scheme first samples the integrated variance increment U from an Inverse Gaussian distribution whose parameters are obtained by a right-endpoint Euler discretization of the integrated process dynamics, then defines the Brownian-integral increment Z deterministically from U, and finally updates V through the exact drift relation. The main theoretical results are Theorem 1.3 (nonnegativity of the simulated V), Proposition 1.4 (exact first conditional moments for V, U, and Z), Proposition 1.5 and Remark 1.6 (the Inverse Gaussian law arises from an implicit Euler discretization of the Riccati equations), and Remark 1.7 (recovery of known Inverse Gaussian limiting laws in high-mean-reversion and long-maturity regimes). Numerical experiments compare the scheme to the QE scheme of Andersen and the Alfonsi scheme, for variance swaps, volatility swaps, Laplace transforms of integrated variance, and Heston call option prices and implied volatilities, across three calibrated parameter sets. The paper reports high accuracy with as few as one to five time steps. Section 4 states extensions to Volterra/rough Heston settings and refers to a companion paper for convergence proofs.

Significance. If the numerical findings are representative, the iVi scheme is a practically attractive addition to the simulation toolbox: it is parameter-free, simple to implement, preserves nonnegativity, matches first conditional moments exactly, and is competitive with, or better than, established schemes in the tested regimes, especially for short-maturity high-mean-reversion calibrations. The nonnegativity and moment-matching proofs are clean and easily verifiable, and the numerical study is extensive and reproducible in structure. The main limitation is that the accuracy that makes the scheme useful is not established theoretically in this manuscript; the right-endpoint discretization is heuristic, and convergence is deferred to a companion paper. The paper is therefore best read as a numerical-method proposal with strong empirical support rather than as a fully analyzed scheme.

major comments (3)
  1. [Section 1.1, Eqs. (1.2)-(1.4)] The defining approximation of the scheme is the replacement of the exact integrated dynamics (1.2) by the right-endpoint rule (1.3), which collapses the path of Z_{t_i,·} to its endpoint. There is no local weak-error estimate in the manuscript for this step, and Section 4 explicitly outsources convergence to Abi Jaber and Attal (2025). Because the accuracy claims in Sections 2 and 3 rest on this approximation, the manuscript does not, on its own, prove the scheme's accuracy. Please add a local error bound for the right-endpoint replacement (or state precisely what remains to be proved), or downgrade the claim to an empirical/numerical one.
  2. [Section 3, Eq. (3.2), Propositions 1.4-1.5] The Heston update (3.2) uses the joint law of (U_{i,i+1}, Z_{i,i+1}) through the term ρ Z_{i,i+1}. Proposition 1.4 only matches first conditional moments, and Proposition 1.5 compares only the marginal Laplace transform of U with the z-argument set to zero. Algorithm 1 makes Ẑ conditionally degenerate given Û, whereas the true Z_{i,i+1} conditional on the single value U_{i,i+1}=u is generally not degenerate, because the path of V inside the interval is not determined by the one integrated value. The numerical tests include ρ = -0.68, -0.70, and -0.9, but the paper gives no bound on the joint weak error, so the reported price accuracy is not guaranteed to persist for, e.g., |ρ| close to 1 or very short maturities. Please analyze the joint-law error or add a systematic stress test over ρ and maturity.
  3. [Abstract and Remark 1.7] The abstract states that the scheme 'yields the exact limiting Inverse Gaussian distributions' in two regimes. The manuscript does not prove these limiting results; it cites Mechkov (2015), Abi Jaber and De Carvalho (2024), McCrickerd (2019), and Forde and Jacquier (2011). Since the one-step distribution is Inverse Gaussian by construction, the matching is a consequence of known limit theorems. Please either give self-contained proofs of the limits (including the parameters of the limiting laws, which are not stated for the long-maturity case) or reformulate the abstract and Remark 1.7 as 'recovers/reproduces' the known limiting distributions.
minor comments (5)
  1. [Section 1.1 and Algorithm 1] For α_i=0, the convention IG(0,0)=0 is stated, but the update (0.5) still divides by σ_i; specify the convention for the boundary case, for example when c=0.
  2. [Proposition 1.5 and Eq. (1.9)] The proposition says that ψ̂ is 'a root' of the quadratic, but only later in Remark 1.6 is it stated that the relevant root has non-positive real part; make this explicit in the proposition.
  3. [Remark 1.6] The statement that φ̂ corresponds to a trapezoidal discretization is argued only asymptotically; a one-line bound on the remainder would make the remark precise.
  4. [Section 3, Eq. (3.2)] The independence of N_i from the random variables used in Algorithm 2 is stated in prose; it should be part of the algorithm specification for reproducibility.
  5. [Introduction and Appendix B] The reference to Duffie, Filipović, and Schachermayer (2003) appears twice in the same sentence; moreover, the branch of the square root in D(u,w) in Eq. (B.2) is not discussed, so the Albrecher et al. (2007) branch choice should be stated for numerical evaluation.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the iVi scheme's Inverse Gaussian law follows from an explicit right-endpoint approximation plus Dambis-Dubins-Schwarz time change, and the numerical accuracy claims are benchmarked against external Fourier references; the sole self-citation (companion convergence theorems) is not load-bearing for those claims.

full rationale

The scheme's construction is not circular. Algorithm 1 first defines alpha_i and sigma_i from the model parameters, then samples U from IG(alpha_i, (alpha_i/sigma_i)^2), sets Z=(U-alpha_i)/sigma_i, and updates V by the exact dynamics relation. The Inverse Gaussian law is derived, not assumed: the paper approximates the integral in Proposition 1.1 by the right-endpoint rule, applies Dambis-Dubins-Schwarz to write Z_{s,t} = Wtilde_{U_{s,t}}, and obtains the implicit first-passage equation U = alpha + sigma Wtilde_U, whose solution is an Inverse Gaussian distribution. This is a self-contained mathematical derivation. Proposition 1.4 (first-moment matching) is a consequence of defining alpha_i as the exact conditional mean of U, but the paper treats it only as a sanity check and benchmarks higher moments and option prices against external Fourier/reference values in Sections 2 and 3. Proposition 1.5 and Remark 1.6 restate the same right-endpoint discretization at the level of the Riccati equation; this is a consistency argument, not an input-output circularity. The only self-referential element is Section 4, where theoretical convergence is deferred to Abi Jaber and Attal (2025), a companion paper with overlapping authorship. That self-citation is a completeness gap for the convergence proof, but it is not load-bearing for the paper's empirical accuracy claims, which are supported by independent benchmark comparisons. The lack of an in-paper weak-error bound for the right-endpoint approximation is a correctness risk, not circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No new entities are postulated; the scheme is a numerical approximation of an existing process and does not introduce particles, forces, or new conserved quantities.

assumptions (4)
  • ad hoc to paper Right-endpoint Euler discretization of the integrated process dynamics and of the Riccati equation is sufficiently accurate for the intended step sizes and parameter regimes.
    Invoked at equations (0.3)-(0.4) and Remark 1.6; the scheme's accuracy depends on this approximation, and no error bound is proved in the paper.
  • standard math Dambis-Dubins-Schwarz theorem: the continuous local martingale Z_{s,*} can be represented as a time-changed Brownian motion.
    Used after (1.3) to write Z_{t_i,s} = W_{U_{t_i,s}} and derive the implicit equation (1.4).
  • ad hoc to paper The Inverse Gaussian law is an adequate conditional proxy for the pair (U_{i,i+1}, Z_{i,i+1}) given V_i.
    This is the defining choice of the scheme; it matches the first moments but the higher-order joint law is not proven.
  • domain assumption Known closed-form conditional Laplace transform and Riccati equations for the Heston model (B.1)-(B.3).
    Used as reference values and in the derivation of the Riccati discretization; standard affine-process result.

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Pith. "Pith review of Simulation of square-root processes made simple: applications to the Heston model." pith.science (2026). https://pith.science/paper/3EQTPQBP

@misc{pith2026241211264,
  author       = {Pith},
  title        = {Pith review of: Simulation of square-root processes made simple: applications to the Heston model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3EQTPQBP}},
  note         = {Machine review of arXiv:2412.11264}
}
read the original abstract

We introduce a simple, efficient and accurate nonnegative preserving numerical scheme for simulating the square-root process. The novel idea is to simulate the integrated square-root process first instead of the square-root process itself. Numerical experiments on realistic parameter sets, applied for the integrated process and the Heston model, display high precision with a very low number of time steps. As a bonus, our scheme yields the exact limiting Inverse Gaussian distributions of the integrated square-root process with only one single time-step in two scenarios: (i) for high mean-reversion and volatility-of-volatility regimes, regardless of maturity; and (ii) for long maturities, independent of the other parameters.

Figures

Figures reproduced from arXiv: 2412.11264 by the authors.

Figure 1
Figure 1. Heston’s Implied volatility surface with the iVi scheme with one single time step and 2 [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. displays sample paths of (V , b U, b Zb) constructed using the iVi scheme in Algorithm 1 for a > 0 (reflecting boundary) and a = 0 (absorbing boundary). Notice how in both cases, all sample paths of Vb remain nicely nonnegative and Ub non-decreasing. For the first row, Vb bounces back when it reaches 0 while it gets absorbed at 0 for the second row. 0.00 0.05 0.10 0.15 0.20 0.000 0.005 0.010 0.015 0.020 0.025 0.030 … view at source ↗
Figure 3
Figure 3. Errors on variance swaps, volatility swaps and Laplace transform of [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: ITM, ATM and OTM call options on S: error in prices in terms of number of time steps for the three cases with 2 million sample paths. 60 80 100 120 140 0.2 0.4 0.6 Implied volatility Number of steps : 1 iVi scheme QE scheme Alfonsi scheme Reference 60 80 100 120 140 St…
Figure 5
Figure 5. Figure 5: Case 1: Implied volatility slice for T = 1 and 2 million sample paths. 60 80 100 120 140 0.1 0.2 0.3 0.4 0.5 Implied volatility Number of steps : 1 iVi scheme QE scheme Alfonsi scheme Reference 60 80 100 120 140 Strike 10 4 10 2 Error MAE iVi MAE QE MAE Alfonsi 60 80 1…
Figure 6
Figure 6. Figure 6: Case 2: Implied volatility slice for T = 1 and 2 million sample paths. Finally, to illustrate the relevance of the iVi scheme for practical applications, for instance, when the Heston model is used as a component of a Local Stochastic Volatility model, a small number o…
Figure 7
Figure 7. Figure 7: Case 3: Implied volatility slice for T = 10 and 2 million sample paths. paths is typically used with a fixed number of time steps [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]
Figure 8
Figure 8. Figure 8: ITM, ATM and OTM call options on S: prices in terms of the number of sample paths for the three cases with 50 time steps. All in all, the iVi scheme seems pretty competitive across several parameter sets, strikes and maturities when pricing call options in the Heston m…
Figure 9
Figure 9. Figure 9: Implied volatility surface of the calibrated Heston model (green) on the market bid and [PITH_FULL_IMAGE:figures/full_fig_p017_9.png]
Figure 10
Figure 10. Figure 10: Implied volatility surface of the calibrated Heston model (green) on the market bid and [PITH_FULL_IMAGE:figures/full_fig_p018_10.png]

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Reference graph

Works this paper leans on

32 extracted references · 30 canonical work pages · cited by 4 Pith papers

  1. [1]

    Simulating integrated V olterra square-root processes and V olterra H eston models via I nverse G aussian

    Eduardo Abi Jaber and Elie Attal. Simulating integrated V olterra square-root processes and V olterra H eston models via I nverse G aussian. arXiv preprint arXiv:2504.19885, 2025

    arXiv 2025

  2. [2]

    Reconciling rough volatility with jumps

    Eduardo Abi Jaber and Nathan De Carvalho. Reconciling rough volatility with jumps. SIAM Journal on Financial Mathematics, 15 0 (3): 0 785--823, 2024

    work page 2024

  3. [3]

    The little H eston trap

    Hansj \"o rg Albrecher, Philipp Mayer, Wim Schoutens, and Jurgen Tistaert. The little H eston trap. Wilmott, 0 (1): 0 83--92, 2007

    work page 2007

  4. [4]

    On the discretization schemes for the CIR (and B essel squared) processes

    Aur \'e lien Alfonsi. On the discretization schemes for the CIR (and B essel squared) processes. Monte Carlo Methods Appl., 11 0 (4): 0 355--384, 2005

    work page 2005

  5. [5]

    High order discretization schemes for the CIR process: application to affine term structure and H eston models

    Aur \'e lien Alfonsi. High order discretization schemes for the CIR process: application to affine term structure and H eston models. Mathematics of computation, 79 0 (269): 0 209--237, 2010

    work page 2010

  6. [6]

    Simple and efficient simulation of the H eston stochastic volatility model

    Leif Andersen. Simple and efficient simulation of the H eston stochastic volatility model. Journal of Computational Finance, 11 0 (3): 0 1--43, 2008

    work page 2008

  7. [7]

    Simulating from the H eston model: A gamma approximation scheme

    Jean-Fran c ois B \'e gin, Myl \`e ne B \'e dard, and Patrice Gaillardetz. Simulating from the H eston model: A gamma approximation scheme. Monte Carlo Methods and Applications, 21 0 (3): 0 205--231, 2015

    work page 2015

  8. [8]

    Euler scheme for SDE s with non- L ipschitz diffusion coefficient: strong convergence

    Abdel Berkaoui, Mireille Bossy, and Awa Diop. Euler scheme for SDE s with non- L ipschitz diffusion coefficient: strong convergence. ESAIM: Probability and Statistics, 12: 0 1--11, 2008

    work page 2008

Show all 32 references
  1. [9]

    Exact simulation of stochastic volatility and other affine jump diffusion processes

    Mark Broadie and \"O zg \"u r Kaya. Exact simulation of stochastic volatility and other affine jump diffusion processes. Operations research, 54 0 (2): 0 217--231, 2006

    2006

  2. [10]

    Simulation schemes for the H eston model with P oisson conditioning

    Jaehyuk Choi and Yue Kuen Kwok. Simulation schemes for the H eston model with P oisson conditioning. European Journal of Operational Research, 314 0 (1): 0 363--376, 2024

    2024

  3. [11]

    A theory of the term structure of interest rates

    John C Cox, Jonathan E Ingersoll Jr, and Stephen A Ross. A theory of the term structure of interest rates. Econometrica: Journal of the Econometric Society, pages 385--407, 1985

    1985

  4. [12]

    Convergence of discretized stochastic (interest rate) processes with stochastic drift term

    Griselda Deelstra and Freddy Delbaen. Convergence of discretized stochastic (interest rate) processes with stochastic drift term. Applied stochastic models and data analysis, 14 0 (1): 0 77--84, 1998

    1998

  5. [13]

    Credit risk: pricing, measurement, and management

    Darrell Duffie and Kenneth J Singleton. Credit risk: pricing, measurement, and management. In Credit Risk. Princeton university press, 2012

    2012

  6. [14]

    Affine processes and applications in finance

    Darrell Duffie, Damir Filipovi \'c , and Walter Schachermayer. Affine processes and applications in finance. The Annals of Applied Probability, 13 0 (3): 0 984--1053, 2003

    2003

  7. [15]

    The large-maturity smile for the H eston model

    Martin Forde and Antoine Jacquier. The large-maturity smile for the H eston model. Finance and Stochastics, 15 0 (4): 0 755--780, 2011

    2011

  8. [16]

    The volatility surface: A Practitioner's Guide

    J Gatheral. The volatility surface: A Practitioner's Guide. John Wiley and Sons, Inc, 2011

    2011

  9. [17]

    Gamma expansion of the H eston stochastic volatility model

    Paul Glasserman and Kyoung-Kuk Kim. Gamma expansion of the H eston stochastic volatility model. Finance and Stochastics, 15: 0 267--296, 2011

    2011

  10. [18]

    o ngy and Mikl \'o s R \'a sonyi. A note on E uler approximations for SDE s with H \

    Istv \'a n Gy \"o ngy and Mikl \'o s R \'a sonyi. A note on E uler approximations for SDE s with H \"o lder continuous diffusion coefficients. Stochastic processes and their applications, 121 0 (10): 0 2189--2200, 2011

    2011

  11. [19]

    A closed-form solution for options with stochastic volatility with applications to bond and currency options

    Steven L Heston. A closed-form solution for options with stochastic volatility with applications to bond and currency options. The review of financial studies, 6 0 (2): 0 327--343, 1993

    1993

  12. [20]

    Convergence of M onte C arlo simulations involving the mean-reverting square root process

    Desmond J Higham and Xuerong Mao. Convergence of M onte C arlo simulations involving the mean-reverting square root process. Journal of Computational Finance, 8 0 (3): 0 35--61, 2005

    2005

  13. [21]

    Fast strong approximation M onte C arlo schemes for stochastic volatility models

    Christian Kahl and Peter J \"a ckel. Fast strong approximation M onte C arlo schemes for stochastic volatility models. Quantitative Finance, 6 0 (6): 0 513--536, 2006

    2006

  14. [22]

    Weak approximation of CKLS and CEV processes by discrete random variables

    Gytenis Lileika and Vigirdas Mackevi c ius. Weak approximation of CKLS and CEV processes by discrete random variables. Lithuanian Mathematical Journal, 60: 0 208--224, 2020

    2020

  15. [23]

    On spatially irregular ordinary differential equations and a pathwise volatility modelling framework

    Ryan McCrickerd. On spatially irregular ordinary differential equations and a pathwise volatility modelling framework. arXiv preprint arXiv:1902.01673, 2019

    1902 arXiv

  16. [24]

    Fast-reversion limit of the H eston model

    Serguei Mechkov. Fast-reversion limit of the H eston model. Available at SSRN 2418631, 2015

    2015

  17. [25]

    Generating random variates using transformations with multiple roots

    John R Michael, William R Schucany, and Roy W Haas. Generating random variates using transformations with multiple roots. The American Statistician, 30 0 (2): 0 88--90, 1976

    1976

  18. [26]

    First order strong approximations of scalar SDE s defined in a domain

    Andreas Neuenkirch and Lukasz Szpruch. First order strong approximations of scalar SDE s defined in a domain. Numerische Mathematik, 128: 0 103--136, 2014

    2014

  19. [27]

    Efficient simulation of the double H eston model

    Dylan Possama \" and Pierre Gauthier. Efficient simulation of the double H eston model. The IUP Journal of Computational Mathematics, 4 0 (3): 0 23--73, 2011

    2011

  20. [28]

    Continuous martingales and B rownian motion , volume 293

    Daniel Revuz and Marc Yor. Continuous martingales and B rownian motion , volume 293. Springer Science & Business Media, 2013

    2013

  21. [29]

    Laplace transforms and suprema of stochastic processes

    Klaus Sch \"u rger. Laplace transforms and suprema of stochastic processes. Advances in finance and stochastics: essays in honour of Dieter Sondermann, pages 285--294, 2002

    2002

  22. [30]

    Low-bias simulation scheme for the H eston model by I nverse G aussian approximation

    Shu Tong Tse and Justin WL Wan. Low-bias simulation scheme for the H eston model by I nverse G aussian approximation. Quantitative finance, 13 0 (6): 0 919--937, 2013

    2013

  23. [31]

    Efficient, almost exact simulation of the H eston stochastic volatility model

    Alexander Van Haastrecht and Antoon Pelsser. Efficient, almost exact simulation of the H eston stochastic volatility model. International Journal of Theoretical and Applied Finance, 13 0 (01): 0 1--43, 2010

    2010

  24. [32]

    A simple and accurate simulation approach to the H eston model

    Jianwei Zhu. A simple and accurate simulation approach to the H eston model. Journal of Derivatives, 18 0 (4): 0 26, 2011

    2011

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