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arxiv: 1907.01846 · v1 · pith:TEXEOC57new · submitted 2019-07-03 · 🧮 math.PR

Option pricing in fractional Heston-type model

Yuliya Mishura , Anton Yurchenko-Tytarenko This is my paper

Pith reviewed 2026-05-25 10:00 UTC · model grok-4.3

classification 🧮 math.PR
keywords fractional Heston modeloption pricingdiscretization schemeMalliavin calculusconvergence ratestochastic volatilityMonte Carlo simulation
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The pith

Discretization schemes for the fractional Heston model converge in expectation to the true option price at a specific rate.

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

The paper develops numerical methods for pricing options when the volatility follows a fractional process with Hurst index greater than 1/2. Explicit closed-form prices are unavailable, so the authors introduce discrete approximations to the volatility and asset price paths. They prove that the expected payoff under these approximations approaches the true expectation as the time steps become finer, and they determine the speed of this convergence. To handle payoffs that jump at certain points, they derive an alternative expression for the expectation using Malliavin calculus that smooths the functional inside.

Core claim

In the fractional Heston-type model with H > 1/2, discretization schemes Ŷ^n and Ŝ^n for the volatility and price processes satisfy E f(Ŝ^n_T) → E f(S_T) as the mesh tends to zero, with a calculable rate of convergence; additionally, Malliavin calculus supplies an alternative formula for E f(S_T) where the functional under the expectation is smooth even when the payoff f has discontinuities.

What carries the argument

The discretization schemes Ŷ^n and Ŝ^n together with Malliavin calculus techniques to smooth the payoff functional.

If this is right

  • Monte Carlo simulation using the discrete schemes yields accurate option prices with known error bounds.
  • The rate of convergence informs how fine the partition must be for a desired precision.
  • The smoothed Malliavin formula enables efficient computation for payoffs with jumps without direct simulation bias.
  • Option pricing becomes feasible in this non-Markovian fractional setting without closed forms.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar discretization and Malliavin approaches could extend to other fractional stochastic volatility models.
  • Testing the convergence rate in numerical experiments would validate the theoretical bounds.
  • The method might apply to risk management calculations beyond simple option payoffs.

Load-bearing premise

The fractional Heston-type model with H > 1/2 has sufficient path regularity so that discretization errors can be controlled and Malliavin calculus applies to the payoff.

What would settle it

Numerical simulations showing that the observed convergence rate of E f(Ŝ^n_T) to E f(S_T) differs from the calculated rate as mesh size decreases would falsify the convergence claim.

Figures

Figures reproduced from arXiv: 1907.01846 by Anton Yurchenko-Tytarenko, Yuliya Mishura.

Figure 1
Figure 1. Figure 1: Three sample trajectories of the process Y obtained by approximation scheme (15); T = 1, κ = 1, θ = 1, ν = 0.14, Y0 = 1, H = 0.7 and ∆n = 0.0001. For the sake of simplicity, instead of linear interpolation between the points of the partition, we put Yˆ n t = Yˆ n t n k for t ∈ [Yˆ n t n k , Yˆ n t n k+1 ). It should be noted that in this case speed of convergence of approximations remains the same as in Th… view at source ↗
Figure 2
Figure 2. Figure 2: Three sample trajectories of the process Yˆ n for H = 0.3, T = 1, κ = 1, θ = 1, ν = 0.14, Y0 = 1 and ∆n = 0.0001. Corollary 5.1. Approximating processes Yˆ n have bounded exponential moments, i.e. for any x > 0 and % < 2: sup n≥1 E exp ( x sup t∈[0,T] (Yˆ n t ) % ) < ∞. Remark 5.3. From Theorem 5.2, Corollary 5.1 and Assumption 2 (ii), using the same argument as in the proof of Theorem 3.5, it is easy to v… view at source ↗
Figure 3
Figure 3. Figure 3: Box-and-whisker plots of Monte-Carlo estimates of Ef(ST ) using smoothed formula; in all cases T = 1, κ = 1, θ = 1, ν = 0.14, µ = 0.5, ρ = 0, σ = 0.5 (x + 0.01)0.9 , H = 0.7; (a) f(x) = (x − 1)+, (b) f(x) = 1[0.5,1](x), (c) f(x) = 1(0.5,∞)(x) + 1 2 P6 k=2 1(0.5k,∞)(x) 7. Proofs Proof of Theorem 3.3. Denote α := 1 − β and let ε > 0 be fixed. By applying the chain rule, we obtain: (20) (Yt + ε) α = (Yt0 + ε)… view at source ↗
read the original abstract

In this paper, we consider option pricing in a framework of the fractional Heston-type model with $H>1/2$. As it is impossible to obtain an explicit formula for the expectation $\mathbb E f(S_T)$ in this case, where $S_T$ is the asset price at maturity time and $f$ is a payoff function, we provide a discretization schemes $\hat Y^n$ and $\hat S^n$ for volatility and price processes correspondingly and study convergence $\mathbb E f(\hat S^n_T) \to \mathbb E f(S_T)$ as the mesh of the partition tends to zero. The rate of convergence is calculated. As we allow $f$ to have discontinuities of the first kind which can cause errors in straightforward Monte-Carlo estimation of the expectation, we use Malliavin calculus techniques to provide an alternative formula for $\mathbb E f(S_T)$ with smooth functional under the expectation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper considers option pricing in a fractional Heston-type model with Hurst parameter H > 1/2. It introduces discretization schemes Ŷ^n and Ŝ^n for the volatility and asset-price processes, establishes convergence of E[f(Ŝ^n_T)] to E[f(S_T)] as the mesh of the time partition tends to zero together with an explicit rate, and derives via Malliavin calculus an alternative representation of E[f(S_T)] in which the payoff functional is replaced by a smooth integrand.

Significance. If the convergence statement and rate are rigorously justified, the discretization result supplies a theoretically controlled numerical method for pricing in fractional-volatility models where closed-form expressions are unavailable. The Malliavin representation would additionally allow Monte-Carlo evaluation of expectations involving discontinuous payoffs without the usual bias from indicator functions.

major comments (2)
  1. [Model setup] Model-setup paragraph (and subsequent sections on well-posedness): existence and uniqueness of strong solutions to the fractional Heston SDE, non-negativity of the volatility process, and the precise Hölder regularity (at least >1/2) needed for both the discretization error bound and Malliavin differentiability are asserted without an explicit theorem, reference, or verification that the fractional-Brownian driver enters the volatility equation in a manner preserving these properties when H > 1/2.
  2. [Convergence analysis] Convergence theorem (presumably §3 or §4): the proof that E[f(Ŝ^n_T)] → E[f(S_T)] with a stated rate relies on the path-regularity and Malliavin-differentiability assumptions identified above; without an independent justification of those assumptions the error estimate cannot be closed.
minor comments (2)
  1. The precise form of the discretization (Euler, Milstein, or other) and the definition of the partition mesh should be stated explicitly before the convergence statement.
  2. Notation for the payoff f and the Malliavin weight should be introduced once and used consistently; the abstract mentions “smooth functional under the expectation” but the precise integration-by-parts formula is not previewed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on the well-posedness and convergence analysis. We address each major comment below and will revise the manuscript to supply the requested justifications and references.

read point-by-point responses

  1. Referee: [Model setup] Model-setup paragraph (and subsequent sections on well-posedness): existence and uniqueness of strong solutions to the fractional Heston SDE, non-negativity of the volatility process, and the precise Hölder regularity (at least >1/2) needed for both the discretization error bound and Malliavin differentiability are asserted without an explicit theorem, reference, or verification that the fractional-Brownian driver enters the volatility equation in a manner preserving these properties when H > 1/2.

    Authors: We agree that the model-setup section requires explicit support. In the revised manuscript we will insert a dedicated well-posedness subsection that (i) cites standard existence-uniqueness results for SDEs driven by fractional Brownian motion with Hurst index H>1/2, (ii) verifies that the volatility process remains non-negative under the chosen drift and diffusion coefficients, and (iii) recalls the Hölder regularity of the solution paths (which follows from the Hölder continuity of the driving fBM and standard estimates for such equations). These additions will also confirm that the Malliavin differentiability needed later is compatible with the same regularity. revision: yes

  2. Referee: [Convergence analysis] Convergence theorem (presumably §3 or §4): the proof that E[f(Ŝ^n_T)] → E[f(S_T)] with a stated rate relies on the path-regularity and Malliavin-differentiability assumptions identified above; without an independent justification of those assumptions the error estimate cannot be closed.

    Authors: The convergence result in Section 3 is indeed conditional on the path-regularity and Malliavin-differentiability properties. Once the well-posedness subsection supplies the missing references and verifications, the assumptions become rigorously justified and the error bound can be closed. In the revision we will add explicit cross-references from the convergence proof back to the new well-posedness statements. revision: yes

Circularity Check

0 steps flagged

No circularity; standard convergence and representation results for an externally defined model.

full rationale

The paper defines discretization schemes Ŷ^n and Ŝ^n for the volatility and price processes in the fractional Heston model (H>1/2) and derives the convergence E f(Ŝ^n_T) → E f(S_T) together with a Malliavin integration-by-parts formula. These steps are mathematical derivations from the assumed SDE dynamics and path regularity; they do not reduce by construction to fitted parameters, self-definitions, or load-bearing self-citations. The model framework is taken as given without internal redefinition of its outputs, so the claims remain independent of the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Review performed on abstract alone; the ledger therefore records only the background assumptions explicitly invoked by the abstract. No free parameters or new entities are mentioned.

axioms (2)
  • domain assumption The fractional Heston-type model with Hurst index H > 1/2 exists and possesses the path regularity needed for discretization error bounds.
    Invoked when the authors introduce the processes S and Y and state that discretization is possible.
  • domain assumption Malliavin calculus applies to the payoff functional and yields an equivalent smooth representation.
    Invoked in the final sentence to justify the alternative formula for discontinuous f.

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