Risk-neutral option pricing under GARCH intensity model

Kyungsub Lee

REVIEW 3 major objections 5 minor 14 references

Risk-neutral option pricing under GARCH intensity model

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A change of measure prices options in the GARCH intensity model.

desk verdict Section 3's constant-size EMM construction is sound and worth knowing, but the generalized Section 4 contains a dimensional error in Eq. (5) and an invalid WLOG step in Lemma 4.7, so its central martingale claim fails as written. read the letter →

arxiv 1908.05405 v1 pith:6EEPSSRD submitted 2019-08-15 q-fin.PR

Kyungsub Lee This is my paper · ORCID

classification q-fin.PR MSC 91G2091G8060G4460G55

keywords risk-neutraloptionpricingGARCHintensitymodelequivalentmartingalemeasurePoissonprocessescompoundvolatilitysmileRadon-Nikodymderivative

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

The paper sets out to show that the GARCH intensity model---where stock prices move by Poisson jump processes with time-varying intensities---admits risk-neutral option pricing. The main tool is a change of measure: a density process $Z(T)$ swaps the physical intensities $\lambda_\pm$ for new intensities $\tilde{\lambda}_\pm$, and when the new intensities satisfy a single drift condition the discounted stock price becomes a martingale under the new measure. In the basic model with fixed jump size $\delta$ this yields a theorem (3.5) that European option prices equal risk-neutral expectations of discounted payoffs. The paper extends the same claim to a generalized model with random jump sizes (Theorem 4.8), and argues the setup can reproduce empirical volatility smiles and volatility spread. A reader should care because this provides a flexible martingale measure for a discrete-time jump model that captures volatility clustering and leverage effects.

What carries the argument

The carrying object is the density process $Z(t)$, defined recursively on each interval by an exponential change of measure: $Z(t)/Z(t_{i-1})$ is a product of exponential martingales of the Poisson processes, with the ratio $\tilde{\lambda}/\lambda$ inside the logarithm and a compensator term outside. In the generalized model the same exponential factor also contains the likelihood ratio $\tilde{f}(\delta)/f(\delta)$ for each jump size. This one object does two jobs: it proves $Z$ is a $P$-martingale, so $Q$ is a genuine probability measure, and it gives the new conditional Poisson intensities $\tilde{\lambda}_\pm$ under $Q$. With the drift condition tying $\tilde{\lambda}_\pm$ to the interest rate, the same exponentials make the discounted stock price a $Q$-martingale.

What would settle it

Set up the generalized model on one time step with $\lambda_+(0)$ small so that $P(N_+(t_1)=0)=e^{-\lambda_+(0)\Delta t}>0$; compute the exact $Q$-Laplace transform of $\delta_+$ by conditioning on the number of jumps in $[0,t_1]$. If the result differs from $\int e^{ux}\tilde{f}_+(x)\,dx$, the claimed jump-size density under $Q$ fails, and Theorem 4.8 with it.

Watch

Extended reading notes

Core claim

Under the physical measure $P$ the stock follows a geometric jump process whose jump directions are governed by conditional Poisson processes with intensities $\lambda_\pm$. The paper constructs an equivalent measure $Q$ by a density process $Z(T)$ that replaces the intensities with new ones $\tilde{\lambda}_\pm$, and chooses these so that $(e^\delta-1)\tilde{\lambda}_+ + (e^{-\delta}-1)\tilde{\lambda}_- = r$. Theorem 3.5 states that under this $Q$ the discounted price $e^{-rt}S(t)$ is a martingale, so European options are priced by the $Q$-expectation of the discounted payoff. For the generalized model with random jump sizes $\delta_\pm$, Theorem 4.8 claims the same martingale property when $Z(T)$ also changes the jump-size densities to $\tilde{f}_\pm$ and the new intensities satisfy an analogous drift condition; in that setting the paper further claims that the $\delta_\pm$ themselves have density $\tilde{f}$ under $Q$. The paper also shows that, when total intensity is preserved, the conditional variance is unchanged by the measure change, and it presents Monte Carlo evidence that the resulting implied volatility has a smile.

Load-bearing premise

For the generalized model, the load-bearing premise is that one density process can change both the jump intensities and the jump-size distributions; the proof assumes that a first positive jump occurs before the first observation time, and since the event of no jump has positive probability this premise is not guaranteed.

Editorial extensions

If this is right

In the constant-jump model, European option prices can be computed by Monte Carlo simulation under $Q$: generate paths with intensities $\tilde{\lambda}_\pm$ and discount payoffs at the risk-free rate $r$.
If the total intensity is kept fixed ($\tilde{\lambda}_+ + \tilde{\lambda}_- = \lambda_+ + \lambda_-$), the conditional variance of returns is identical under $P$ and $Q$, so the model can produce implied-volatility smiles without moving the overall volatility level.
In the small-$\delta$, constant-intensity limit, the pricing equation reduces to the classical European pricing PDE with variance $h = \delta^2(\lambda_+ + \lambda_-)$.
In the generalized model, choosing different risk-neutral jump-size densities $\tilde{f}_\pm$ shifts the risk-neutral skewness of log-returns, giving a direct lever for matching observed skew in option prices.

Reading between the lines

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

Because the density $Z(T)$ factors across the two jump directions, the Section 3 construction should extend to models with more than two independent jump components or to multi-asset intensity models; the martingale argument does not use the fact that there are exactly two directions.
A sharper test than a single smile plot would be a full cross-sectional fit: estimate $\tilde{\lambda}_\pm$ and $\tilde{f}_\pm$ from option prices and check whether the implied risk-neutral jump distribution moves in the direction of the physical skew.
The generalized theorem as written depends on the first-jump reduction in Lemma 4.7; conditioning on realized jump times rather than assuming a jump before $t_1$ would complete the argument, and the constant-$\delta$ pricing result does not rely on that step.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Quick take: Section 3 is a clean, if modest, result; Section 4 is broken as written. The constant-size EMM construction is correct: with intensities satisfying (e^δ−1)λ~+ + (e^{−δ}−1)λ~− = r, the usual exponential-martingale change of measure yields a Q under which the discounted price is a martingale. The variance-preserving special case is a nice touch, and the smile curves are a sensible numerical illustration, though thin (one parameter set, no error bars, no market comparison).

The problems are in the generalized model. Equation (5) sets (~φδ+−1)λ~+ + (~φδ−−1)λ~− equal to rΔt. The left side has units of inverse time; the right side is dimensionless. The correct condition, from E_Q[S(t_i)/S(t_{i-1})] = e^{rΔt}, is that the bracket equals r. As stated, the one-step martingale condition would give S(t_{i-1}) e^{r (Δt)^2}. The proof of Theorem 4.8 then uses Definition 4.4(i) as if the bracket were r, so the final equality only works if Eq. (5) has r on the right. That is an internal contradiction, not just a typo, because the theorem's stated conclusion fails for Δt≠1.

The second soft spot is Lemma 4.7. The proof says, without loss of generality, that the first up-jump occurs before t1. That is not WLOG: with positive probability there are no up-jumps in the first interval, and on those paths the density ratio does not reweight the mark. The mark-change argument needs a proper treatment of the joint distribution of jump time and mark. The constant-size result in Section 3 does not depend on either defect, and the literature references are appropriate. But the generalized extension, which the paper advertises as the main advancement, is not established.

Who is this for? Researchers who want a worked example of EMM construction for a Poisson intensity GARCH model. The constant-size part is worth a short paper; the generalized part needs real corrected proofs. I would send it to peer review because the core is salvageable and the errors are instructive, not fatal to the whole project. The referee should ask for a corrected Eq. (5), a rigorous treatment of mark distributions under the measure change, and a stronger empirical section.

Referee Report

3 major / 5 minor

Summary. The paper develops a risk-neutral option-pricing framework for the GARCH intensity model of Choe and Lee. Under the physical measure, log-returns are driven by two Poisson-type counting processes with GARCH-type intensity processes. Section 3 constructs an equivalent measure Q by a Doléans-Dade exponential density Z(T) with new intensities ~λ± satisfying (e^δ−1)~λ+ + (e^{−δ}−1)~λ− = r, proves that the discounted stock price is a Q-martingale (Theorem 3.5), and reports a simulated implied-volatility smile under a variance-preserving choice of ~λ±. Section 4 extends the construction to random jump sizes with densities f± and changed densities ~f±, claiming analogous intensity and mark changes (Lemma 4.7) and a martingale property (Theorem 4.8). The paper concludes that the framework is consistent with volatility smile and spread and that the extension handles conditional skewness.

Significance. If the Section 3 construction is taken on its own, it is a clean and largely correct equivalent-martingale-measure construction for a Poisson-intensity asset price with constant jump size; Theorem 3.5 is the central deliverable and is mathematically sound modulo typographical errors. The variance-preserving restriction (3) is a sensible modelling choice, and the construction is not circular: Eq. (2) is a choice of ~λ±, and the martingale property is then verified directly. The generalized extension in Section 4 is natural and potentially useful for capturing skewness, but as written it contains a dimensional inconsistency in the risk-neutral condition and a flawed argument for the mark change, and the proof of Theorem 4.8 has conditioning errors. Because these defects touch exactly the statements that the generalized model is a valid EMM, the extension needs substantial revision. The numerical evidence is illustrative only and lacks the detail needed to support a strong empirical claim.

major comments (3)

[Section 4, Definition 4.4(i), Eq. (5)] Eq. (5) requires (~φδ+−1)~λ+(ti) + (~φδ−−1)~λ−(ti) = rΔt. Under Assumption 2.1(ii), ~λ± are intensities per unit time, so the left-hand side has units of inverse time, while rΔt is dimensionless. The martingale condition for a compound-Poisson price over an interval of length t−u is [(~φδ+−1)~λ+ + (~φδ−−1)~λ−](t−u) = r(t−u), i.e. the bracketed expression must equal r, not rΔt. Consequently the last step of Theorem 4.8, which invokes Definition 4.4(i) to replace the exponent by r(t−u), is invalid for general Δt; with Eq. (5) as written the one-step conditional expectation would be S(ti−1)exp(rΔt^2) rather than S(ti−1)e^{rΔt}. This is a load-bearing error in the generalized EMM claim.
[Section 4, Lemma 4.7] The proof of the mark-distribution claim begins with 'Without loss of generality, assume that first jump ... is less than t1'. This reduction is invalid because P(N+(t1)=0)=exp(−λ+(t0)Δt)>0, and on the no-jump branch the factor Z+,1(t1) contains no mark term, so the calculation of E_Q[e^{uδ+,1}] cannot be reduced to the case where the first jump occurs in the first interval. A correct proof must condition on the jump time or use the joint distribution of the Poisson process and its marks. The displayed computation also contains typos (e.g., λ−(t0) in the exponent and log(λ+(t)/λ−(t)) in place of the full mark ratio), which further obscure the argument. This gap affects the generalized model's claim that δ±,j have density ~f± under Q, although it does not affect the constant-jump-size results of Section 3.
[Section 4, Theorem 4.8] The proof as printed has conditioning errors: the first displayed equality is for E_Q[S(u)|F(t)], whereas the theorem needs E_Q[S(t)|F(u)] or the equivalent discounted statement; the subsequent expectation is then taken conditionally on F(t) while the sums and the factor Z(t)/Z(u) involve jumps after u, which are not F(t)-measurable. The exponent also contains a typo (λ−(ti−1)−λ−(ti−1) in the first line should involve ~λ−(ti−1)), and the final line uses ~φδ+ in the down-jump term where ~φδ− is required. These errors make it impossible to verify the claimed martingale property from the written proof, even after correcting Eq. (5).

minor comments (5)

[Section 3, Theorem 3.2] The tower-property chain contains the typo 'Z(tt−2)' in the display; it should be Z(ti−2).
[Section 4, Lemma 4.6] The proof has a missing closing bracket in 'E[Z−,i(t)|F(ti−1]' and writes 'Fi−1' later; these should be F(ti−1).
[Section 4, Definition 4.4(i)] The symbols ~φδ+ and ~φδ− are defined as Q-expectations before Q is constructed; they should be defined directly as integrals against ~f+ and ~f− to avoid an apparent circularity, even though the integral formulas make the intended meaning clear.
[Section 3, Remark 3.6] The sentence 'Hence A(t) is a martingale' is imprecise: A(t) is absolutely continuous in t, so the argument should state that the finite-variation term must vanish because the sum of the local martingale part and A(t) is a martingale.
[Section 5 and Figure 1] The empirical claim of consistency with volatility smile and spread is supported only by one figure and a parameter table, with no description of standard errors, strike grid, or comparison with alternative models; the claim is therefore suggestive rather than demonstrated.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the EMM is explicitly constructed via drift-matching conditions and verified directly; the smile is an output, not a fitted target.

full rationale

The option-pricing derivation is self-contained. Definition 3.1 chooses candidate intensities satisfying Eq. (2), which is exactly the drift condition needed for the discounted stock price to be a Q-martingale; Theorem 3.2 and Lemma 3.4 verify that Z is a P-martingale and that the intensities under Q are ~λ±, and Theorem 3.5 then computes EQ[S(t)|F(u)] = S(u)e^{r(t−u)} by invoking Eq. (2). This is the standard construction of an equivalent martingale measure, not a prediction extracted from a fit: Eq. (2) is a defining condition, and the additional variance-preserving condition (3) is an independent modeling choice whose consequences, such as the volatility smile in Figure 1, are generated by Monte Carlo from Table 1 rather than used to fit the measure. The generalized model likewise defines Z from ~f± and condition (5), and Theorem 4.8 invokes that condition; the mathematical gaps in Lemma 4.7 (invalid 'without loss of generality' about the first jump time) and Theorem 4.8 (units of rΔt versus r) are internal correctness issues, not circularity. The GARCH intensity model is imported from the author's prior paper [5] as the starting model, but the risk-neutral pricing argument does not reduce to [5], and no uniqueness theorem or fitted parameter is used to force the conclusion. Therefore no circular step is present.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the GARCH intensity model (imported from [5]), on the assumed existence of chosen risk-neutral intensities, and on the true-martingale property of the density process. In the generalized model, the density process also includes a change of the jump-size distribution, whose proof is incomplete. The paper introduces no new physical or economic entities; the degrees of freedom are the risk-neutral intensities and jump densities.

free parameters (5)

Risk-neutral up intensity ~λ+(t) = Unspecified; constrained by (2) and optionally (3)
The EMM is parameterized by a choice of new intensities. The paper does not give a specific functional form; the smile example uses the variance-preserving choice (3), which still leaves one degree of freedom.
Risk-neutral down intensity ~λ-(t) = Unspecified; constrained by (2) and optionally (3)
Same as ~λ+.
Risk-neutral up jump density ~f+(x) = Unspecified
In the generalized model, the measure change also chooses a new density for the up-jump sizes; no guidance is given for selecting it.
Risk-neutral down jump density ~f-(x) = Unspecified
Same as ~f+.
GARCH parameters (ω±, α±, β±, γ±, δ) in Table 1 = Table 1 values
Used for the Monte Carlo smile illustration. The origin of the values is not given in this paper; they are taken as inputs, presumably from prior estimation in [5].

assumptions (4)

domain assumption Assumption 2.1 (from [5]): N± are conditionally independent Poisson processes with piecewise constant intensities λ±, and the price is S(t)=S(0)exp(δ(N+ - N-)).
The entire pricing framework starts from this model. It is imported from the author's earlier work [5], not derived.
ad hoc to paper Existence of positive r.c.l.l. step processes ~λ± satisfying (e^δ-1)~λ+ + (e^{-δ}-1)~λ- = r, and optionally ~λ+ + ~λ- = λ+ + λ-.
The EMM is defined by choosing such processes; the paper asserts they are taken and does not prove existence for a general GARCH intensity path.
domain assumption Finiteness of moment generating functions φδ± = E[e^{δ±}] and ~φδ± = E_Q[e^{±δ±}] in the generalized model.
Needed for the compound Poisson expectations in Definition 4.3, 4.4, and the proofs of Lemmas 4.6-4.7. Not stated as a condition.
domain assumption The density process Z is a true P-martingale (E[Z(t)]=1 for all t), not just a local martingale.
Proved in Theorem 3.2 for constant δ, but the generalized proof in Lemma 4.6 contains gaps; the WLOG step in Lemma 4.7 leaves the mark-distribution change unproved.

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Pith. "Pith review of Risk-neutral option pricing under GARCH intensity model." pith.science (2026). https://pith.science/paper/6EEPSSRD

@misc{pith2026190805405,
  author       = {Pith},
  title        = {Pith review of: Risk-neutral option pricing under GARCH intensity model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6EEPSSRD}},
  note         = {Machine review of arXiv:1908.05405}
}

read the original abstract

The risk-neutral option pricing method under GARCH intensity model is examined. The GARCH intensity model incorporates the characteristics of financial return series such as volatility clustering, leverage effect and conditional asymmetry. The GARCH intensity option pricing model has flexibility in changing the volatility according to the probability measure change.

Figures

Figures reproduced from arXiv: 1908.05405 by the authors.

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Reference graph

Works this paper leans on

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

E., and Jean-Michel Zakoian

Babsiri, M. E., and Jean-Michel Zakoian. 2001. Journal of Econo- metrics 101:257 – 294

work page 2001

[2] [2]

Bakshi, G., and Dilip Madan. 2006. Management Science 52:1945– 1956

work page 2006

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Black, F., and Myron S. Scholes. 1973. Journal of Political Eco- nomics 81:637–654. 19

work page 1973

[4] [4]

Bollerslev, T. 1986. Journal of Econometrics 31:307–327

work page 1986

[5] [5]

H., and Kyungsub Lee.2014

Choe, G. H., and Kyungsub Lee.2014. AStA Advances in Statistical Analysis 98:197–224

work page 2014

[6] [6]

Duan, J.-C. 1995. Mathematical Finance 5:13–32

work page 1995

[7] [7]

J., and P

Elliott, R. J., and P. Ekkehard Kopp. 1990. Stochastic Analysis and Applications 8:157 – 167

work page 1990

[8] [8]

R., Ravi Jagannathan, and David E

Glosten, L. R., Ravi Jagannathan, and David E. Runkle. 1993. Journal of Finance 48:1779–1801

work page 1993