Papers

We study finite-maturity American puts in the Heston model without the Feller condition. At positive variance, the exercise boundary is smooth and strictly decreasing in remaining maturity and variance. At zero variance, the endpoint is locally Lipschitz in remaining maturity, the boundary moves with bounded speed in maturity uniformly down to zero variance, and it extends continuously to the axis. Each boundary slice at positive maturity has a regular $C^1$ closure tangent to the zero-variance axis, its stock threshold has variance derivative tending to minus infinity, and both partial derivatives of the inverse boundary graph vanish along the slice. The continuation premium at the axis contact stock is of order variance, and as a consequence smooth fit fails in the variance direction at the axis. At expiry the whole boundary, including its axis edge, converges to the strike. We give quantitative boundary gap bounds, an exact limiting weighted boundary Hessian, and vertical identities retaining the maturity term. Exact boundary gap equivalents remain open.

We study a local stochastic volatility (LSV) model in which the variance factor follows a generalized Langevin equation (GLE) with colored noise, and past returns act on variance through a memory kernel with state-dependent amplitudes. For kernels given by finite sums of exponentials, we prove existence, uniqueness, and non-explosion under a growth condition linking the volatility map to the potential. In a one-mode family with a polynomial potential, feedback through log returns can cause explosion exactly when the squared volatility factor grows faster than the restoring force. Under structural and nondegeneracy conditions, an exact reduced Markovian state has a smooth density solving a deterministic forward equation. The Dupire conditional expectation therefore requires no stochastic partial differential equation. Under a bounded Sharpe ratio, exponential utility selects the minimal entropy martingale measure, whose premium solves a quadratic backward stochastic differential equation. Under a regularity hypothesis, which we verify in special cases with and without leverage, the premium depends only on the current state, and the squared VIX equals the expected average future local stochastic variance. Finally, a regularized calibration problem with bid–ask constraints admits a minimizer whenever a feasible point has finite objective value.

We present a statically and dynamically arbitrage-free European option market model with smooth interpolation between option prices, based on the SANOS framework, and prove that its dynamics admit a unique strong solution. In spirit, it is a stochastic model for local volatility when the latter is used as a representation of a European option price surface, not as a pricing model. We illustrate the strength of our approach on S&P Index Option data from January 2022 to August 2025. We show that the model's initial state fits within bid/ask for more than 95% of relevant market option prices. We then train the model dynamics under $\mathbb{P}$ and $\mathbb{Q}$ and show that the model is indeed free of arbitrage while fitting the dynamics of the market well. We also propose a range of BHW-tests to test for absence of arbitrage.

FLEX options are customized exchange-traded contracts allowing participants to choose nonstandard strikes, expirations, exercise styles, and settlement terms while retaining central clearing. We investigate whether single-name FLEX activity leaves measurable footprints in regular listed-option markets and underlying equities. Using a linked dataset of FLEX consolidations, daily open-interest files, listed-option chains, equity returns, and corporate actions over 250 trading days (March 2025-March 2026) across 41 underlyings and 53 event dates, we document three primary empirical findings: (1) Matched-control difference-in-differences estimates reveal a robust post-event increase in listed-option open interest (beta = 0.3061, p = 0.0042, Benjamini-Hochberg p = 0.0168). (2) FLEX activity is strongly selected into high implied-volatility (IV) environments, concentrated in mid-price underlyings (date-fixed-effect ATM IV premium beta = 0.1235, p = 0.0030). (3) Event-day IV-minus-realized-volatility spreads predict execution-stressed short-straddle returns (beta = 0.0366, p = 0.0100). Short-lived equity return effects (CAR_0 difference of -0.85%, p = 0.0442) do not persist over 1-to-5-day horizons. Overall, FLEX activity serves primarily as an observable marker of institutional volatility demand and open-interest positioning rather than a directional stock-return predictor.

Catastrophe bonds have become a central instrument for transferring catastrophic risk to capital markets, yet their payout structures are typically specified exogenously rather than optimized. In this paper, we treat the indemnity function as the decision variable and study the optimal design of CAT bonds from the sponsor's perspective. We establish the optimality of a piecewise linear indemnity function under both expected-disutility and spectral-risk-measure frameworks and thereby reduce an infinite-dimensional design problem to a one-dimensional optimization over the attachment point. The optimal design also eliminates the potential moral hazard associated with the discontinuity of threshold payout structures. We characterize the optimal attachment point under six risk criteria and obtain explicit or semi-explicit solutions. We further derive bounds for the optimal attachment point and develop comparative statics that provide practical design insights. Using U.S. severe storm losses from NOAA's Billion-Dollar Weather and Climate Disasters database for calibration, we show that the optimal CAT bond designs achieve substantial risk reductions across most risk criteria.

European option smiles determine the risk-neutral marginal laws of an asset, but not their intertemporal coupling, which is decisive for many applications. The Bass martingale construction selects, among all calibrated martingales, the one closest to Bachelier dynamics; it permits fast calibration at discrete maturities and recovers the Dupire local-volatility (LV) model as the maturity grid is refined. This article develops a modular calibration overlay for existing stochastic and path-dependent volatility models. We sequentially construct a martingale that matches all prescribed marginals exactly while remaining as close as possible, in an adapted Knothe-Rosenblatt sense, to the reference dynamics. As with the Bass LV model, each calibration step is amenable to an efficient Martingale Sinkhorn algorithm. We develop the theoretical foundations, numerical implementation and consider convergence to the SLV model. We also benchmark the method for Heston and Bergomi finite-factor path-dependent volatility dynamics and develop the multi-asset extension.

We derive a short-maturity expansion for up-and-out put barrier option prices under continuous stochastic volatility when the strike and the barrier approach the spot at the diffusive scale. Assuming joint weak convergence of the normalized terminal return, the relative volatility fluctuation, and the running maximum, together with uniform integrability, we show that the leading term is the time-inhomogeneous Black-Scholes barrier price fitted to the forward variance curve. The first model-dependent correction is of order $\theta^{H+1/2}$, where $\theta^H$, $H \in (0,1/2]$, is the order of the relative volatility fluctuation, and is represented explicitly through killed Brownian transition densities. For regular volatility models, where $H= 1/2$, the coefficient is determined by the short-maturity at-the-money implied-volatility skew. For rough volatility models with $H < 1/2$, the coefficient reduces to a one-dimensional integral. Numerical experiments show that the correction materially improves the Black-Scholes approximation.

We develop a non-perturbative framework for stochastic-volatility option pricing built on the two-particle-irreducible (2PI) effective action and the Dyson-Schwinger gap equations of quantum field theory. In log-price, log-volatility or Lamperti coordinates, the joint law of the state variables is approximated by a self-consistent Gaussian whose mean and effective diffusion follow from the 2PI stationarity conditions, with the exponential and CEV vertices evaluated through the exact Gaussian moment-generating function rather than a Taylor cut. For exponential vertices the skeletons beyond this level sum in closed form: in exp-OU the evaluated melon level cuts the Hartree error by 20-130x, to 0.01-0.11 bp. Whether the dressed inverse propagator is local in time separates Markovian models, where the gap equation collapses to a few ODEs, from rough (Volterra) models, where the full two-time propagator is retained. At nonzero correlation the field is integrated deterministically by conditioning on its two leading Gaussian modes, with exact conditional moments and a moment-matched law for the integrated variance and leverage integral; in the CEV models the correlated drift enters a Dyson equation around the exact absorbed-CEV propagator, resummed by jumps along the variance clock. These closures match Monte Carlo references for rough Bergomi approximately at their resolution (0.01-0.1 bp) down to H=0.07, rho=-0.9, and price SABR to 0.97 bp over a 72-cell grid with maturities to ten years (75 bp for Hagan's expansion) and rough SABR to 0.92 bp. Conditional on the volatility field, forward-start smiles and continuously-monitored barriers reduce to field-only quadratures on the native Volterra field, and the causal response block yields the impulse-vega curve in one adjoint contraction. Derivations, extended benchmarks and secondary applications are in an ancillary technical supplement.

We study the short-maturity implied volatility and the skew stickiness ratio for baskets of assets with continuous, possibly rough, stochastic volatility. The fluctuation of the instantaneous basket variance has two sources: fluctuations of the constituent variances and fluctuations of the basket weights. We derive a near-the-money implied volatility expansion that separates these contributions. We then specialize the result to volatility models given by general functions of Gaussian Volterra factors and obtain an explicit basket skew coefficient in terms of the short-time kernel asymptotics, the factor sensitivities, and the return-factor correlations. A density expansion justifies differentiation of the near-the-money expansion at the money. Finally, using a Malliavin representation of the dynamics of total implied variance, we prove that the short-maturity skew stickiness ratio converges to the universal limit $H + 3/2$ for Gaussian factor basket models with $H \in (0, 1/2]$.

Catastrophe (cat) bonds overcome some lack of reinsurance by sourcing capacity from the wider capital markets. We present a new type of cat bond addressing the known trade-off between moral hazard and basis risk. As our main contributions we propose a trigger mechanism which is entirely transparent and simpler to evaluate compared to indemnity modeling techniques, as well as a methodology to price this cat bond. This is relevant for insurers and public authorities in a world where natural disasters are occurring with increasing frequency and severity due to climate change, but also for players willing to enter the cat bond market for whom the lack of transparency of this asset class has been a significant obstacle. Our trigger is derived from a cost random field which separates the physical hazard, a vulnerability function and the exposure. This allows the trigger to take a flexible form between parametric and modeled loss, in case exposure is taken into account. We present a case study based on historical windstorm events impacting Germany. Using wind speed data from historical storms, we fit a max-stable random field on a resolution which is standard in the reinsurance industry. The availability of industry loss and exposure data allows us to calibrate the vulnerability component to historical observations. Besides measuring the basis risk associated with our trigger, we perform a full model assessment and discuss numerical results.

We derive a first-order representation of Black-Scholes implied variance in a continuous local martingale model. Total implied variance is the conditional expectation of the quadratic variation of the log price given its terminal value, up to a smaller-order term, for bounded standardized log-strikes. The framework incorporates small volatility-of-volatility, fast mean-reverting, and short-maturity asymptotics.

We give a two-period counterexample to the absence of arbitrage for the posterior-weighted pricing rule in Estimate nothing by Duembgen and Rogers. Both physical models have strictly positive transition densities, and each model is equipped with an equivalent martingale measure. Nevertheless, the mixed price of a single derivative falls deterministically from $5/2$ to $2$ between two trading dates. If these prices are tradable, shorting the derivative and closing the position one period later yields a certain profit. A finite-state appendix also illustrates the failure of recursive consistency.

We develop semi-closed pricing formulas and lifted-model hedging methods for discretely monitored geometric and arithmetic Asian options in the Volterra-Heston stochastic volatility model. Exploiting the affine Volterra structure, we derive a tractable transform for the joint law of the terminal log-price and the discretely monitored geometric average. This transform yields semi-closed pricing formulas for geometric Asian options, which in turn provide effective control variates for Monte Carlo valuation of arithmetic Asian options. Under the stated real-moment and affine-transform hypotheses, we also derive the Galtchouk-Kunita-Watanabe decomposition for Fourier-representable payoffs and obtain a variance-optimal hedge in terms of the Riccati-Volterra equation and the forward-variance curve. Using N-factor Markovian approximations, we obtain a finite-dimensional numerical implementation for hedging Asian options. Our numerical experiments document factor convergence for a regular non-Markovian kernel and the effect of rebalancing frequency on hedging error. In the Heston benchmark, geometric Asian controls substantially reduce the variance of arithmetic-Asian price estimates and improve the finite-sample stability of regression-based hedging relative to direct regression.

Pathwise differentiation of Monte Carlo estimators fails at payoff discontinuities, producing zero or biased sensitivities for barriers, autocallables, and digital options. The industry workaround — smoothing the indicator functions — introduces bias and requires per-product calibration. We derive a correction formula that restores unbiased Greeks without smoothing. For a payoff $F(Z,\theta)$ that is piecewise smooth with discontinuities on surfaces $\{g_i = 0\}$, we show that the sensitivity decomposes into a pathwise term (computed by standard AAD) plus a sum of boundary corrections, each involving the payoff jump, the Gaussian density at the boundary, and the sensitivity of the boundary to the parameter. The correction is computed by Newton root-finding in the normal-random space, with the jump evaluated by two forward replays of the pricing kernel. The implementation uses AADC (pip install aadc), whose tape replay and automatic discontinuity tracking make the method fully automatic — the quant writes standard pricing code, and the correction driver identifies and handles all discontinuities. We prove the formula for arbitrary compositions of smooth functions and indicator functions (not just outer products), covering real autocallable payoff structures with recursive alive/dead logic. Benchmarks on QuantLib models (GBM, Heston, Hull-White) show all Greeks within 0.1–4% of analytic or bump-and-revalue references.

We propose a deterministic numerical method for pricing and hedging surrenderable equity-linked life-insurance contracts with periodic premiums and fund contributions, maturity and death guarantees, and Bermudan surrender under correlated stochastic volatility and stochastic interest rates. The main computational challenge is the non-recombining accumulated fund, which couples with multiple stochastic factors and early exercise. Our key idea is to avoid a full multidimensional fund lattice: variance and interest-rate factors are discretized on recombining lattices, while at each factor node the contract value is represented as an adaptive one-dimensional function of the fund. Periodic contributions then act as translations of the fund argument, whereas surrender is handled directly through a backward obstacle condition. Piecewise-cubic representations propagate payoff and exercise singularities and are compressed by continuous pruning criteria that control the representation error. We establish weak convergence of the financial chains, convergence of the adaptive valuation under vanishing representation error, and Delta consistency on regular fund regions. For the strict binomial scheme, additional regularity yields first-order weak accuracy and a Talay-Tubaro expansion supporting Richardson extrapolation. Numerical experiments show compact representations, favorable cost–accuracy, and close agreement with independent Monte Carlo and cross-fitted least-squares Monte Carlo benchmarks. Hedging results further show that contracts with similar values can generate materially different exposures to equity, volatility, and interest-rate risk.

A companion paper \cite{ItkinDF2026} introduced the Diagonal Frog (DF) positivity-preserving schemes for anisotropic Fokker–Planck equations, advancing each directional substep by a Krylov-computed matrix exponential, which dominates the cost. Replacing that exponential by a rational map $r(\gamma L)$ reduces the substep to a banded solve, but the positivity argument no longer applies. We prove that for eventually exponentially positive generators the entrywise sign of $r(\gamma L)$ at large steps is decided by a single number, the value $r(\infty)$ taken on infinitely stiff modes. Nonnegativity holds above a computable threshold when $0\le r(\infty)<1$, and at most on a bounded interval, empty or vanishingly narrow in all our tests, when $r(\infty)<0$. The criterion rejects the Crank–Nicolson (trapezoidal) method, where $r(\infty)=-1$, and selects the subdiagonal Padé$(0,2)$ method, which is second order, L-stable and provably positive above an explicit threshold. The resulting DF-ADI scheme costs $O(N)$ per step, keeps the implicit factorized mixed derivative unchanged, is second order in space and time, and conserves discrete mass exactly. In the strong cross-diffusion regime, however, the directional factors demand a step larger than the mixed derivative permits, so the composite second-order scheme is only empirically positive there, and the criterion serves to discriminate the well-behaved multiplicative factors from the stabilizing-correction schemes rather than to guarantee positivity. Against the Krylov exponential it runs ten to thirty-two times faster at matched accuracy in our tests with the gain growing with the mesh. We extend the construction to the backward Kolmogorov equation and to jump-diffusion models.

Option-implied rates are often treated as frictionless because completed boxes deliver riskless payoffs. I show that this interpretation requires option- and benchmark-side implementation wedges to offset. Using SPX and RUT options from 2012-2025, I find a 2-3 bp unannualized increase in the option funding basis when maturity first crosses December 31. The effect behaves as a fixed price wedge, strengthens in the mid-2010s, survives alternative contracts and benchmarks, appears independently in government-bond CIP, and is reproduced in an independently constructed option panel. Put-call parity can identify discount rates precisely without establishing their economic purity.

Basket options are difficult to value under correlated lognormal dynamics because weighted sums and differences of lognormal variables have no tractable distribution. This paper develops a probability-based four-moment framework that separates the exact pricing representation from the distributional approximation. A change of measure first writes a basket price as a linear combination of probabilities. For a standard basket with one positive weight, these probabilities become CDF values of positive correlated lognormal sums. Each sum is approximated by a shifted lognormal variance mixture matched to its first four moments. For an unrestricted mixed-sign basket, a signed shifted lognormal proxy gives an analytical call-price formula. We state admissibility conditions, provide a practical root-selection rule, establish the main strike-based financial properties of the direct proxy, and derive exact pricing-error identities in terms of cumulative distribution function (CDF) discrepancies. The numerical analysis combines standard-basket benchmarks with an empirical application to a normalized $3{:}2{:}1$ crack spread constructed from RBOB gasoline, ULSD or heating oil, and WTI futures. The results show that the probability reformulation and the fourth-moment condition improve the distributional fit and pricing accuracy, particularly when maturity and tail asymmetry increase. The framework remains analytical, transparent, and suitable for repeated valuation across strikes and maturities.

The Marketron model of \cite{HalperinItkin2025Mark} and its option pricing extension in \cite{HalperinItkinMarketron2} suffer from structural non-identifiability: an eighteen-parameter space traps solvers in suboptimal local minima and renders economic quantities unmeasurable. By removing exact scaling gauges and sign symmetries, freezing non-financial parameters by explicit criteria, and adiabatically eliminating the fast hidden signal, we derive a robust nine-parameter reduced model. A Gauss-Newton Hessian with empty null space and a manifold-boundary analysis confirm that the reduced core carries no exact symmetry and admits no further reduction. A diffusive correlation between flow and return innovations captures the short-maturity skew. A staged calibration from the physical measure to the risk-neutral measure, illustrated on SPX options, fits the whole surface with a single parameter set. The same reduction turns the wedge between the physical and pricing values of the flow block into a well-defined market price of flow risk rather than a ridge artifact, identifiable here for the first time, though a single surface constrains its level only weakly. Finally, our analysis reveals that in the Marketron model the log-price obeys a generalized Langevin equation with a closed-form, state-modulated memory kernel, and that the memory variable itself is the exact Markovian lift of this kernel. This mapping also yields a testable condition, the equality of the signal and memory relaxation rates, which on the SPX surface come out well separated, though both weakly identified, placing the fitted market tentatively in the driven, non-equilibrium regime and turning the active-matter reading from an analogy into a falsifiable constraint.

Hedge ratios, factor models and diversified portfolios all rest on an estimate of which firms move together. That estimate is not stable: firms migrate between the groupings the market treats as coherent, and when enough migrate the organizing axes of the cross-section turn. We measure the rate of that turning as the mean squared sine of the principal angles between subdominant eigenspaces of consecutive twelve-month S&P 500 correlation matrices. A typical month rewrites a fifth of the structure and carries four-fifths forward. That rate is priced: it couples to the aggregate variance risk premium at t = 5.40, no level measure correlates above 0.32, and the implied-correlation surface spans at most 6.7 percent of it. Only the persistent component is priced - the premium compensates the pace of revision, not the distance traveled. The mechanism is prepayment: implied variance rises on impact, volatility follows two to three quarters later (simulated-null p < 0.03 at h = 1-9), and the premium converges as it arrives. Three pre-registered boundaries hold: no timing alpha, no crash protection, and a downside version inseparable from intensity. The premium is, in part, rent on exposure held over a map still being redrawn.

We consider the question of the optimal timing of the sale of an asset with stochastic dynamics. Our analysis is based on the method of the distribution builder introduced by Sharpe, Goldstein and Blythe [SGB00] for the purpose of optimal portfolio selection. Instead of specifying a utility function or risk aversion coefficient, this tool directly elicits the target distribution of the investor. We show how the problem of an optimal asset sale is in this setting linked to the problem of finding a Skorokhod embedding of a distribution into a diffusion process. In the case where the asset process follows a geometric Brownian motion and a specific family of distributions is targeted, one can observe a risk-return tradeoff.

This paper considers European multi-asset option pricing under Lévy and affine characteristic-function models. The main obstruction is the curse of dimensionality: direct multidimensional COS pricing forms tensor-product coefficient arrays whose size grows exponentially with the number of assets. We study and extend COS-TT-CHF, a low-rank construction that uses TT-cross to compress sampled characteristic-function tensors into tensor-train COS coefficients for arithmetic basket and min/max option pricing. Once built, the compressed representation gives fast post-setup strike-grid and selected component Delta/Vega calculations. The numerical study compares with adaptive-quadrature Fourier benchmarks, direct COS, a tensor-Fourier min-option benchmark, and quasi-Monte Carlo (QMC) references based on randomized Sobol points. The reported timings show a low-dimensional crossover against direct COS as the benchmark moves from $d=2$ to $d=4$, favorable timings against the tensor-Fourier min-option benchmark from $d=3$ onward, and favorable timings against the QMC common-Heston reference already at $d=2$. The reported tests reach $d=30$ for GBM and $d=20$ for VG, NIG, and common-Heston benchmark families, with accuracy, rank, runtime, control-sensitivity, and component Delta/Vega diagnostics reported throughout.

We propose a novel valuation framework for contingent convertible (CoCo) bonds based on the issuing bank's Common Equity Tier 1 (CET1) ratio, which is widely acknowledged as an indicator of a bank's solvency. Our approach develops a bivariate jump-diffusion model that captures the dynamic relationship linking the CET1 ratios, share prices, and CoCo bond prices, incorporating both continuous market movements and correlated jump risk. The model advances existing literature through three key innovations: (1) a hybrid mechanism for modeling regulatory discretion in trigger decisions, (2) a class of power conversion schemes that generalizes traditional approaches while maintaining analytical tractability, and (3) a method to overcome the temporal discrepancy between high-frequency market data and low-frequency regulatory reporting. We derive semi-closed form formulas for both write-down and equity-convertible CoCo bonds and validate our model through five case studies spanning from 2009 to 2023, including an in-depth analysis of the 2023 Credit Suisse collapse. The results demonstrate a significant improvement in pricing and hedging performance while highlighting the model's data-adaptive nature that enables short-term predictions.

Using the local time-space calculus of Peskir (2005) and the method developed in Mijatovic (2010), we derive a new integral representation for the backward probability associated with the first-passage time (FPT) of a diffusion process through a time-dependent barrier. We present a complete three-step numerical algorithm: first, the problem is reduced to a Volterra-type integral equation; second, its kernel is approximated by a Markov chain; finally, the resulting equation is solved using a quadrature rule. The method is implemented for several representative examples, and its convergence properties are established. An extension to double barrier problems is carried out

Thousands of SOFR derivatives are available in exchanges and OTC, but the market remains illiquid and incomplete. Such a market is beyond the scope of classic risk-neutral approaches that imply linear pricing rules and, at best, approximate hedging strategies whose hedging error may be difficult to quantify. This paper develops an indifference pricing model which is consistent with observed derivative quotes, the agent's financial position and views about the uncertain future as well as risk preferences as described by a convex risk measure. In addition to prices and hedging strategies, the model gives an explicit description of the hedging error and the associated risk. The approach is illustrated numerically using hundreds of CME-listed derivatives to price and hedge unreplicable OTC SOFR derivatives. The indifference prices are computed in less than a minute on a regular PC. We find that the optimal hedging portfolios tend to be sparse but still provide good approximations of the derivative payouts.

The impact of web datasets on market prices has suggested the development of new sources of information, such as social media and web portals, indicating the possibility of an emergent phenomenon. We propose a defining perspective, termed open information, that adds to the existing types of public and private information. We demonstrate their existence and justify material significance for pricing. In this respect, we present statistical hypotheses to test for a web dataset, GDELT, integrated for carbon pricing, that is represented by EU Allowance spot prices. Tests are designed with VAR and GARCH-X formulations, and return forecasting. The outcomes cannot rule out the material existence of open information. The result is significant for providing a conceptual basis to integrate a vast number of web datasets as alternative data in investment decisions.

Informed traders are supposed to need anonymity: they profit by hiding among the uninformed. A decentralized exchange now publishes the counterparty. Every committed order, cancellation, rejection, and fill carries a persistent pseudonymous wallet address. We reconstruct the full-depth limit order book from a record of 17.1 billion messages and 14.3 million aggressive orders by 147,113 wallets, covering $84.3 billion in taker notional. We report three findings. First, informativeness is a persistent wallet attribute. Wallets ranked by the price movement following their aggressive orders retain that ordering across adjacent ten-day windows, with a rank correlation of 0.52. Second, the ranking predicts returns. Adding the live activity of the highest-ranked wallets to a standard anonymous benchmark of prices, quotes, and order flow raises the out-of-sample R2 for one-second returns to 12.31%, a 13.2% gain (t = 9.2) that is 1.6 times the largest of 200 activity-matched placebo cohorts. Third, measured at realized trades rather than at every sampled moment, the increment grows from 1.43 to 2.47 percentage points of R2. Public wallet histories therefore carry short-horizon price information that anonymous order-book data leave unmeasured.

Fractional Brownian motion (fBm) exhibits attractive features for financial modeling, including long-range dependence, path roughness, and anomalous diffusion. However, its non-semimartingale nature precludes the use of conventional no-arbitrage approaches to option pricing. We address this limitation by introducing a time-changed fBm, obtained by evaluating fBm at stochastic gamma activity time, where activity time represents cumulative executed trading time. The resulting process retains the defining properties of fBm while recovering the semimartingale structure. Building on this construction, we develop the fractional Variance Gamma (fVG) model and propose a generalized method of moments (GMM) estimation procedure for option pricing. An empirical analysis of the S&P 500 yields an estimated Hurst exponent of approximately 0.45, consistent with mildly sublinear temporal scaling of return moments.