Mixed Fractional Black-Scholes Model

• The Mixed Fractional Black-Scholes Model is defined by combining Brownian and fractional Brownian motions to incorporate memory effects into asset pricing.
• It exhibits regime-dependent behavior, showing complete markets with no arbitrage for H > 3/4 and approximate arbitrage with simple strategies for H in (1/2, 3/4).
• Numerical methods like PDE solvers, meshless collocation, and neural networks are applied to hedging, option valuation, and parameter estimation in this framework.

The mixed fractional Black-Scholes model is a Black-Scholes-type asset-pricing framework in which the driving noise is a linear combination of standard Brownian motion and fractional Brownian motion. In the survey literature, the mixed fractional Brownian motion is written as $X_t=\sigma W_t+\nu B_t$, with $W_t$ and $B_t$ independent, and the corresponding mixed fractional Black-Scholes price model is

$$S_t=S_0\exp\left(\mu t+\sigma W_t-\frac{1}{2}\sigma^2 t+\nu B_t\right).$$

This construction is used to combine the classical geometric Brownian component with a fractional component that carries memory or long-range dependence. In a distinct but related numerical-PDE usage, “mixed fractional Black-Scholes” also denotes time-space-fractional Black-Scholes equations with both time and space fractional derivatives (Bender et al., 2010, Torres-Hernandez et al., 2020).

1. Definition and mathematical structure

Fractional Brownian motion is a centered Gaussian process $B_t$ with Hurst index $H\in(0,1)$ and covariance

$$\operatorname{Cov}(B_t,B_s)=\frac{1}{2}\left(|t|^{2H}+|s|^{2H}-|t-s|^{2H}\right).$$

For $H=\tfrac12$, it reduces to standard Brownian motion; for $H\neq\tfrac12$, it is not a semimartingale; and for $H>\tfrac12$, it exhibits long-range dependence and has zero quadratic variation (Bender et al., 2010).

The mixed fractional Brownian motion is obtained by adding an independent Brownian term and a fractional Brownian term,

$W_t$0

and the mixed fractional Black-Scholes model uses this mixed noise in the exponent of the asset price process. In the risk-neutral form reported in the survey literature, one sets $W_t$1, so that

$W_t$2

The stated motivation is that standard Black-Scholes assumes independent Gaussian increments, whereas fractional models introduce long-range dependence; the mixed model is designed to blend realism with tractability (Bender et al., 2010).

A separate branch of the literature uses “mixed fractional” in the PDE sense. In that setting, the Black-Scholes equation is generalized to include both time and space fractional derivatives: $W_t$3 with

$W_t$4

This is a different mathematical object from the mixed Brownian/fractional Brownian asset-price model, even though both are described as mixed fractional Black-Scholes formulations in the literature (Torres-Hernandez et al., 2020).

2. Arbitrage, semimartingales, and completeness

The central theoretical issue for the mixed fractional Black-Scholes model is whether adding a Brownian component neutralizes the arbitrage pathologies of the pure fractional model. The survey literature gives a regime-dependent answer. In the pure fractional Black-Scholes model, arbitrage is present for $W_t$5. In the mixed model, however, the outcome depends sharply on the Hurst index $W_t$6 (Bender et al., 2010).

If $W_t$7, the mixed fractional Black-Scholes model is stated to be “in law equivalent to the standard BS model.” In this regime, there is no arbitrage for admissible strategies, and the market is complete. If $W_t$8, the same survey reports strong approximate arbitrage opportunities using simple strategies, while also stating that there is no arbitrage for strategies that are admissible, meaning bounded from below (Bender et al., 2010).

This yields an important correction to a common simplification. It is not accurate to say that a Brownian component removes arbitrage for every $W_t$9. The literature instead distinguishes between a law-equivalent, complete regime for $B_t$0 and an intermediate regime $B_t$1 in which approximate arbitrage can occur for simple strategies, even though no arbitrage holds for admissible strategies (Bender et al., 2010).

Hurst regime	Structural status	Financial consequence
$B_t$2	Semimartingale; law-equivalent to BS	No arbitrage for admissible strategies; market complete
$B_t$3	Not a semimartingale	Strong approximate arbitrage with simple strategies; no arbitrage for admissible strategies

The intermediate regime is one of the main sources of controversy in the area. The mixed model is neither uniformly identical to Black-Scholes nor uniformly pathological. Its behavior depends on the interaction between the Brownian part, the fractional part, and the admissibility class of trading strategies (Bender et al., 2010).

3. Hedging and option valuation

A basic structural property emphasized in the survey literature is that the mixed fractional Black-Scholes model has the same quadratic variation as the standard Black-Scholes model: $B_t$4 This fact is central because it underpins a Black-Scholes-type hedging and pricing structure, even though the sample-path behavior also contains a fractional memory component (Bender et al., 2010).

For sufficiently regular payoffs and admissible hedging strategies of the smooth type, the survey reports that “the optimal hedging cost equals the Black-Scholes price.” It further states that these hedges, viewed as functionals on paths, and the corresponding option prices are the same as in the Black-Scholes model. For convex European payoffs $B_t$5, the replication identity is written as

$B_t$6

where the integral is interpreted as a forward or Riemann-Stieltjes integral (Bender et al., 2010).

This result is frequently interpreted as showing that, for smooth strategies, option valuation in the mixed model depends on quadratic variation rather than on the long-memory component itself. A plausible implication is that the Brownian contribution is doing more than merely regularizing the model: it preserves the quadratic-variation-based replication structure that is absent in the pure fractional Brownian setting. The survey literature also frames this as an open empirical question, asking whether option prices depend only on the quadratic variation of stock returns (Bender et al., 2010).

The hedging picture should nevertheless be stated carefully. The transfer of Black-Scholes pricing formulas is reported relative to “smooth” admissible strategies, not as an unrestricted statement for every possible strategy class. That restriction is part of the model’s mathematical content rather than a technical footnote (Bender et al., 2010).

4. Transaction costs and discrete trading

Transaction costs play a decisive role in the fractional and mixed fractional literature. The survey states that proportional transaction costs eliminate arbitrage in both the fractional Black-Scholes and mixed fractional Black-Scholes models, and relates this to the existence of $B_t$7-consistent price systems (Bender et al., 2010).

The pure fractional case has been analyzed in detail under discrete trading. In the model

$B_t$8

with trading only at equidistant times $B_t$9 and proportional transaction cost rate

$$S_t=S_0\exp\left(\mu t+\sigma W_t-\frac{1}{2}\sigma^2 t+\nu B_t\right).$$0

the terminal wealth of the discrete delta strategy converges to

$$S_t=S_0\exp\left(\mu t+\sigma W_t-\frac{1}{2}\sigma^2 t+\nu B_t\right).$$1

where the limiting hedging error $$S_t=S_0\exp\left(\mu t+\sigma W_t-\frac{1}{2}\sigma^2 t+\nu B_t\right).$$2 is expressed through the local time of fractional Brownian motion. For a call payoff $$S_t=S_0\exp\left(\mu t+\sigma W_t-\frac{1}{2}\sigma^2 t+\nu B_t\right).$$3, the error simplifies to

$$S_t=S_0\exp\left(\mu t+\sigma W_t-\frac{1}{2}\sigma^2 t+\nu B_t\right).$$4

If the transaction costs decrease faster, $$S_t=S_0\exp\left(\mu t+\sigma W_t-\frac{1}{2}\sigma^2 t+\nu B_t\right).$$5 with $$S_t=S_0\exp\left(\mu t+\sigma W_t-\frac{1}{2}\sigma^2 t+\nu B_t\right).$$6, perfect replication is recovered in the limit (Azmoodeh, 2010).

The same source explicitly contrasts this with mixed-fractional models, noting that “in models including both Brownian and fractional components, the pathwise quadratic variation is nonzero,” and its summary table states that mixed-fractional models “can reach perfect hedge” and have “No subhedge if QV” (Azmoodeh, 2010).

This contrast is conceptually important. In the pure fractional setting, the limiting error is tied to local times and to the absence of genuine quadratic variation. In the mixed setting, the nonzero quadratic variation changes the hedging geometry and supports the view that transaction-cost asymptotics differ fundamentally from those of the pure fractional model (Azmoodeh, 2010, Bender et al., 2010).

5. Numerical formulations and statistical estimation

A substantial computational literature studies mixed fractional Black-Scholes equations in the PDE sense. For time-space-fractional Black-Scholes problems, one numerical approach uses meshless radial basis function collocation with Caputo time discretization. The method is described as effective in one and two space dimensions, flexible with respect to node choice, and supported by a preconditioning procedure that reduces condition numbers from the order $$S_t=S_0\exp\left(\mu t+\sigma W_t-\frac{1}{2}\sigma^2 t+\nu B_t\right).$$7 down to single or double digits. An example reported in the numerical results gives $$S_t=S_0\exp\left(\mu t+\sigma W_t-\frac{1}{2}\sigma^2 t+\nu B_t\right).$$8, preconditioned condition number $$S_t=S_0\exp\left(\mu t+\sigma W_t-\frac{1}{2}\sigma^2 t+\nu B_t\right).$$9, and $B_t$0 (Torres-Hernandez et al., 2020).

Another line of work uses a two-layered artificial neural network to solve ordinary and fractional Black-Scholes equations after time discretization into a sequence of ODEs. The method combines Caputo finite differences, Adam optimization, fine tuning across time steps, and arctangent domain mapping for infinite spatial domains. The reported errors range from $B_t$1 to $B_t$2, average epoch times are reported as low as $B_t$3 to $B_t$4, and the paper states that the method performs especially well for the mixed fractional Black-Scholes model because it can handle the memory effect caused by the fractionality (Bajalan et al., 2021).

For the mixed Brownian/fractional Brownian model itself, recent work has focused on inference from discrete-time observations. One estimation framework considers $B_t$5 independent processes

$B_t$6

where $B_t$7 are unobserved random effects and $B_t$8, $B_t$9, and $H\in(0,1)$0 are global parameters. The proposed hybrid procedure constructs generalized-method-of-moments estimators for the Brownian volatility, the fractional scaling parameter, and the Hurst parameter, proves strong consistency and joint asymptotic normality, and then estimates the random effects by a plug-in method (Chebli et al., 11 Aug 2025).

The same work also develops a nonparametric estimator for the distribution of random effects using Lagrange interpolation at Chebyshev-Gauss nodes. Its asymptotic mean squared error is reported as

$H\in(0,1)$1

compared with the standard kernel estimator’s $H\in(0,1)$2, and it is shown to satisfy uniform strong consistency under the stated growth conditions for $H\in(0,1)$3, $H\in(0,1)$4, and $H\in(0,1)$5 (Chebli et al., 11 Aug 2025).

6. Related fractional generalizations and conceptual boundaries

The mixed fractional Black-Scholes model belongs to a broader class of Black-Scholes extensions driven by non-Markovian or anomalous-diffusion mechanisms, but these extensions are not interchangeable. One nearby construction is the generalized fractional Brownian motion model, where

$H\in(0,1)$6

This process includes standard Brownian motion, fractional Brownian motion, and sub-fractional Brownian motion as special cases. In that framework, because gfBm is not a semimartingale for $H\in(0,1)$7, there is no risk-neutral measure, and an actuarial approach is adopted for option pricing (Araneda, 2021).

Another important family keeps Brownian motion in the price equation but introduces fractionality through volatility. In the constrained fractional stochastic volatility model, volatility is generated by a differential equation driven by fractional Brownian motion with Hurst parameter greater than $H\in(0,1)$8, while viability conditions enforce positivity. The paper establishes absence of arbitrages, market completeness, and a pricing formula, provided the fractional driver is constructed through the same Brownian filtration via the Decreusefond-Ustünel representation (Marie, 2016). Rough fractional volatility models sharpen this distinction further: when volatility is driven by fractional Brownian motion with $H\in(0,1)$9, the short-time at-the-money implied-volatility skew obeys a power law in time-to-maturity, and standard local-stochastic volatility models are reported not to be dynamically consistent with that power law (Fukasawa, 2015).

Subdiffusive Black-Scholes models define yet another direction. They introduce an inverse-stable or more general inverse-subordinator time change, producing time-fractional Black-Scholes PDEs. In one such framework, the market is arbitrage-free but in general incomplete, and European call prices are represented as averages of Black-Scholes prices over a random effective activity time (Zhang et al., 13 Nov 2025). In a related model with a stochastic short rate under a subdiffusive fractional Brownian motion regime, explicit formulas for European call and put options are derived together with a fractional Black-Scholes equation, and the special cases $$\operatorname{Cov}(B_t,B_s)=\frac{1}{2}\left(|t|^{2H}+|s|^{2H}-|t-s|^{2H}\right).$$0 and $$\operatorname{Cov}(B_t,B_s)=\frac{1}{2}\left(|t|^{2H}+|s|^{2H}-|t-s|^{2H}\right).$$1 recover subdiffusive or classical limits (Shokrollahi, 2018).

These neighboring models clarify the scope of the mixed fractional Black-Scholes model. The mixed Brownian/fractional Brownian formulation modifies the return driver by adding a fractional component to Brownian motion. Time-space-fractional Black-Scholes equations modify the governing PDE through fractional derivatives. Fractional stochastic volatility models place memory in volatility while keeping Brownian price noise. Subdiffusive models randomize operational time. The literature treats these as related but mathematically distinct mechanisms for introducing memory, anomalous diffusion, or long-range dependence into option pricing (Bender et al., 2010, Torres-Hernandez et al., 2020, Marie, 2016, Zhang et al., 13 Nov 2025).