Jacobi Stochastic Volatility Model

• The Jacobi stochastic volatility model is a bounded variance process defined on a compact interval, ensuring volatility remains within fixed limits.
• Its polynomial diffusion structure enables closed-form moment formulas, Gram–Charlier A density expansions, and efficient series representations for option pricing.
• By interpolating between Black–Scholes and Heston models through bounded variance and state-dependent leverage, it offers a tractable alternative for pricing derivatives.

Searching arXiv for the primary Jacobi stochastic volatility paper and closely related references. The Jacobi stochastic volatility model is a stochastic volatility specification in which the squared volatility of the asset return follows a Jacobi process on a compact interval $[v_{\min},v_{\max}]$. In the formulation introduced in "The Jacobi Stochastic Volatility Model" (Ackerer et al., 2016), the model is designed as a tractable alternative to the Heston model: the instantaneous variance is bounded and mean-reverting, leverage is retained, the Heston model appears as a limit case, and the polynomial diffusion structure yields closed-form moment formulas together with Gram–Charlier A expansions and series representations for option prices (Ackerer et al., 2016).

1. Definition and core state dynamics

The model is built around a variance factor $V_t$ constrained to satisfy

$0 \le v_{\min} < v_{\max}.$

Its defining quadratic function is

$$Q(v)=\frac{(v-v_{\min})(v_{\max}-v)}{\left(\sqrt{v_{\max}}-\sqrt{v_{\min}}\right)^2}.$$

This function is nonnegative on $[v_{\min},v_{\max}]$, vanishes at the boundaries, and satisfies

$v \ge Q(v),$

with equality at $v=\sqrt{v_{\min}v_{\max}}$ (Ackerer et al., 2016).

The coupled dynamics of variance and log price are

$$\begin{aligned} dV_t &= \kappa(\theta - V_t)\,dt + \sigma \sqrt{Q(V_t)}\,dW_{1t}, \ dX_t &= \left(r-\delta-\frac{V_t}{2}\right)dt + \rho \sqrt{Q(V_t)}\,dW_{1t} + \sqrt{V_t-\rho^2 Q(V_t)}\,dW_{2t}, \end{aligned}$$

with parameters

$$\kappa>0,\qquad \theta\in(v_{\min},v_{\max}],\qquad \sigma>0,\qquad \rho\in[-1,1],$$

and independent Brownian motions $W_{1t},W_{2t}$. The traded asset price is

$V_t$0

Moreover,

$V_t$1

so $V_t$2 is the instantaneous volatility of log returns, and the discounted dividend-adjusted price $V_t$3 is a martingale under the risk-neutral measure (Ackerer et al., 2016).

A central structural feature is that the model is a polynomial diffusion. This gives direct access to moments and to generator-based recursions for pricing formulas. A plausible implication is that the bounded state space and polynomial structure are jointly responsible for the tractability emphasized throughout the model’s analysis (Ackerer et al., 2016).

2. Bounded variance, leverage, and boundary behavior

The variance process is a Jacobi/Wright–Fisher diffusion on $V_t$4. The paper states the following properties: $V_t$5 for all $V_t$6; no state $V_t$7 is absorbing, in the sense that

$V_t$8

the drift $V_t$9 pulls $0 \le v_{\min} < v_{\max}.$0 toward $0 \le v_{\min} < v_{\max}.$1; and the Jacobi process has a stationary beta distribution in the classical setting, although the paper’s emphasis is on bounded support and tractability rather than on stationary analysis (Ackerer et al., 2016).

The instantaneous correlation between the return and variance shocks is

$0 \le v_{\min} < v_{\max}.$2

This equals $0 \le v_{\min} < v_{\max}.$3 at $0 \le v_{\min} < v_{\max}.$4, and is generally bounded in magnitude by $0 \le v_{\min} < v_{\max}.$5 (Ackerer et al., 2016). Thus the model incorporates leverage through a state-dependent correlation structure rather than a constant instantaneous return–variance correlation.

The boundedness of $0 \le v_{\min} < v_{\max}.$6 has an immediate pricing implication: Black–Scholes implied volatility is bounded between $0 \le v_{\min} < v_{\max}.$7 and $0 \le v_{\min} < v_{\max}.$8 (Ackerer et al., 2016). This sharply distinguishes the model from unbounded positive-state volatility diffusions. In comparison, the combined multiplicative-Heston model of (Moghaddam et al., 2018) is a positive, mean-reverting, unbounded volatility diffusion on $0 \le v_{\min} < v_{\max}.$9 with diffusion coefficient

$$Q(v)=\frac{(v-v_{\min})(v_{\max}-v)}{\left(\sqrt{v_{\max}}-\sqrt{v_{\min}}\right)^2}.$$0

whereas the Jacobi-type diffusion coefficient is boundary-vanishing and keeps the variance in a compact interval (Moghaddam et al., 2018). The analogy emphasized there is Beta distribution for bounded Jacobi state space versus Beta Prime distribution for an unbounded positive-state counterpart (Moghaddam et al., 2018).

3. Relation to Black–Scholes and Heston

Two limiting regimes organize the model’s interpretation. First, Black–Scholes is recovered as a special case: if

$$Q(v)=\frac{(v-v_{\min})(v_{\max}-v)}{\left(\sqrt{v_{\max}}-\sqrt{v_{\min}}\right)^2}.$$1

then $$Q(v)=\frac{(v-v_{\min})(v_{\max}-v)}{\left(\sqrt{v_{\max}}-\sqrt{v_{\min}}\right)^2}.$$2, and the log-price dynamics reduce to

$$Q(v)=\frac{(v-v_{\min})(v_{\max}-v)}{\left(\sqrt{v_{\max}}-\sqrt{v_{\min}}\right)^2}.$$3

(Ackerer et al., 2016).

Second, the Heston model arises as a limit. If

$$Q(v)=\frac{(v-v_{\min})(v_{\max}-v)}{\left(\sqrt{v_{\max}}-\sqrt{v_{\min}}\right)^2}.$$4

then

$$Q(v)=\frac{(v-v_{\min})(v_{\max}-v)}{\left(\sqrt{v_{\max}}-\sqrt{v_{\min}}\right)^2}.$$5

and the Jacobi model formally converges to

$$Q(v)=\frac{(v-v_{\min})(v_{\max}-v)}{\left(\sqrt{v_{\max}}-\sqrt{v_{\min}}\right)^2}.$$6

which is the Heston model (Ackerer et al., 2016).

The convergence is not only heuristic. The paper proves weak convergence in path space,

$$Q(v)=\frac{(v-v_{\min})(v_{\max}-v)}{\left(\sqrt{v_{\max}}-\sqrt{v_{\min}}\right)^2}.$$7

as $$Q(v)=\frac{(v-v_{\min})(v_{\max}-v)}{\left(\sqrt{v_{\max}}-\sqrt{v_{\min}}\right)^2}.$$8 and the initial conditions converge appropriately, with Heston included as the $$Q(v)=\frac{(v-v_{\min})(v_{\max}-v)}{\left(\sqrt{v_{\max}}-\sqrt{v_{\min}}\right)^2}.$$9 limit (Ackerer et al., 2016).

The boundary condition ensuring the process stays strictly inside the interval is

$[v_{\min},v_{\max}]$0

When $[v_{\min},v_{\max}]$1 and $[v_{\min},v_{\max}]$2, this reduces to the familiar CIR/Heston positivity condition

$[v_{\min},v_{\max}]$3

(Ackerer et al., 2016).

A common misconception is to regard the Jacobi specification as unrelated to Heston because of bounded variance. The limit result shows the opposite: the model is constructed so that bounded-volatility dynamics interpolate toward Heston as the upper bound is removed (Ackerer et al., 2016).

4. Density expansions and polynomial tractability

A major contribution of the model is that the density of $[v_{\min},v_{\max}]$4, and more generally the joint density of finite sequences of log returns, admits a Gram–Charlier A expansion with closed-form coefficients (Ackerer et al., 2016).

For a Gaussian density $[v_{\min},v_{\max}]$5 with mean $[v_{\min},v_{\max}]$6 and variance $[v_{\min},v_{\max}]$7, the weighted space is

$[v_{\min},v_{\max}]$8

If $[v_{\min},v_{\max}]$9 denotes the density of $v \ge Q(v),$0, the likelihood ratio

$v \ge Q(v),$1

belongs to $v \ge Q(v),$2 under the paper’s conditions, so it has the Hermite expansion

$v \ge Q(v),$3

where $v \ge Q(v),$4 are generalized Hermite polynomials orthonormal in $v \ge Q(v),$5. The coefficients are the Hermite moments

$v \ge Q(v),$6

The paper proves that for suitable $v \ge Q(v),$7, specifically

$v \ge Q(v),$8

the density satisfies

$v \ge Q(v),$9

which guarantees the expansion (Ackerer et al., 2016).

For log returns

$v=\sqrt{v_{\min}v_{\max}}$0

the joint density $v=\sqrt{v_{\min}v_{\max}}$1 admits the analogous multivariate Hermite expansion relative to a product Gaussian weight

$v=\sqrt{v_{\min}v_{\max}}$2

The coefficients are

$v=\sqrt{v_{\min}v_{\max}}$3

with

$v=\sqrt{v_{\min}v_{\max}}$4

(Ackerer et al., 2016).

The mechanism behind the expansion is that conditional on the volatility path, the return vector is Gaussian with diagonal covariance: $v=\sqrt{v_{\min}v_{\max}}$5 where

$v=\sqrt{v_{\min}v_{\max}}$6

and then

$v=\sqrt{v_{\min}v_{\max}}$7

The paper stresses that this convergence generally fails for Heston, because Heston does not satisfy the required exponential integrability (Ackerer et al., 2016). That contrast identifies bounded variance not merely as a modeling restriction, but as an analytic condition enabling convergent density expansions.

5. Option pricing by closed-form series

The density expansion leads directly to series representations for option prices whose discounted payoffs depend on the asset price trajectory at finitely many time points. For a discounted payoff $v=\sqrt{v_{\min}v_{\max}}$8,

$v=\sqrt{v_{\min}v_{\max}}$9

where

$$\begin{aligned} dV_t &= \kappa(\theta - V_t)\,dt + \sigma \sqrt{Q(V_t)}\,dW_{1t}, \ dX_t &= \left(r-\delta-\frac{V_t}{2}\right)dt + \rho \sqrt{Q(V_t)}\,dW_{1t} + \sqrt{V_t-\rho^2 Q(V_t)}\,dW_{2t}, \end{aligned}$$0

The truncated approximation is

$$\begin{aligned} dV_t &= \kappa(\theta - V_t)\,dt + \sigma \sqrt{Q(V_t)}\,dW_{1t}, \ dX_t &= \left(r-\delta-\frac{V_t}{2}\right)dt + \rho \sqrt{Q(V_t)}\,dW_{1t} + \sqrt{V_t-\rho^2 Q(V_t)}\,dW_{2t}, \end{aligned}$$1

and it converges to $$\begin{aligned} dV_t &= \kappa(\theta - V_t)\,dt + \sigma \sqrt{Q(V_t)}\,dW_{1t}, \ dX_t &= \left(r-\delta-\frac{V_t}{2}\right)dt + \rho \sqrt{Q(V_t)}\,dW_{1t} + \sqrt{V_t-\rho^2 Q(V_t)}\,dW_{2t}, \end{aligned}$$2 as $$\begin{aligned} dV_t &= \kappa(\theta - V_t)\,dt + \sigma \sqrt{Q(V_t)}\,dW_{1t}, \ dX_t &= \left(r-\delta-\frac{V_t}{2}\right)dt + \rho \sqrt{Q(V_t)}\,dW_{1t} + \sqrt{V_t-\rho^2 Q(V_t)}\,dW_{2t}, \end{aligned}$$3 (Ackerer et al., 2016).

For a call payoff

$$\begin{aligned} dV_t &= \kappa(\theta - V_t)\,dt + \sigma \sqrt{Q(V_t)}\,dW_{1t}, \ dX_t &= \left(r-\delta-\frac{V_t}{2}\right)dt + \rho \sqrt{Q(V_t)}\,dW_{1t} + \sqrt{V_t-\rho^2 Q(V_t)}\,dW_{2t}, \end{aligned}$$4

the coefficients are

$$\begin{aligned} dV_t &= \kappa(\theta - V_t)\,dt + \sigma \sqrt{Q(V_t)}\,dW_{1t}, \ dX_t &= \left(r-\delta-\frac{V_t}{2}\right)dt + \rho \sqrt{Q(V_t)}\,dW_{1t} + \sqrt{V_t-\rho^2 Q(V_t)}\,dW_{2t}, \end{aligned}$$5

with

$$\begin{aligned} dV_t &= \kappa(\theta - V_t)\,dt + \sigma \sqrt{Q(V_t)}\,dW_{1t}, \ dX_t &= \left(r-\delta-\frac{V_t}{2}\right)dt + \rho \sqrt{Q(V_t)}\,dW_{1t} + \sqrt{V_t-\rho^2 Q(V_t)}\,dW_{2t}, \end{aligned}$$6

Put coefficients follow from put-call parity. For a digital payoff

$$\begin{aligned} dV_t &= \kappa(\theta - V_t)\,dt + \sigma \sqrt{Q(V_t)}\,dW_{1t}, \ dX_t &= \left(r-\delta-\frac{V_t}{2}\right)dt + \rho \sqrt{Q(V_t)}\,dW_{1t} + \sqrt{V_t-\rho^2 Q(V_t)}\,dW_{2t}, \end{aligned}$$7

the coefficients are

$$\begin{aligned} dV_t &= \kappa(\theta - V_t)\,dt + \sigma \sqrt{Q(V_t)}\,dW_{1t}, \ dX_t &= \left(r-\delta-\frac{V_t}{2}\right)dt + \rho \sqrt{Q(V_t)}\,dW_{1t} + \sqrt{V_t-\rho^2 Q(V_t)}\,dW_{2t}, \end{aligned}$$8

All of these are stated in closed form (Ackerer et al., 2016).

The framework extends beyond plain-vanilla options. The paper states that it includes European call, put, and digital options, forward start options, and can be applied to discretely monitored Asian options (Ackerer et al., 2016). For the forward start call on the underlying return between $$\begin{aligned} dV_t &= \kappa(\theta - V_t)\,dt + \sigma \sqrt{Q(V_t)}\,dW_{1t}, \ dX_t &= \left(r-\delta-\frac{V_t}{2}\right)dt + \rho \sqrt{Q(V_t)}\,dW_{1t} + \sqrt{V_t-\rho^2 Q(V_t)}\,dW_{2t}, \end{aligned}$$9 and $$\kappa>0,\qquad \theta\in(v_{\min},v_{\max}],\qquad \sigma>0,\qquad \rho\in[-1,1],$$0,

$$\kappa>0,\qquad \theta\in(v_{\min},v_{\max}],\qquad \sigma>0,\qquad \rho\in[-1,1],$$1

the discounted payoff reduces to a univariate call in the return increment. For the proportional-strike forward start call

$$\kappa>0,\qquad \theta\in(v_{\min},v_{\max}],\qquad \sigma>0,\qquad \rho\in[-1,1],$$2

the Fourier coefficients factor in a convenient way, so no numerical integration is needed. For fixed-strike and floating-strike discretely monitored Asian options, closed-form Fourier coefficients are not available, but the price can still be approximated using the Gram–Charlier density approximation and numerical Gaussian cubature (Ackerer et al., 2016).

6. Numerical behavior, implementation, and analytical context

The numerical analysis reported in the paper shows that the truncated series converges rapidly for European options. For ATM European calls in the calibration example, truncation at around $$\kappa>0,\qquad \theta\in(v_{\min},v_{\max}],\qquad \sigma>0,\qquad \rho\in[-1,1],$$3 already gives implied volatilities accurate to within about 10 basis points. The Hermite moments satisfy an exponential decay bound under suitable conditions,

$$\kappa>0,\qquad \theta\in(v_{\min},v_{\max}],\qquad \sigma>0,\qquad \rho\in[-1,1],$$4

and the truncation error is bounded by

$$\kappa>0,\qquad \theta\in(v_{\min},v_{\max}],\qquad \sigma>0,\qquad \rho\in[-1,1],$$5

(Ackerer et al., 2016).

Implementation is organized around the polynomial generator matrix $$\kappa>0,\qquad \theta\in(v_{\min},v_{\max}],\qquad \sigma>0,\qquad \rho\in[-1,1],$$6. The paper gives the explicit basis

$$\kappa>0,\qquad \theta\in(v_{\min},v_{\max}],\qquad \sigma>0,\qquad \rho\in[-1,1],$$7

which makes $$\kappa>0,\qquad \theta\in(v_{\min},v_{\max}],\qquad \sigma>0,\qquad \rho\in[-1,1],$$8 sparse, with at most seven nonzero entries per column in the chosen basis. Hermite moments are computed through this polynomial generator matrix; in the multivariate case they are obtained recursively using matrix exponentials $$\kappa>0,\qquad \theta\in(v_{\min},v_{\max}],\qquad \sigma>0,\qquad \rho\in[-1,1],$$9 and selection matrices $W_{1t},W_{2t}$0 (Ackerer et al., 2016). The paper also notes that Fourier coefficients for calls and digitals are computed in less than a millisecond via recursion, while Hermite moments are more expensive but can be reused for all payoffs with the same maturity (Ackerer et al., 2016).

The model’s boundedness has consequences for PDE analysis as well. In the general stochastic-volatility framework of "Valuation equations for stochastic volatility models" (Bayraktar et al., 2010), uniqueness of the valuation PDE among nonnegative classical solutions with at most linear growth holds if and only if the stock price is a martingale. That paper allows volatility factors that may hit the boundary $W_{1t},W_{2t}$1, with absorbing behavior if $W_{1t},W_{2t}$2 and instantaneously reflecting behavior if $W_{1t},W_{2t}$3, and emphasizes that boundedness of the volatility factor does not by itself settle PDE uniqueness (Bayraktar et al., 2010). A plausible implication for bounded Jacobi variance models is that compact state space simplifies growth issues but does not eliminate the need to verify the martingale property when pricing through the valuation PDE.

A separate asymptotic perspective is given by the martingale expansion framework of (Fukasawa, 14 Jan 2026), which treats continuous stochastic volatility models in small volatility-of-volatility and fast mean-reversion regimes. That paper does not explicitly work out the Jacobi model, but states that its framework can be applied to a Jacobi stochastic volatility model if the integrated variance admits an expansion of the form

$W_{1t},W_{2t}$4

with suitable limit theory for the normalized martingale and fluctuation pair (Fukasawa, 14 Jan 2026). This suggests that the Jacobi model is compatible not only with exact polynomial methods but also with perturbative implied-volatility expansions when an appropriate scaling is imposed.

The main limitation stated in (Ackerer et al., 2016) is equally clear: the expansion machinery relies on bounded variance and does not apply to Heston in the same convergent form. For some exotic payoffs, notably Asians, numerical quadrature remains necessary, and larger truncation orders may be needed for stability (Ackerer et al., 2016). Within those limits, the Jacobi stochastic volatility model occupies a specific position: it is a bounded-volatility, mean-reverting diffusion model that sits between Black–Scholes and Heston, preserving leverage and analytical tractability while replacing unbounded variance with compact support (Ackerer et al., 2016).