1. Start With the Core Stochastic System
Under risk-neutral dynamics, the Heston model is usually written as:
\[
dS_t = r S_t \, dt + \sqrt{v_t}\, S_t \, dW_t^S
\]
\[
dv_t = \kappa (\theta - v_t)\, dt + \sigma \sqrt{v_t}\, dW_t^v
\]
\[
d\langle W^S, W^v \rangle_t = \rho \, dt
\]
Here, sigma is the vol-of-vol parameter. It controls how random the variance process is.
Interpretation: when sigma > 0, variance itself is diffusing randomly.
When sigma = 0, that random shock vanishes.
2. The Full Heston PDE
If V = V(S, v, t) is the option price, then the two-factor Heston PDE is:
\[
V_t
+ \frac{1}{2} v S^2 V_{SS}
+ \rho \sigma v S V_{Sv}
+ \frac{1}{2} \sigma^2 v V_{vv}
+ r S V_S
+ \kappa (\theta - v) V_v
- r V
= 0
\]
This is a genuine stochastic-volatility PDE: second order in both S and v.
3. What Sigma = 0 Removes
Set sigma = 0. Then two terms disappear immediately:
\[
\rho \sigma v S V_{Sv} = 0
\]
\[
\frac{1}{2} \sigma^2 v V_{vv} = 0
\]
So the PDE becomes:
\[
V_t
+ \frac{1}{2} v S^2 V_{SS}
+ r S V_S
+ \kappa (\theta - v) V_v
- r V
= 0
\]
4. Why This Is a Major Mathematical Change
This is not a tiny perturbation. It changes the structure of the problem.
- The mixed derivative term
V_Sv disappears.
- The second derivative in variance
V_vv disappears.
rho drops out of pricing entirely.- The variance process stops being stochastic and becomes deterministic.
The variance dynamics reduce from an SDE to an ODE:
\[
dv_t = \kappa (\theta - v_t)\, dt
\]
with closed-form solution:
\[
v(t) = \theta + (v_0 - \theta)e^{-\kappa t}
\]
So once sigma = 0, variance is no longer a random state variable. It is a known curve in time.
5. What This Means for FFT / Characteristic Functions
Your usual Heston characteristic function contains terms with sigma^2 in the denominator.
That formula is derived for the generic stochastic-vol case.
\[
\frac{C(\cdots)}{\sigma^2}, \quad \frac{D(\cdots)}{\sigma^2}, \quad \ldots
\]
So plugging in sigma = 0 directly makes that representation blow up.
Key point: the formula becomes invalid at the boundary, but the model itself does not.
6. The Boundary Model Is Simpler
Since variance is deterministic, the only thing you need for pricing over maturity T is the
integrated variance:
\[
I(T) = \int_0^T v(s)\, ds
\]
Using the deterministic variance path above:
\[
I(T) = \theta T + (v_0 - \theta)\frac{1 - e^{-\kappa T}}{\kappa}
\]
and if kappa = 0, then:
\[
I(T) = v_0 T
\]
7. The Pricing Problem Collapses to a One-Factor Problem
Once you know I(T), the effective variance over [0, T] is:
\[
\sigma_{\mathrm{eff}}^2 = \frac{I(T)}{T}
\]
So option pricing is no longer a full two-factor Heston pricing problem. It becomes Black-Scholes-style
pricing with deterministic time-dependent variance.
That is the clean interpretation of the sigma = 0 boundary: not a broken model, but a reduced model.
8. What You Should Take Away
sigma = 0 changes the math significantly.- The variance shock vanishes completely.
rho becomes unidentifiable from prices.- The full Heston representation with
1 / sigma^2 is no longer the right formula.
- The correct thing is to treat
sigma = 0 as an exact boundary case with its own closed-form structure.
9. Practical Coding Interpretation
In calibration terms, that means:
- If a candidate has
sigma > 0, evaluate it with the usual Heston CF plus FFT machinery.
- If a candidate has
sigma = 0, do not feed it into the singular formula.
- Instead, evaluate that candidate using the deterministic-variance pricing branch.
This is not a heuristic floor. It is the mathematically correct boundary model.