Brownian Bridge™

1. In the Heston model $dS_t = rS_t\,dt + \sqrt{v_t}\,S_t\,dW_t^1$ and $dv_t = \kappa(\bar{v}-v_t)\,dt + \xi\sqrt{v_t}\,dW_t^2$ , what does the parameter $\rho$ (the correlation between $W^1$ and $W^2$ ) control in the implied vol smile?

The skew: negative ρ makes downside puts more expensive than upside calls (equity smile shape)The smile curvature: larger ρ widens the smile symmetricallyThe ATM vol level: ρ scales the short-dated ATM implied volThe term structure: negative ρ causes vol to be higher for longer maturities

2. The Feller condition for the Heston variance process is $2\kappa\bar{v} \ge \xi^2$ . For calibrated parameters $\kappa=1.5$ , $\bar{v}=0.04$ , $\xi=0.4$ : is the condition satisfied?

No: 2×1.5×0.04 = 0.12 < 0.16 = 0.4². The variance process may reach zeroYes: 2×1.5×0.04 = 0.12 > 0.4² = 0.016Yes: the Feller condition is automatically satisfied by any calibrated HestonCannot determine without knowing the current variance v₀

3. The Albrecher (2007) stable form of the Heston characteristic function is preferred over the original Heston (1993) form. The source of instability in the original form is:

A complex logarithm ln(1 − g·e^{−dτ}) whose argument crosses the branch cut as u variesThe characteristic function grows exponentially for large uThe formula uses the wrong measure (physical instead of risk-neutral)The parameter d can be complex, causing overflow in exponential computations

4. You calibrate Heston to today's vol surface and get an excellent fit (RMSE = 0.3 vol points). You re-run with 5 different starting points and get parameter sets with $\kappa$ ranging from 0.8 to 3.5. What does this tell you?

Multiple local minima: the vol surface does not uniquely identify Heston parameters — κ and v̄ are poorly identified separatelyThe calibration is unstable and the model is wrongThe RMSE is too low — the model is overfittingThe different starting points found different global minima

5. In the Lewis (2000) formula $C = S - \frac{\sqrt{FK}\,e^{-r\tau}}{\pi}\int_0^\infty \text{Re}\left[\frac{e^{-iuk}\,\phi(u-i/2)}{u^2+1/4}\right]du$ , why is the characteristic function evaluated at $u - i/2$ (shifted by $-i/2$ ) rather than at $u$ ?

The shift ensures the integrand decays for large u: e^{−iuk}φ(u−i/2) decays when k > 0, making the integral convergeThe shift converts the risk-neutral measure to the physical measureThe shift eliminates the need for a dampening parameter αThe shift corrects for the discrete dividend yield

6. A vol quant observes that Heston cannot fit the 1-week implied vol smile — the model smile is too flat compared to the market. The most likely explanation is:

Heston smile curvature at short dates scales as ξ²T, which is small for T = 1/52The Feller condition is violated for short maturitiesThe Albrecher formula is inaccurate for T < 0.1The mean reversion κ is too slow for short-dated options

7. During Heston calibration, $\xi$ is constrained to $\xi > 0$ via the bound $\xi \in [10^{-4}, 10]$ . At convergence, $\xi$ is at its lower bound $10^{-4}$ . What does this indicate?

The model is trying to reduce vol of vol to near zero — the market smile has minimal curvature and the optimizer is collapsing Heston toward the local vol limitThe calibration has failed and must be restarted with a different initial ξThe Feller condition is satisfied and the variance process is well-behavedThe instruments used do not include sufficient strikes to identify ξ

8. A desk wants to calibrate Heston intraday to freshly observed quotes, running every minute. The calibration must complete in under 100ms. Assuming 40 instruments, 5 parameters, and 1ms per Lewis formula evaluation, which approach is feasible?

Warm-start from the prior calibration: 3–10 LM iterations × 10 FD evaluations × 1ms = 30–100msCold-start grid search: 432 grid points × 40 instruments × 1ms = 17 seconds — too slowNewton's method directly on the 5-parameter system: 1 iteration = 5msReplace the Lewis formula with a closed-form approximation

Quiz: Heston Model Calibration in Practice — Fitting the Smile

Quick Quiz