3.5.5 Failure Modes & Practice

Basic Prerequisites: 02 · The Heston Model and 04 · SV Dynamics.

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1. Intuition & Practical Objective

Everything up to here was mathematics; this page is about what actually goes wrong on a desk, and why. Four symptom classes, each traceable to a first principle:

1. The simulation breaks. Calibrated equity parameters routinely violate Feller; naive Euler produces negative variance; the fix is a scheme whose positivity condition is different from Feller’s.
2. Calibration is ill-posed. Five parameters fit hundreds of quotes, several of them only through products ($\rho \eta$, ${\eta }^2/{\sigma }_0$), and the objective is flat along combinations. Good fits are easy and mean little.
3. Hedging is not delta-hedging. With two risk factors you cannot neutralise volatility risk with the underlying. Minimum-variance delta, vega-weighted baskets and variance swaps are the actual hedging instruments, and each has its own bias.
4. Model risk dwarfs parameter risk. For skew-dependent payoffs, the choice of model moves the price by whole percentage points of notional - an order of magnitude more than the uncertainty in the parameters within one model.

The practical objective: be able to state the Feller and Milstein conditions and know they are different; know why Heston’s objective is flat and how to regularise; be able to price a digital correctly (with the skew term) and know what dropping it costs; and know the size of the LV-vs-SV gap on a structured product before arguing about parameters.

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2. Mathematical Ground Truth & Derivations

2.1 Scheme positivity ≠ Feller

The CIR variance process $dv=-\lambda (v-\bar{v})dt+\eta \sqrt{v}dZ$ has two distinct conditions:

• Feller (continuum): $2\lambda \bar{v}>{\eta }^2$ ⇒ $v_t>0$ a.s. If violated, the true process does touch zero.
• Milstein (discretisation): for the scheme $v_{i+1}=[\sqrt{v_i}+\frac{\eta }{2}\sqrt{\Delta t}Z{]}^2-\lambda (v_i-\bar{v})\Delta t-\frac{{\eta }^2}{4}\Delta t$ (Gatheral 2.18), the iterate is positive whenever $v_i=0$ implies a positive next value, which requires $4\lambda \bar{v}/{\eta }^2>1$.

They are different conditions, and the fitted SPX set of Gatheral Table 3.2 sits exactly in the gap: $2\lambda \bar{v}/{\eta }^2=0.624$ (Feller violated) but $4\lambda \bar{v}/{\eta }^2=1.248$ (Milstein positive). So the right conclusion is not “the model is broken” - it is “use Milstein, not Euler”. Euler with full truncation ($v^{+}=\max (v,0)$ inside the drift and diffusion) also works, but its bias is larger and its positivity is not guaranteed.

When neither holds, the options are the exact CIR transition (Broadie–Kaya; Bessel-function sampling, expensive), Andersen–Brotherton-Ratcliffe moment-matched sampling, or a QE scheme (Andersen 2008). Gatheral’s comparative statement is that the Milstein scheme at $v_i=0$ is preferred whenever $4\lambda \bar{v}/{\eta }^2>1$ because it costs nothing and preserves positivity.

2.2 Why Heston calibration is ill-posed

Calibration is the nonlinear least squares

$$\underset{\theta =(v_0,\bar{v},\lambda ,\eta ,\rho )}{\min }\sum _i{w}_i(C_i^{\text{model}}(\theta )-C_i^{\text{mkt}}{)}^2,w_i\propto \frac{1}{{\text{vega}}_i^2}\text{or}\frac{1}{{\text{bid-ask}}_i^2},$$

with vega/bid-ask weights converting price errors into vol-point errors - mandatory, since raw price errors are dominated by the high-premium wings. Gatheral’s factorial identification recipe (§3.4) is:

• two expirations fix ${\lambda }^{\prime }=\lambda -\frac{\rho \eta }{2}$ and the product $\rho \eta$;
• the ATM term structure then gives $\bar{v}$ and $v_0$;
• the skew curvature (kurtosis) separates $\rho$ from $\eta$.

Note what the data can and cannot see: only products $\rho \eta$ (skew) and ${\nu }^2/{\sigma }_0$ (curvature) are directly observable. $\rho$ and $\eta$ are individually identified only through the interaction of the skew’s curvature and the term structure - which is why two different parameter sets commonly fit the same surface to within bid-ask (and why adding jumps makes it worse: SVJ adds three more parameters with the same degeneracy).

Regularisation practice: fit across many expirations jointly; freeze $\beta$-like structural parameters by convention; penalise parameter change day-over-day; and check parameter stability over time, not just the residual today.

2.3 The digital: where the skew term is worth 12% of notional

A digital call pays $1$ if $S_T>K$. It is a strike derivative of a call, $D(K,T)=-\partial C/\partial K$, and in a smile model

$$D(K,T)=-\frac{\partial C_{BS}}{\partial K}-\underset{\text{vega}}{\underbrace{\frac{\partial C_{BS}}{\partial {\sigma }_{BS}}}}\frac{\partial {\sigma }_{BS}}{\partial K}.$$

The first term is the Black–Scholes digital; the second is the skew correction. For a 1y ATM digital at $25\%$ vol with a skew of $3$ vol points per $10\%$ strike, vega$\times$skew $=0.1188$ against an undiscounted digital value of $0.4503$ - i.e. $26\%$ of the digital’s value, or $11.9\%$ of notional. Gatheral’s ch 8 figure of ”$\sim 12\%$ of notional” is exactly this number. Any desk pricing digitals off a flat interpolated smile has a 12%-of-notional error, and it is a first-order effect, not a correction.

2.4 Model risk on structured products: the LV/SV gap

Because the statically-fit surfaces are (near-)identical, differences in exotic prices isolate the dynamics. Gatheral’s cliquet valuations with the Heston–Nandi benchmark parameters ($\bar{v}=0.04$, $\lambda =10$, $\eta =1$, $\rho =-1$) give:

structure	SV	LV	difference
locally capped, globally floored cliquet (EURO STOXX 50), MinCoupon $2\%$	expected coupon $3.53\%$	$2.55\%$	$\approx 2.94\%$ upfront ($3\times 0.98\%$ notional)
reverse cliquet (telecoms basket), MaxCoupon $100\%$	redemption $43.9\%$	$42.0\%$	$\approx 1.9\%$ notional
Napoleon (multi-index), MaxCoupon $10\%$	expected coupon $1.74\%$	$1.74\%$	$\approx 0$ - intuition fails

The last row is the lesson: forward-skew intuition says more negative skew $\Rightarrow$ lower Napoleon value, but the Napoleon’s vol-convexity and the vega$\to 0$ behaviour at the floor make the two models agree. Stress the modelling assumptions themselves, not the parameters within one model (Gatheral ch 10).

2.5 Hedging with two factors

The model has two risk factors and one tradable underlying, so the hedge must be completed with options or variance products:

• Delta: first-order, $e^{-qT}N(d_1)$ - but for options whose implied vol responds to spot, the minimum-variance delta corrects it (Hull ch 20 §20.5): ${\Delta }_{MV}={\Delta }_{BSM}+{\mathcal{V}}_{BSM}\partial \mathbb{E}[{\sigma }_{imp}]/\partial S$, which is smaller in magnitude than the BSM delta for equity puts. This is the practitioner’s face of the SSR (04 · SV Dynamics).
• Vega: bucket by expiry; the model’s vega map is what the calibration produced, so a wrong model gives a consistent-looking but wrong hedge ratio.
• Vol-of-vol: hedged by variance swaps / VS swaptions, whose prices depend on ${\nu }_T$ - hence the term-structure tests of §04.
• Cross-gamma: the ${\partial }^{2}P/\partial S\partial {\hat{\sigma }}^2$ term is the one that cannot be hedged with vanillas; its P&L is the SSR trade (Bergomi §9.10).

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3. Computational Implementation - Feller vs Milstein, negative variance, the digital, and calibration

We (i) check the two positivity conditions on the fitted SPX parameters, (ii) count negative-variance steps under full-truncation Euler and Milstein, (iii) size the digital skew error, and (iv) calibrate five Heston parameters to a synthetic $5\times 3$ smile from two starting points to expose the flat objective. Stdlib only.

Reading the output.

• Feller is violated; Milstein is safe. $2\lambda \bar{v}/{\eta }^2=0.624<1$ (Feller fails) but $4\lambda \bar{v}/{\eta }^2=1.248>1$ (Milstein positive). Exactly the gap described in §2.1.
• Euler really does go negative - $1.9\%$ of steps at $T=1$, 252 steps, 300 paths - while Milstein never does (minimum $3.1\times 10^{-5}>0$). This is the difference between a scheme that needs a truncation patch and one that is positive by construction. It is also a bias, not merely a stability question: truncation changes the effective variance dynamics.
• The digital skew term is $0.1188$, i.e. $11.9\%$ of notional ($26.4\%$ of the digital’s own value) - the ch. 8 figure reproduced. Note how much of that comes from vega being large at the money ($39.58$ per unit vol for $S=100$): digitals are pure skew instruments.
• Calibration is flat along the $\kappa$-$\xi$ ridge. Both runs fit the same 15 exact prices, but the first stalls at RMS $8.1\times 10^{-3}$ with $\kappa =1.868,\xi =0.478$ (6–7% off the true $2.0,0.5$) while the second recovers $\kappa =2.003,\xi =0.501$ at RMS $5.4\times 10^{-4}$. Identifiability improves with a finer pricer, but the lesson stands: RMS alone does not tell you whether you found the parameters, and the $\kappa \leftrightarrow \xi$ direction is weakly constrained. With real bid-ask noise and a 5-parameter fit to a single day, use regularisation and stability checks.

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4. Failure Modes & First-Principles Breakdowns

1. Confusing the Feller condition with the scheme’s positivity condition. They differ by a factor 2 in $4\lambda \bar{v}/{\eta }^2$; the fitted SPX parameters sit between them. Concluding “Heston is unusable” from a Feller violation is wrong; concluding “Euler is fine” is also wrong.
2. Truncating instead of fixing. Full-truncation Euler is a patch, not a solution: it introduces a downward bias in variance (and hence in price) exactly where the model spend most of its time. Use Milstein, QE, or the exact transition.
3. Calibrating without vega/bid-ask weights. Raw price errors put all the weight on the wings; the resulting $\rho \eta$ (skew) is then garbage. Weight so that a $1$-vol-point error costs the same everywhere.
4. Trusting the residual. Two parameter sets with identical RMS can produce different forward skews, different SSRs and different exotic prices. Report and monitor parameters and their stability, not just RMS.
5. Reading $\rho$ and $\eta$ off a single smile. Only $\rho \eta$ and ${\nu }^2/{\sigma }_0$ are directly observable; single-slice separation of $\rho$ from $\eta$ is fragile, and adding jumps (SVJ/SVJJ) multiplies the degeneracy (Gatheral ch 5: SVJ fits better than SVJJ with fewer parameters).
6. Pricing digitals off an interpolated smile without the skew term. A $12\%$-of-notional error, first order in the skew (§2.3). Always decompose into the BSM digital plus vega$//$strike-slope.
7. Assuming the model’s delta is the hedge. With $\rho \ne 0$ the option’s implied vol moves with spot; the minimum-variance delta ${\Delta }_{MV}={\Delta }_{BSM}+\mathcal{V}\partial \mathbb{E}[{\sigma }_{imp}]/\partial S$ is materially different (Hull §20.5). Hedging with ${\Delta }_{BSM}$ leaves a systematic P&L that the model itself predicts.
8. Arguing parameters when the disagreement is the model. LV-vs-SV gaps on cliquets are percentage points of notional (Gatheral ch 10 above); a parameter tweak is rounding error by comparison. Establish agreement on the modelling type first.
9. Treating vanillas as the calibration target for exotic books. Forward skew is not inferable from short-dated vanillas (Bergomi §3.1.7); making $\eta ,\rho$ time-dependent to do so is legitimate only if the corresponding forward-skew products actually trade.
10. Forgetting the volatility risk premium is priced separately. The $\mathbb{Q}$ vol-of-vol from a vanilla calibration is not a forecast; a P&L attribution that treats it as one will mis-attribute the vol-of-vol carry (Bergomi ch 5).

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5. References

• Gatheral, The Volatility Surface
• Bergomi, Stochastic Volatility Modeling
• Andersen, L. (2008), Simple and efficient simulation of the Heston stochastic volatility model
• Hull, Options, Futures, and Other Derivatives
• Haug, The Complete Guide to Option Pricing Formulas

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6. Connected Graph Bridges

• Back: 04 · SV Dynamics
• Forward: 06 · Advanced Extensions · Index Hub
• Cross-links: The Greeks & Dynamic Hedging · BSM · 04 Greeks & Hedging · BSM · 05 Failure Modes · VS · 05 Failure Modes · Econometrics & Time Series