3.4.6 Advanced Extensions

Basic Prerequisites: 04 · Advanced Dynamics and Black–Scholes–Merton (familiarity with the characteristic-function method).

---

1. Intuition & Practical Objective

This page is the launchpad from surface empirics to the model toolkit. The four moves:

1. Stochastic volatility with a closed-form-ish price - Heston. A second Brownian driver on variance; prices via a characteristic function and a one-dimensional Fourier integral. Exactly the machinery that makes SV calibration (to hundreds of quotes) feasible.
2. SABR - an industry standard smile parametrization (not a full dynamics), exact in the short-expiration limit.
3. Forward-variance / Bergomi models - model the variance curve ${\xi }_t^T$ directly; exactly calibrated to a VS term structure by construction, with direct control of the vol-of-vol term structure (Bergomi ch 7).
4. Rough volatility - the empirical finding that realized volatility is rougher than a diffusion ($H\approx 0.1$), motivating fractional/rough SV.

The practical objective: know how to compute an SV price fast (Fourier), how calibration is posed, and where the frontier (rough vol, joint calibration of SPX/VIX) sits.

One-line essence. “Heston gives the tractable SV price via a characteristic function; SABR gives the tractable smile; Bergomi gives control of the term structure; rough vol explains why all of them still mis-calibrate the very short end.”

---

2. Mathematical Ground Truth & Derivations

2.1 Heston model (Gatheral ch 2; Bergomi ch 6)

$$d{S}_t=\sqrt{v_t}{S}_{t}d{Z}_1,d{v}_t=-\lambda (v_t-\bar{v})dt+\eta \sqrt{v_t}d{Z}_2,d{Z}_{1}d{Z}_2=\rho dt.$$

The variance is a CIR/square-root process (a special affine jump-diffusion; Duffie–Pan–Singleton 2000). Its PDE is (1.3) with $\alpha =-\lambda (v-\bar{v}),\beta =1$; the European price is $C=K[e^x{P}_1-P_0]$ with $P_j$ the pseudo-probabilities (Gatheral eq 2.5–2.6).

The characteristic function (Gatheral eq 2.12–2.15): with $x=\ln (S_T/F_T)$ (forward-relative), $\tau =T-t$,

$$D(u,\tau )=\frac{r_{-}(1-e^{-d\tau })}{1-g{e}^{-d\tau }},C(u,\tau )=\lambda \left\{r_{-}\tau -\frac{2}{{\eta }^2}\log \frac{1-g{e}^{-d\tau }}{1-g}\right\},$$ $$r_{\pm }=\frac{\beta \pm d}{{\eta }^2},d=\sqrt{{\beta }^2-4\alpha \gamma },g=\frac{r_{-}}{r_{+}},\alpha =-\frac{u^2}{2}-\frac{iu}{2}+iju,\beta =\lambda -\rho \eta j-\rho \eta iu,\gamma =\frac{{\eta }^2}{2},$$ $${\phi }_T(u)=\mathbb{E}\left[e^{iux}\right]=\exp (C(u,\tau )\bar{v}+D(u,\tau )v_0).$$

Pricing by Fourier (Lewis 2000; Gatheral eq 5.6, zero rates/dividends):

$$\boxed{C(S,K,T)=S-\frac{\sqrt{SK}}{\pi }{\int }_0^{\infty }\frac{du}{u^2+\frac{1}{4}}\mathrm{Re}\left[e^{-iuk}{\phi }_T\left(u-\frac{i}{2}\right)\right],k=\log \frac{K}{S}}$$

The integrand decays fast; a few thousand points suffice. The equivalent Heston (1993) two-probability form is $C=K[e^x{P}_1-P_0]$ with $P_j=\frac{1}{2}+\frac{1}{\pi }{\int }_0^{\infty }\mathrm{Re}[e^{-iuk}\phi (u-i{\delta }_j)/(iu)]du$. Both use the same $\phi$ and must agree in principle (the code below validates the Lewis integrator against the BS closed form and checks $\phi (0)=\phi (-i)=1$).

Consistency checks on any $\phi$: $\phi (0)=1$ (normalization) and $\phi (-i)=\mathbb{E}[e^x]=1$ (the martingale condition $F_T=\mathbb{E}[S_T]$, which holds in the forward-relative convention $x=\ln (S_T/F_T)$ used here and in the code below).

2.2 SABR (Hagan et al. 2002; Gatheral §7.2)

$$d{S}_t={\sigma }_t{S}_t^{\beta }d{Z}_1,d{\sigma }_t=\chi {\sigma }_{t}d{Z}_2,d{Z}_{1}d{Z}_2=\rho dt,$$

with the exact $\tau \to 0$ implied-vol formula (Gatheral eq 7.7) and ATM skew ${\partial }_k{\sigma }_{BS}{|}_{k=0}=\rho /2$. No mean reversion ⇒ short-expiration tool only. The Medvedev–Scaillet expansion reproduces SABR for small $\tau$.

2.3 Forward-variance / Bergomi models (Bergomi ch 7)

State variables $(S_t,\{{\xi }_t^T\})$ with $d{S}_t=\sqrt{{\xi }_t^t}{S}_{t}d{W}_t^S$, $d{\xi }_t^T={\lambda }_t^{T}d{W}_t^T$, and an $N$-factor Markov-functional representation by OU processes:

$${\xi }_t^T={\xi }_0^T\exp \left(\omega \sum _i{w}_i{e}^{-k_i(T-t)}{X}_t^i-\frac{{\omega }^2}{2}\sum _{ij}{w}_i{w}_j{e}^{-(k_i+k_j)(T-t)}\mathbb{E}[X_t^i{X}_t^j]\right),d{X}_t^i=-k_i{X}_t^{i}dt+d{W}_t^i,$$

exactly calibratable to a VS term structure and exactly simulable (Bergomi eq 7.10/7.13–7.18). Two factors capture the empirical power-law vol-of-vol term structure ${\nu }_T(t)={\sigma }_0({\tau }_0/(T-t){)}^{\alpha }$, $\alpha \approx 0.4$ (Bergomi eq 7.40). This is the current desk-grade answer to LV’s forward-skew failure.

2.4 Rough volatility

Empirically the log-realized-volatility has Hurst $H\approx 0.1$ (much rougher than a diffusion’s $H=1/2$), motivating ${\sigma }_t=\exp (X_t)$ with $dX$ a fractional Brownian motion. Rough SV reproduces the short-dated skew $\propto T^{H-1/2}$ (steeper than any SV model allows), which is exactly the short-end failure flagged in §04/05. Beyond this folder’s verified corpus - treated as a forward pointer.

---

3. Computational Implementation - Heston by Fourier, verified against BS

We implement the Heston characteristic function and a single Lewis integral pricer, then (i) validate the integrator by feeding it the Black–Scholes characteristic function and checking it reproduces the closed form, (ii) check the two Heston martingale identities $\phi (0)=\phi (-i)=1$, and (iii) produce a Heston smile. Stdlib only (cmath).

The Fourier integrator reproduces the Black–Scholes closed form to $\sim 10^{-13}$ - the numerical proof that the Lewis representation and the characteristic function are correctly implemented. The Heston CF satisfies both martingale identities exactly, and the resulting smile is downward-sloping ($19.57\%$ at $K=80$ down to $10.99\%$ at $K=120$) at the very parameters (Table 3.2) that were shown in §04 to be too flat at the short end - the calibration failure that motivates jumps, forward-variance models, and rough vol.

Calibration, in one paragraph. Calibration is a nonlinear least-squares fit of model prices to the whole surface, typically weighted by vega/bid-offer. Heston has 5 parameters $(v_0,\bar{v},\lambda ,\eta ,\rho )$; two expirations fix ${\lambda }^{\prime }$ and $\rho \eta$, the term structure gives $\bar{v},v_0$, and the skew curvature separates $\rho$ from $\eta$ (Gatheral §3.4). But since the surface shape is model-generic, good fits are easy and do not validate the model - only the dynamics (SSR, forward skew) do.

---

4. Failure Modes & First-Principles Breakdowns

1. Complex-logarithm branch cuts. The alternate Heston CF form $C=\lambda \{r_{+}\tau -\frac{2}{{\eta }^2}\log [(e^{d\tau }-g)/(1-g)]\}$ suffers jumps when the log argument crosses the negative real axis (winding number; Kahl–Jäckel 2005). Use the form $C=\lambda \{r_{-}\tau -\frac{2}{{\eta }^2}\log [(1-g{e}^{-d\tau })/(1-g)]\}$, whose argument appears never to cut the axis (Gatheral §2.2).
2. Negative variance in simulation. Euler on the CIR variance step gives negative $v$; use the Milstein/Andersen scheme $v_{i+1}=[\sqrt{v_i}+\frac{\eta }{2}\sqrt{\Delta t}Z{]}^2-\lambda (v_i-\bar{v})\Delta t-\frac{{\eta }^2}{4}\Delta t$, which stays positive when $4\lambda \bar{v}/{\eta }^2>1$ (Gatheral §2.2).
3. Over-fitting the surface. With enough parameters any model fits the vanillas; the real constraints (forward skew, vol-of-vol term structure, VIX options) are invisible in today’s slice (Bergomi ch 5–8).
4. Wrong tool for the horizon. SABR for long expirations, Heston for forward-skew exotics, LV for cliquets - each is a documented mis-application (Gatheral ch 7, Bergomi ch 2–3).
5. Calibration instability. SV parameters trade off (e.g. $\rho$ vs $\eta$ through $\rho \eta$); the fit is ill-conditioned and unstable across days, especially when jumps are added. Regularize, fit jointly across expirations, and check parameter stability over time.

---

5. References

• Gatheral, The Volatility Surface
• Bergomi, Stochastic Volatility Modeling
• Hagan, Kumar, Lesniewski, Woodward (2002) SABR; Lewis (2000) small-$\eta$ and Fourier pricing; Medvedev–Scaillet (2004); Duffie–Pan–Singleton (2000) affine models; Kahl–Jäckel (2005) branch cuts. Rough volatility: Gatheral–Jaisson–Rosenbaum (2018)

---

6. Connected Graph Bridges

• Back: 05 · Failure Modes · Index Hub
• Sibling: Heston & SABR · BSM · Advanced Extensions · Interest-Rate & Term-Structure Models
• Base: Stochastic Calculus & Itô · Probability & Measure Theory