Brownian Bridge™

1. The Breeden-Litzenberger identity states that, for a twice-differentiable call price surface $C(K, T)$ : $\frac{\partial^2 C}{\partial K^2}(K, T) = e^{-rT}\, p(K, T)$ What does $p(K, T) $represent, and what market condition ensures$ p(K, T) \geq 0$?

p(K,T) is the risk-neutral density of S_T at level K; the no-butterfly-arbitrage condition ensures non-negativity.p(K,T) is the physical-measure density of S_T; no-calendar-spread arbitrage ensures non-negativity.p(K,T) is the implied vol at strike K and maturity T; the no-static-arbitrage condition ensures positivity.p(K,T) is the delta of the call option; put-call parity ensures non-negativity.

2. Dupire's formula for local volatility is: $\sigma_{loc}^2(K, T) = \frac{\partial C/\partial T + qC + (r-q)K\,\partial C/\partial K}{\frac{1}{2}K^2\,\partial^2 C/\partial K^2}$ Where does this formula break down and why?

It breaks down where the risk-neutral density p(K,T) = 0 (far OTM), because the denominator vanishes and local vol is undefined.It breaks down at the forward price K = F(T), because the numerator is always zero at ATM.It breaks down for T > 2 years because the forward Kolmogorov equation is unstable for long maturities.It breaks down for dividend-paying underlyings because the (r-q) term makes the equation non-elliptic.

3. A smooth equity implied vol surface is parameterised as $\sigma_{impl}(K, T) = 0.20 - 0.10 \cdot \ln(K/F_T)/\sqrt{T}$ , with $r = q = 0$ and $S_0 = 100$ . At $K = F_T = 100$ , $T = 1$ , using the Berestycki-Busca-Florent approximation, what is the approximate local volatility $\sigma_{loc}(100, 0.5)$ ?

20% — because log-moneyness is zero at ATM, so BBF gives σ_impl(F, T) = σ_loc(F, T/2) = ATM vol.10% — because the local vol is half the implied vol at ATM.40% — because the local vol skew is double the implied vol skew, pushing ATM local vol up.15% — because local vol averages the surface over the path from S₀ to K.

4. A structuring desk observes that the 1-year ATM implied vol skew is $-5\%$ per unit of log-moneyness. According to the Gatheral half-skew approximation, what is the slope of the local vol surface at the forward, half a year in?

−10% per unit of log-moneyness — because the implied vol skew is approximately half the local vol slope.−5% per unit of log-moneyness — because local vol and implied vol skews are equal at ATM.−2.5% per unit of log-moneyness — because the local vol slope is half the implied vol skew.−5% per unit of strike — because the conversion from log-moneyness to strike space halves the slope.

5. A quant desk is asked to price a 3-year cliquet paying $\sum_{i=1}^{3} \max(S_{T_i}/S_{T_{i-1}} - 1, 0)$ where $T_i = i$ years. They calibrate a local vol model to the current vanilla surface. What is the primary risk of using this model for the cliquet?

Local vol generates overly flat forward smiles: the implied vol surface for forward-starting options collapses toward ATM, underpricing the cliquet's skew exposure.Local vol overestimates the forward smile skew, causing the cliquet to be significantly overpriced.Local vol cannot price path-dependent products because the Feynman-Kac theorem does not apply to Markovian diffusions.Local vol misses the risk-free rate dynamics, which dominates cliquet pricing for long maturities.

6. In the Gyöngy (1986) theorem, what is the relationship between an arbitrary Itô process and its 'mimicking' diffusion?

The mimicking diffusion has the same one-dimensional marginal distributions as the original process; its diffusion coefficient is E[σ²_t | S_t = K].The mimicking diffusion has the same joint distribution as the original process at all times; it requires matching all finite-dimensional distributions.The mimicking diffusion has the same quadratic variation as the original process; its drift equals the local vol slope.The mimicking diffusion is the unique process whose transition density solves the same Fokker-Planck equation as the original with a constant volatility.

7. The Dupire formula (Theorem 3.2) is derived from the forward Kolmogorov equation for the transition density $p(S, T)$ . What is the key integration step that connects the Kolmogorov equation to the call price surface?

Multiply the Kolmogorov equation by the payoff (S − K)⁺ and integrate over S from K to ∞, then integrate by parts twice to isolate σ_loc.Take the Fourier transform of the Kolmogorov equation, equate it to the characteristic function of the log-price, and invert.Apply Itô's lemma to (S_t − K)⁺ and take expectations, yielding the Dupire equation directly.Differentiate the Black-Scholes PDE with respect to maturity T to obtain the forward equation, then identify local vol as the diffusion coefficient.

8. A desk is calibrating a Local-Stochastic Volatility (LSV) model of the form $dS = (r-q)S\,dt + \sigma_{loc}(S,t) \cdot Z_t \cdot S\,dW$ where $Z_t$ is a mean-reverting vol factor. What condition must $\sigma_{loc}(S,t)$ satisfy to ensure the LSV model reprices the vanilla surface exactly, and why is this calibration computationally demanding?

σ_loc must satisfy E[Z²_t | S_t = K] = 1 for all (K,T); calibration requires solving a McKean-Vlasov PDE or particle methods because the leverage function depends on the law of (S_t, Z_t).σ_loc must match the Dupire local vol pointwise; calibration is straightforward because Z_t cancels at the calibration points.σ_loc must be chosen so that the ATM implied vol equals σ_atm for all maturities; the skew is automatically matched by Z_t.σ_loc must satisfy ∂σ_loc/∂S = 0 everywhere; the stochastic component Z_t is solely responsible for generating the smile.

Quiz: Dupire Local Volatility

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