FX Smile Calibration from Delta Quotes

Worked Solution

How to Think About It: FX options are quoted in delta space because traders think in terms of hedge ratios, not strikes. The three quotes -- ATM vol, risk reversal, and butterfly -- fully characterize the smile's level, skew, and curvature at the 25-delta points. Before writing any formulas, understand what each quote means economically: ATM vol sets the overall level, the risk reversal measures how much the smile tilts (calls richer or puts richer), and the butterfly measures how much extra convexity the wings carry over ATM. A trader hearing "RR25 = -1.5 vol, BF25 = 0.3 vol" immediately knows puts are richer than calls (negative skew) and the smile has moderate curvature.

Quick Sanity Checks: - $\text{RR}_{25} > 0$ means 25-delta calls have higher vol than 25-delta puts (right-skewed smile, common in some EM pairs). $\text{RR}_{25} < 0$ means puts are richer (typical for risk-off pairs like EUR/USD). - $\text{BF}_{25} \geq 0$ always -- wings should cost at least as much as ATM in vol terms. A negative butterfly would mean the smile dips below ATM, which is possible but unusual. - ATM strike should be near the forward for the forward-delta-neutral convention.

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Part 1: Delta Convention and Mapping to Strikes/Vols

We use the forward delta (also called "pips delta" or "Black delta") convention, which is standard for maturities beyond about 1 year. Forward delta for a call is:

$\Delta_C^{\text{fwd}} = \Phi(d_1)$

where $d_1 = \frac{-k + \frac{1}{2}\sigma^2 T}{\sigma\sqrt{T}}$, $k = \ln(K/F_{0,T})$, and $\Phi$ is the standard normal CDF. For a put: $\Delta_P^{\text{fwd}} = \Phi(d_1) - 1$.

ATM is defined as delta-neutral straddle (DNS): the strike where $|\Delta_C| = |\Delta_P| = 0.5$. This gives $d_1 = 0$, so:

$K_{\text{ATM}} = F_{0,T} \, e^{\frac{1}{2}\sigma_{\text{ATM}}^2 T}$

The market quotes give us wing vols:

$\sigma_{25C} = \sigma_{\text{ATM}} + \text{BF}_{25} + \frac{1}{2}\text{RR}_{25}$

$\sigma_{25P} = \sigma_{\text{ATM}} + \text{BF}_{25} - \frac{1}{2}\text{RR}_{25}$

This comes from the definitions: $\text{RR}_{25} = \sigma_{25C} - \sigma_{25P}$ and $\text{BF}_{25} = \frac{1}{2}(\sigma_{25C} + \sigma_{25P}) - \sigma_{\text{ATM}}$.

Given $\sigma_{25C}$, the 25-delta call strike satisfies $\Phi(d_1(K_{25C}, \sigma_{25C})) = 0.25$. Since $\Phi^{-1}(0.25) \approx -0.6745$, we solve:

$\frac{-\ln(K_{25C}/F_{0,T}) + \frac{1}{2}\sigma_{25C}^2 T}{\sigma_{25C}\sqrt{T}} = -0.6745$

$K_{25C} = F_{0,T} \exp\left(\frac{1}{2}\sigma_{25C}^2 T + 0.6745 \, \sigma_{25C}\sqrt{T}\right)$

Similarly, the 25-delta put has $|\Delta_P| = 0.25$, i.e., $\Phi(d_1) = 0.75$, so $d_1 = 0.6745$:

$K_{25P} = F_{0,T} \exp\left(\frac{1}{2}\sigma_{25P}^2 T - 0.6745 \, \sigma_{25P}\sqrt{T}\right)$

We now have three $(K, \sigma)$ pairs: $(K_{25P}, \sigma_{25P})$, $(K_{\text{ATM}}, \sigma_{\text{ATM}})$, $(K_{25C}, \sigma_{25C})$.

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Part 2: Smooth Smile Calibration

A clean and arbitrage-friendly parameterization is the SVI (Stochastic Volatility Inspired) model of Gatheral. Total variance as a function of log-moneyness $k = \ln(K/F_{0,T})$ is:

$w(k) = a + b\left(\rho(k - m) + \sqrt{(k - m)^2 + s^2}\right)$

where $a, b, m, \rho, s$ are parameters. This has five degrees of freedom, which is ideal for fitting three anchor points plus controlling wing behavior.

Calibration procedure: 1. Convert the three $(K_i, \sigma_i)$ pairs to $(k_i, w_i)$ where $w_i = \sigma_i^2 T$. 2. Minimize the sum of squared errors $\sum_{i} (w_{\text{SVI}}(k_i) - w_i)^2$ subject to the no-arbitrage constraints below. 3. Use initial guesses: $a \approx \sigma_{\text{ATM}}^2 T$, $m \approx 0$, $\rho$ from the sign of $\text{RR}_{25}$.

With only three data points and five parameters, regularization or fixing some parameters (e.g., $m = 0$) is needed to avoid overfitting. In practice, 10-delta quotes are added for five anchor points.

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Part 3: No-Arbitrage Conditions

Butterfly arbitrage (single maturity):

The key condition is that the implied probability density must be non-negative everywhere. In terms of total variance $w(k)$, this is equivalent to the condition that the function:

$g(k) = \left(1 - \frac{k \, w'(k)}{2w(k)}\right)^2 - \frac{w'(k)^2}{4}\left(\frac{1}{w(k)} + \frac{1}{4}\right) + \frac{w''(k)}{2} \geq 0 \quad \forall k$

This is the Gatheral-Jacquier condition. In plain terms: - $w(k) > 0$ for all $k$ (positive total variance). - $w(k)$ cannot be too concave -- excessive concavity means negative density, which means you can construct a butterfly spread with negative price. - The right wing slope must satisfy $\lim_{k \to \infty} w'(k) \leq 2$ (Roger Lee's moment formula).

Calendar arbitrage (across maturities):

Total variance must be non-decreasing in maturity at every strike:

$w(k, T_1) \leq w(k, T_2) \quad \text{for all } k, \; T_1 < T_2$

Equivalently, $\partial w / \partial T \geq 0$ for every $k$. This ensures that longer-dated options are worth at least as much as shorter-dated ones at the same strike -- otherwise you could sell the longer-dated and buy the shorter-dated for a free calendar spread.

In practice, after fitting SVI slices at each maturity, you check that the total variance surfaces do not cross when plotted as a function of $k$ across $T$.

Answer: The three delta quotes map to wing vols via $\sigma_{25C/P} = \sigma_{\text{ATM}} + \text{BF}_{25} \pm \frac{1}{2}\text{RR}_{25}$, and strikes are recovered by inverting the forward-delta formula. The smile is calibrated using SVI parameterization of total variance $w(k)$. No-butterfly-arbitrage requires the Gatheral-Jacquier density condition $g(k) \geq 0$ everywhere; no-calendar-arbitrage requires total variance to be non-decreasing in maturity at every strike.