Calculator
Volatility smile
Implied volatility vs
ATM IV term structure (days to expiry)
Strike detail
Method
Black-76 on the forward F, discount D = e^(−rT), T = days / 365
C = D·[F·N(d1) − K·N(d2)] P = D·[K·N(−d2) − F·N(−d1)]
d1 = [ln(F/K) + σ²T/2] / (σ√T), d2 = d1 − σ√T
(identical to Black-Scholes-Merton with F = S·e^((r−q)T))
Forward from put-call parity (strike K* minimising |C − P|):
F = K* + (C − P) / D
No-arbitrage bounds (price must lie strictly inside):
D·max(F − K, 0) < C < D·F D·max(K − F, 0) < P < D·K
Solver: Newton σ ← σ − (V(σ) − V_mkt) / vega(σ)
keep bracket [lo, hi] ∋ root, start [1e−6, 10]; bisect when Newton leaves it
stop when |V(σ) − V_mkt| < 1e−12 or step < 1e−12
Smile summary per expiry, k = ln(K/F), OTM points:
IV(k) ≈ a + b·k + c·k² (least squares)
ATM IV = a, skew = b, curvature = c
The composite smile uses puts for strikes below the forward and calls at or above it, falling back to the other side when one is missing. Rows whose price breaches a bound are flagged in the detail table and excluded. Use mid prices; a stale or one-sided quote is the most common cause of a flagged row.
Summary for AI agents (Markdown mirror)
# Implied Volatility Smile & Skew Calculator
> Client-side implied volatility solver and smile plotter for European options across strikes and expiries. URL: https://quantcore-labs.com/tools/iv-smile/
## Input
- - Rows: days, strike, call price, put price (either price optional)
- - Spot, rate and dividend yield (used when the forward cannot be inferred from parity)
## Output
- - Call and put IV per strike, with no-arbitrage flags
- - Per expiry: forward, ATM IV, skew (dIV/dk), curvature, from a quadratic fit in ln(K/F)
- - Charts: smile per expiry, ATM term structure
## Method
- - Black-76 inversion; safeguarded Newton-Raphson with bisection fallback; converges for any price inside the bounds
- - Forward from put-call parity at the strike minimising |C − P|
FAQ
- Why use out-of-the-money options for the smile?
- OTM options are more liquid and their prices are almost entirely time value, so their implied volatility is less sensitive to small errors in the forward or in bid-ask midpoints. ITM options carry large intrinsic value, so a one-tick price error can move their IV a lot. Under put-call parity with the correct forward, call and put IVs at the same strike are identical anyway.
- Why does plain Newton-Raphson fail for implied volatility?
- Vega approaches zero for deep in- or out-of-the-money options and very short expiries, so a Newton step can shoot to negative or huge volatilities. This solver keeps a bracket that always contains the root and falls back to bisection whenever Newton would leave it, so it converges for every price within the no-arbitrage bounds.
- Does this work for China ETF options such as 50ETF or 300ETF?
- Yes. These are European-style, so Black-76 on the forward is the standard model. Inferring the forward from put-call parity is recommended because ETF dividends and borrow costs are otherwise hard to specify. Use mid prices and calendar days to expiry.
- What does the skew number mean?
- It is the slope b of the fitted IV curve in log-moneyness at the money. A value of −0.30 means IV rises about 3 volatility points for a strike 10% below the forward (k = −0.10). Equity and index smiles usually show negative skew because downside puts are in demand as crash protection.