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IntermediateOptions, Derivatives & FinancePython

Run this module

cd "Finance - Implied Volatility Surface"
python "implied_vol_surface.py"

View source on GitHub

Implied Volatility & the Volatility Surface

Black-Scholes turns volatility into a price. The market runs the formula backwards: it quotes a price and we solve for the volatility that reproduces it. That number is the implied volatility (IV) — the market's forward-looking estimate of how much the underlying will move.

Collect IV across every strike and maturity and you get the volatility surface. It is never flat, and the shape of that non-flatness (the smile, the skew, the term structure) is what options traders actually trade.

Functions

Function Description
bs_price(S, K, T, r, sigma, option, q) Black-Scholes-Merton European price
bs_vega(S, K, T, r, sigma, q) Vega — ∂price/∂σ, the engine of the IV solver
implied_vol(price, S, K, T, r, option, q) Invert BS for IV (Newton + bisection)
VolSurface Build & bilinearly interpolate a surface from a price grid
VolSurface.smile(T) The IV smile across strikes at one maturity

Inverting Black-Scholes

There is no closed form for IV, so we solve BS(σ) = market_price numerically. Newton's method converges fast because we know the derivative (vega):

σ_{n+1} = σ_n - (BS(σ_n) - price) / vega(σ_n)

Pure Newton can overshoot for deep in/out-of-the-money options where vega is tiny, so this implementation keeps a bracket [lo, hi] and falls back to bisection whenever a Newton step would leave it. The result is a solver that is both fast and can't diverge.

The surface

import numpy as np
from implied_vol_surface import VolSurface, bs_price

S, r = 100.0, 0.02
strikes = np.array([80, 90, 100, 110, 120], dtype=float)
maturities = np.array([0.25, 0.5, 1.0], dtype=float)

# price_grid[i, j] = market price of the option at maturities[i], strikes[j]
surf = VolSurface(S, r).fit(strikes, maturities, price_grid, option="call")

surf.iv(105, 0.4)   # interpolated IV anywhere on the surface
surf.smile(0.5)     # the smile at the 6-month maturity

Reading the shape

  • Skew (equities): IV rises as strike falls. Crash protection (low-strike puts) is in demand, so it is priced richer — a fatter left tail than log-normal assumes.
  • Smile (FX, single stocks): IV is high at both wings and lowest near the money.
  • Term structure: IV usually rises with maturity in calm markets and inverts (short-dated richer) during stress.

Practical notes

  • Garbage in, garbage out. IV inversion amplifies quote noise; always sanity check that prices respect no-arbitrage bounds (intrinsic ≤ price ≤ S for a call). implied_vol returns nan when they don't.
  • Bilinear interpolation is fine for queries inside the quoted grid. For a production surface you would fit a smooth parametric form (e.g. SVI) and enforce no calendar/butterfly arbitrage — a natural next step from here.
  • This pairs directly with Finance - Greeks Calculator, Black-Scholes Option Pricing and Finance - Volatility Calculator (realised vs. implied).
  • IV is an annualised number; to get an expected daily move, scale by σ * sqrt(1/252).

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