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IV ExpertWeek 23 • Lesson 65Duration: 50 min

SURF Volatility Surfaces

The 3D map of market fear — and why Black-Scholes is visibly wrong here

Learning Objectives

  • •Understand the volatility smile and skew
  • •Learn how to construct and interpret volatility surfaces
  • •Know why the surface shape contains tradeable information

Explain Like I'm 5

If Black-Scholes were actually correct, all options on the same stock would have the same implied vol. They don't. Not even close. Plotting implied vol against strike gives you a curve called the "volatility smile" — OTM puts and calls have HIGHER implied vol than ATM options. The smile is the market saying "we know returns have fat tails." It IS the market's correction to Black-Scholes' flawed assumptions.

Think of It This Way

The vol surface is like a topographic map of fear. High points (high IV) represent areas where the market is pricing in danger. Low points represent calm zones. The map changes shape constantly as sentiment shifts. After a crash the terrain becomes jagged and tall. During calm markets it flattens out.

1The Volatility Smile

Pre-1987: Options were priced close to Black-Scholes. Implied vol was roughly flat across strikes. The market largely believed returns were normally distributed. Black Monday (October 19, 1987): The S&P 500 dropped 22.6% in ONE DAY. That's a 20+ sigma move under the normal distribution. Probability of occurrence under normality: essentially zero. And yet it happened. Post-1987: The volatility SKEW appeared overnight. OTM puts became far more expensive (higher implied vol) because traders now paid a premium for crash protection. The smile has never disappeared. The smile shape varies by asset class: - Equity index: Skewed left (put vol >> call vol). Pure crash protection demand. - FX: Symmetric smile. Currencies can blow up in either direction. - Commodities: Can be right-skewed. Supply disruption = price spike risk.

2The Surface in 3D

The full volatility surface has three dimensions: 1. Strike (or moneyness): Determines the smile/skew shape 2. Maturity: Different expirations have different vol levels 3. Implied volatility: The height of the surface Key features to watch: - Term structure: Usually upward-sloping in calm markets (longer-dated = higher vol). INVERTS during crises (short-dated > long-dated) because near-term fear spikes. - Smile dynamics: The smile becomes more pronounced in high-vol regimes — it "stiffens." - Sticky strike vs sticky delta: Does the smile anchor to strike prices or to moneyness? This determines how your hedges behave. The vol surface is one of the richest data structures in all of finance. Every point represents a market consensus about a specific probability distribution.

3Trading the Vol Surface

If you think the surface is "wrong" somewhere, you can trade it: - Skew trade: Think OTM puts are overpriced? Sell OTM puts, buy ATM puts. You're betting the skew will flatten. - Calendar trade: Think term structure will flatten? Sell far-dated, buy near-dated. You're betting near-term vol will rise relative to long-term. - Butterfly: Think the wings are overpriced? Sell OTM calls and puts, buy ATM. You're betting the smile will flatten. These are sophisticated trades. You need to understand not just the LEVEL of vol but how the SHAPE of the surface will change. Most retail traders don't even know this dimension of trading exists.

4Vol Surface as an Information Signal

Even if you never trade options, the vol surface tells you things about market expectations: - Steep skew = market pricing in crash risk. Consider being cautious with long positions. - Flat smile = relatively calm market. Lower hedging costs. - Inverted term structure = near-term event risk (earnings, FOMC, elections). - VIX summarizes the vol surface into a single number — production systems use it as a regime indicator. The vol surface is free information about what the options market thinks is going to happen. Options traders tend to be among the most informed participants. Ignoring this data is leaving alpha on the table.

Equity Implied Volatility Smile (30-Day Options)

Key Formulas

Moneyness

Normalized measure of how far the strike is from spot. m=0 is ATM. |m|>2 is deep OTM. The smile is plotted against moneyness for consistency across assets.

Hands-On Code

Volatility Surface Construction

python
import numpy as np

def build_vol_surface(strikes, maturities, market_prices, S, r):
    """Build implied volatility surface from market prices."""
    from scipy.stats import norm
    
    def bs_price(S, K, T, r, sigma):
        d1 = (np.log(S/K) + (r + sigma**2/2)*T) / (sigma*np.sqrt(T))
        d2 = d1 - sigma*np.sqrt(T)
        return S*norm.cdf(d1) - K*np.exp(-r*T)*norm.cdf(d2)
    
    def find_iv(price, S, K, T, r):
        sigma = 0.20
        for _ in range(50):
            bs = bs_price(S, K, T, r, sigma)
            d1 = (np.log(S/K) + (r + sigma**2/2)*T) / (sigma*np.sqrt(T))
            vega = S * norm.pdf(d1) * np.sqrt(T)
            if vega < 1e-10: break
            sigma -= (bs - price) / vega
        return sigma
    
    surface = np.zeros((len(maturities), len(strikes)))
    
    for i, T in enumerate(maturities):
        for j, K in enumerate(strikes):
            if market_prices[i][j] > 0:
                surface[i][j] = find_iv(market_prices[i][j], S, K, T, r)
    
    print(f"=== VOLATILITY SURFACE ===")
    print(f"Strikes: {strikes}")
    print(f"Maturities: {maturities}")
    print(f"ATM vol (shortest): {surface[0][len(strikes)//2]:.2%}")
    print(f"ATM vol (longest):  {surface[-1][len(strikes)//2]:.2%}")
    
    return surface

Builds an implied volatility surface from market option prices by computing IV for every strike-maturity combination via Newton's method.

Knowledge Check

Q1.Why do OTM puts on the S&P 500 have higher implied vol than ATM options?

Assignment

Build a volatility surface from option market data. Plot the smile at different maturities. Identify the skew direction and compare across asset types.

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