The Geometry of Market Risk
In the Black-Scholes world, volatility (σ) is treated as a constant parameter. However, in reality, volatility is a dynamic surface that moves as the underlying price (S) moves. This creates a significant problem: if you calculate your hedge (Delta) assuming volatility is constant, you are missing a massive component of your risk.
The Total Derivative
To calculate the true risk of an option, we must use the Total Derivative. This mathematical concept states that the change in option price isn't just about the spot price moving; it's also about the volatility changing because the spot price moved.
The Total Derivative
Model Delta (∂V/∂S)
The standard Delta found in textbooks. It assumes Σ is frozen.
Vega (∂V/∂Σ)
How much money you make/lose if volatility rises by 1 point.
Asset-Vol Correlation (dΣ/dS)
The link. Does vol crash when the market rallies? (Usually yes).
The "Shadow Delta" Trap
The term Vega × (dΣ/dS) acts as a "Shadow Delta." It modifies your effective exposure.
Trader's Intuition: The Long Call Example
Suppose you own a Call option on the S&P 500.
- Scenario:The market rallies +1%.
- BS Delta:Make money on Delta.
- Reality:When S&P 500 rallies, panic subsides, and Volatility Drops (dΣ/dS < 0).
- Net P&L:You make money on price, but lose money on Vega.
Regime 1: Sticky Strike
The "Painted on the Wall" Theory
Imagine the volatility skew is a physical curve painted onto the price axis. It is static. It does not move. When the stock price (S) moves, we simply look up the volatility at the fixed strike (K) on this unmoving curve.
The Mechanics
Under Sticky Strike, implied volatility Σ(K, S) is a function of Strike K only.
Sticky Strike Math
Even though the curve is fixed, the At-The-Money (ATM) volatility changes. If the market rallies (moves right) on a downward sloping skew, the new ATM strike is higher, which has a lower volatility on the fixed curve.
Trader's Intuition
- 1.Psychological Anchors: Investors often view round numbers ($100, $150) as permanent support/resistance levels.
- 2.Range-Bound Markets: This regime works best when the market is chopping sideways.
The Skew Trap
Trading a Risk Reversal expecting Sticky Strike:
Regime 2: Sticky Delta
The "Floating Smile" Theory
Imagine the volatility skew is a kite tied to the stock price. As the stock price moves, the entire curve floats along with it. Volatility is not determined by the absolute price, but by how far the price is from the current spot (Moneyness).
The Mechanics
Under Sticky Delta, implied volatility is a function of Moneyness (M = K/S).
Sticky Delta Math
If Spot moves +10%, the entire Vol curve moves +10% to the right. The "ATM Vol" remains constant.
Why FX Markets Love This
In Foreign Exchange, there is no natural "Up" or "Down" (is USD/JPY going up or is JPY/USD going down?).
Therefore, volatility is quoted in Delta. By definition, if the spot moves, the "25-Delta" strike changes location. This structure forces a Sticky Delta regime.
The Hedging Implication (Short Put)
Interactive Simulator: Strike vs. Delta
The curve is rigid. Vol at $100 remains fixed even if Spot goes to $90.
The curve floats. ATM Vol follows the spot price.
The Skew Stickiness Ratio (SSR)
Real markets exist on a spectrum between Sticky Strike and Sticky Delta. The SSR quantifies exactly where we are:
Skew Stickiness Ratio
SSR = 0
Pure Sticky Delta
SSR = 1
Pure Sticky Strike
SSR ≈ 1.5-2.0
Real Market (S&P 500)