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.
∂V/∂S: The standard Delta found in textbooks. It assumes Σ is frozen.
∂V/∂Σ: How much money you make/lose if volatility rises by 1 point.
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, T) is a function of Strike K only.
This means if Spot moves from $100 to $110, the volatility of the $100 Strike Put does not change.
What happens to ATM Vol?
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.
Result: Market Rally = ATM Vol Drop.
Trader's Intuition
- 1.Psychological Anchors: Investors often view round numbers ($100, $150) as permanent support/resistance levels. The fear (volatility) associated with breaking $100 stays constant regardless of where the stock is today.
- 2.Range-Bound Markets: This regime works best when the market is chopping sideways. Supply and demand for specific strikes (e.g., call overwriting at $110) keeps those vols pinned.
The Skew Trap
If you trade a Risk Reversal (Long Call / Short Put) expecting Sticky Strike behavior:
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 price itself ($100), 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).
If Spot moves +10%, the entire Vol curve moves +10% to the right. The "ATM Vol" remains constant.
Because Equity Skew is downward sloping (Slope < 0), the Shadow Delta term is usually Positive. This means Sticky Delta deltas are less negative for Puts than Sticky Strike deltas.
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 (e.g., "25-Delta Call"). By definition, if the spot moves, the "25-Delta" strike changes location. This structure forces a Sticky Delta regime.
The Hedging Implication
For a Short Put position:
"You don't need to sell as much stock to hedge, because the rising volatility helps cushion the blow."
Interactive Simulator: Strike vs. Delta
Adjust the Spot Price to see how the Volatility Surface reacts under different regimes. Notice how the Sticky Delta curve shifts sideways, while the Sticky Strike curve remains frozen.
The curve is rigid. Vol at $100 remains fixed even if Spot goes to $90.
The curve floats. ATM Vol follows the spot price.
Key Concepts
Sticky Strike
The "Painted on the Wall" theory. Volatility is fixed to absolute strike prices. When the stock moves, we look up volatility at the same strike level.
Sticky Delta
The "Floating Smile" theory. Volatility is tied to moneyness (K/S), not absolute price. The entire curve moves with 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:
Practical Implications
Risk Management
Getting the regime wrong leads to "P&L Leakage" - mysterious profits or losses in attribution reports. Naive VaR calculations can underestimate losses by ~40% during market crashes.
Rule of Thumb Adjustments
Adjusted Delta
Delta_adj ≈ Delta_BS + Vega × (Skew_slope / Spot)
Adjusted Gamma
Gamma_adj ≈ Gamma_BS + 2 × Vanna × (dσ/dS)