Static Portfolio Theory for Variance and Gamma Swaps
The Theory Behind Static Replication
The theory relies on replicating the payoffs using option pricing mathematics. Here are the key formulas:
Variance Swap Theory
The realized variance of a stock can be replicated by a portfolio weighted by 1/K²:
\[ \text{Variance} \approx \int \frac{1}{K^2} \times \text{Option}(K) \, dK \]
Where:
- K = strike price
- You buy both calls (K > Sâ‚€) and puts (K < Sâ‚€)
- Weight each option by 1/strike²
- Sâ‚€ = current stock price
Why 1/K²? This comes from the mathematical relationship between option prices and realized variance in the Black-Scholes framework.
Gamma Swap Theory
Gamma can be replicated with a different weighting - typically 1/K:
\[ \text{Gamma exposure} \approx \int \frac{1}{K} \times \text{Option}(K) \, dK \]
The Mathematical Intuition:
- Options naturally capture convexity (gamma/variance effects)
- By weighting options across all strikes with these specific formulas, the portfolio's payout automatically matches what variance/gamma swaps should pay
- It's like the options "naturally aggregate" into the desired exposure
The Problem in Practice:
You need options at every possible strike, but:
- Exchanges only list strikes in increments (5, 10, 25 point intervals)
- Far out-of-the-money options are illiquid
- Very low/high strikes might not exist at all
So the "static" theory assumes infinite liquidity across all strikes, which doesn't exist in reality.
How Theory Translates to Actual Trades
Variance Swap - Long Position (receiving realized variance)
What you buy:
- Puts at all strikes below current stock price (K < Sâ‚€)
- Calls at all strikes above current stock price (K > Sâ‚€)
- Each weighted by 1/K²
Example: Stock at $100
- Buy put at $95 strike: quantity = 1/95² = 0.000111
- Buy put at $90 strike: quantity = 1/90² = 0.000123
- Buy call at $105 strike: quantity = 1/105² = 0.000091
- Buy call at $110 strike: quantity = 1/110² = 0.000083
- etc.
Gamma Swap - Long Position
What you buy:
- Same structure (puts below, calls above current price)
- But weighted by 1/K instead of 1/K²
Example: Stock at $100
- Buy put at $95: quantity = 1/95 = 0.0105
- Buy put at $90: quantity = 1/90 = 0.0111
- Buy call at $105: quantity = 1/105 = 0.0095
- etc.
Key Points:
- You're always buying options (long gamma/convexity)
- The weighting makes far OTM options have smaller positions
- Lower strikes get higher weightings due to the 1/K or 1/K² factor
- You need both puts and calls to capture moves in both directions
In Practice: Dealers approximate this by buying options at available strikes and interpolating.
Quick Comparison
| Aspect |
Variance Swap |
Gamma Swap |
| Weighting Formula |
1/K² |
1/K |
| Tail Exposure |
Lower (due to 1/K² weighting) |
Higher (due to 1/K weighting) |
| Example Quantity at $95 Strike |
0.000111 |
0.0105 |
| Hedging Approach |
Static replication possible |
Static replication possible |
| Market Liquidity |
Was popular, collapsed in 2008 |
Never gained meaningful traction |