How does continuous-time delta hedging work and why does it break down in practice?

Question

AcadiFi · Accepted Answer

Continuous delta hedging replicates an option by holding Delta shares of the underlying at every instant. Theoretically, the hedged portfolio earns the risk-free rate with zero variance.

In practice, Nadia Kowalski at Bergamo Quant Capital rebalances her 620-contract SPX call book at fixed intervals. With gamma of 0.084 and underlying volatility of 18%, discrete hedging introduces gamma PnL proportional to (dS)^2 - sigma^2 S^2 dt. Over a day with 8 rebalances instead of continuous, hedge slippage averaged 0.27% of premium.

```mermaid
flowchart TD
  A[Option Position] --> B[Compute Delta]
  B --> C[Buy/Sell Shares]
  C --> D[Wait dt]
  D --> E[Market Moves]
  E --> B
  E --> F[Gamma PnL Leak]
```

Three frictions kill continuous hedging:
1. Transaction costs — each rebalance pays bid-ask spread and commission
2. Gamma slippage — large jumps mean Delta changes faster than you trade
3. Discrete time — between rebalances, you're exposed to the second-order term

Optimal hedging frequency balances gamma risk against transaction costs. Leland's model adjusts implied volatility upward to compensate: sigma_adjusted = sigma * sqrt(1 + sqrt(8/pi) * k / (sigma * sqrt(dt))), where k is proportional cost. Nadia's desk targets rebalancing when |Delta drift| exceeds 0.015 rather than on a fixed clock.

How does continuous-time delta hedging work and why does it break down in practice?

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