Delta hedging — details
About this app
The hedge-frequency trade-off over 300 paths: the histogram of hedged P/L tightens with N (gamma residual risk ~ 1/√N), but transaction costs push the mean linearly negative — the curves over N reveal the RMSE optimum N★. Debunks “hedging more often is always better”.
Discrete hedging leaves a residual
When an option is delta-hedged only at discrete times, an error builds up between rebalances that depends on the curvature of the option value, roughly ½·Γ·(ΔS)². Across many price paths this error nearly averages out, but it scatters. The standard deviation of the hedged P&L falls about as 1/√N, where N is the number of rebalances. Every trade, however, costs a fee on the traded notional. So as N grows, the spread shrinks while costs push the mean below zero. The RMSE √(μ² + std²) combines both into one criterion.
What the app shows
The app hedges a short call (K = 100, r = 3 %, T = 1 year) on 300 fixed random paths that stay the same for every setting. The top chart is a histogram of final P&L with the mean and a ±1 std band. Below, curves over N show how mean and spread depend on hedge frequency, and a star marks the RMSE optimum N*. The values panel lists μ, std, RMSE, N*, average costs paid and average notional traded.
Try it
Keep transaction costs at zero and push N up: the histogram tightens and the optimum sits at the right edge. Then set costs to 15–20 basis points and σ to 10–20 %: the mean drifts negative as N rises and the RMSE curve develops an interior minimum. See how N* moves when you change costs or volatility.
Frequently asked questions
- How often should a delta hedge be rebalanced?
- Without transaction costs, more often always means less residual risk. With costs there is a trade-off, because every adjustment costs money. The best frequency is where the combination of P&L spread and cost drag, for example measured as RMSE, is smallest.
- What is gamma risk in hedging?
- Gamma describes how fast delta changes with the price. Between two rebalances the hedge therefore drifts out of line, leaving an error of roughly ½·Γ·(ΔS)². The larger gamma and the larger the price move, the larger this error.
- How does hedging error scale with rebalancing frequency?
- With N evenly spaced rebalances, the standard deviation of the hedged P&L falls roughly in proportion to 1/√N. Doubling the frequency cuts the spread by only about 30 %. Transaction costs, by contrast, grow with the number of trades.
- What does the star in the app mark?
- The star marks the hedge frequency N* with the smallest RMSE of hedged P&L across the 300 paths. Without transaction costs it sits at the right edge; with costs it moves inwards.
Subject: Financial derivatives
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