Delta hedge lab

A delta-hedged option is usually described as a position on the difference between realized and implied volatility. That is true on average, and a single path can tell a very different story.

This lab buys a European call, hedges it at the implied volatility, and simulates what happens at a chosen realized volatility, hedging frequency and transaction cost. Set both volatilities to 20% and look at how wide the result still is.

What I found

For a three-month at-the-money call at 20% implied, daily hedges and 20,000 paths, the average result matches the first-order estimate of vega × (realized − implied) very closely: +0.99 against +1.00 at 25% realized, and −1.00 against −1.00 at 15%.

The noise is what the average hides. With realized exactly equal to implied, the expected result is zero, and the standard deviation is 0.44 on a premium of 3.99. That matches the Derman-Kamal approximation for discrete hedging, and it halves when you hedge four times as often. At 25% realized, a view that is five points right, about 2% of paths still lose money.

The last result surprised me the most. Plain realized variance explains between 35% and 67% of the variation between paths. Gamma-weighted variance explains between 97% and 98%. The market pays you for moves while you hold gamma, so a large move far from the strike is worth much less than the same move at the strike. Two paths with the same realized volatility can end in different places, and the formula shows why.

What it leaves out

Volatility is constant within a path, returns are lognormal, and the hedge uses the implied volatility. There are no jumps, no skew and no stochastic volatility, and real books have all of these.

Personal project built with synthetic data. It uses nothing from any employer, and it is not investment advice.