Discrete Delta Hedging Adds P&L Dispersion Around Black–Scholes Value

Summary

The document explains a passage about hedging an option when future volatility is known but delta adjustments occur only at discrete intervals. Under the idealized Black–Scholes framework, continuous rebalancing with the correct volatility replicates the option value. With discrete rebalancing, the hedge is imperfect, so the final profit and loss can vary around that theoretical value.

The response characterizes this variation as noise from trading the underlying to maintain the hedge, rather than a systematic gain or loss. Less frequent adjustments leave more room for hedge error and therefore a wider distribution of outcomes; more frequent adjustments compress that distribution toward the theoretical value. The explanation is qualitative and relies on the stated idealized assumptions. It does not quantify the dispersion or address practical frictions such as transaction costs, jumps, volatility misspecification, or market impact, so it should not be read as a complete model of real hedging results.

Key ideas

• Continuous delta hedging with the correct volatility replicates the option value in the idealized model.
• Discrete rebalancing introduces variation in the final hedge profit and loss.
• The response describes the resulting dispersion as centered around the theoretical value, not as systematic profit or loss.
• More frequent adjustments reduce the width of the outcome distribution under the assumptions described.
• The explanation does not quantify hedge error or include market frictions.

Tags

# Known future volatility and difficulty in predicting final P/L


# Known future volatility and difficulty in predicting final P/L












I have started Chapter 1 of Dynamic Hedging by Taleb and it starts by saying

> "Even if traders knew the exact future volatility but hedged themselves (rebalanced the gamma) at discretely spaced increments, they would have difficulty predicting the final P/L". "If they traded every millionth of a second they would get a P/L with certainty. Increasing the frequency of adjustments would compress the results as shown"

> "The mean is the Black Scholes Merton value of the security"

I find this very much starts at the deep end and is fairly poorly explained sentence. Could someone please help me make sense of this.

## Answer by dm63 (score 3, accepted)

https://quant.stackexchange.com/a/24908

He's saying that if you know the volatility, and you hedge continuously, you can lock in the exact Black-Scholes price. Any deviation from that delta hedging scheme must result in noise. ie the replicated price must have a distribution with some width around the theoretical value. This noise does not create systematic profit or loss, because it's just doing market transactions in the underlying stock. Hence the distribution of the replicated price is still centered around the theoretical value. The greater the deviation from continuous hedging, the wider the distribution of the replicated price. Nothing earth shattering.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.