How Changing Implied Volatility Affects Black–Scholes Delta Hedging

Summary

The document asks how to choose volatility when using Black–Scholes deltas to hedge an option, and whether updating the volatility input with each day’s implied volatility changes expected profit and loss or its variance. It contrasts implied volatility, realized volatility, and an arbitrary assumption, then proposes recalculating delta as market implied volatility changes over the option’s life.

The discussion is framed as an open question rather than a worked analysis. It points to a paper comparing hedging with different volatility inputs, but provides no simulation results, derivation, or conclusion about the proposed daily update. The practical takeaway is that the choice of volatility input is central to delta hedging and merits empirical comparison; results will depend on the option, market path, and rehedging assumptions.

Key ideas

• Black–Scholes delta hedging requires a volatility input, and the document considers implied, realized, or assumed volatility.
• The proposed approach recalculates delta using current implied volatility as it changes over the option’s life.
• The author asks how this dynamic approach affects expected hedging profit and loss and its variance.
• The document poses the question but does not provide results or a conclusion.

Tags

# black Scholes model hedging without constant volatility


# black Scholes model hedging without constant volatility












I have started to look deeply in the hedging and I have created some simulations to simulate delta hedging strategies. I use BS model to calculate delta. The only issue was, which Volatility should I use?

- Implied volatility

- Realised volatility/Actual volatility (the actual volatility, which I can not predict in real world)

- Arbitrary volatility (which I guess)

I found a great answer to this question in this paper: http://web.math.ku.dk/~rolf/Wilmott_WhichFreeLunch.pdf

But, then I wanted to know, how things will change, if I do not use constant volatility in BS model. What about I will use the most actual IV as parameter to BS to calculate the delta for hedging during the life of the option? How the expected PnL and variance of PnL change compare to constant hedging with IV or realised volatility (as you can see it on mentioned paper in Chapter 5 Hedging with Different Volatilities)?

For example: After I sell option on Monday and I calculate the delta based on actual IV and hedge it. The next day (Tuesday), I will recalculate my delta based on the IV on Tuesday (which will be different compare to Monday IV) and hedge it.

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