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.

• The simulations you created are the tool to give the answer. In the simple case where you simulate with a GBM having constant vol $\sigma$ and hedge with BS having another constant vol $s$ it is possible to derive a formula for the hedging error. This is well known. In a very general case I recommend simulation and creating histograms of the heding errors. Jul 6, 2022 at 17:29
• Strictly speaking, the fact that the IV on Tuesday is different from the IV on Monday is evidence that the Black Scholes assumption of constant vol is wrong and a "new paradigm" is needed. In practice however most traders do what you say: they myopically adjust to the new market vol (reasoning that the market represents a resonable forecast of constant vol going forward) only to discover on Wednesday the the IV has again changed. It is a logically inconsistent approach (constantly revising the estimate of the constant vol), but in practice it seems to work not too badly. Jul 6, 2022 at 20:42