How to simulate a delta hedged option strategy

I'd like to do a montecarlo simulation of a $$\Delta$$ hedged strategy (long OTM call) to see how the PnL distributes on cases like:

• $$\sigma_{bought} < \sigma_{realized}$$
• $$\sigma_{bought} > \sigma_{realized}$$
• $$\sigma_{bought} = \sigma_{realized}$$.

For this, I calculate the purchased option price with $$\sigma_{bought}$$ at $$t_0$$ and then do a random walk for the underlying asset $$S$$ so I can modify the hedge amount on each step.

My problem is that in order to calculate the hedge amount variations on $$t$$, I do not only need a new $$S_t$$ but also a volatility as $$\Delta$$ depends on vol (at least with BSM formula: $$N(d_1)$$ where $$d_1$$ depends on vol).

Question: What volatility should I use on each step to calculate the new $$\Delta$$ of the option?

• The simplest, but still interesting, case: at time 0 you (and everyone else) believe that the vol will be $\sigma_b$ so this is how the option is priced and how you will calculate $\Delta$ for hedging. But from 0 to T the actual movements of the stock are controlled by $\sigma_r$ (and you don't know this). Commented Feb 19, 2023 at 9:59
• So, for example If I want to compare $\sigma_b = 10\%$ with $\sigma_r = 15\%$. That means I should calculate initial option value with $\sigma = 10\%$, but then on each step: montecarlo simulation as well as $d1 (for \Delta)$ will use $\sigma = 15\%$? Commented Feb 19, 2023 at 18:05
• See below "Calculate initial option value with vol(b) = .10, and the hedging delta at each step with vol(b) = .10. But then use vol (r) = .15 to calculate the path of the stock." Commented Feb 20, 2023 at 6:57