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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?

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  • $\begingroup$ 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). $\endgroup$
    – nbbo2
    Feb 19 at 9:59
  • $\begingroup$ 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 $\sigma = 15\%$? $\endgroup$
    – Oliver Mohr Bonometti
    Feb 19 at 18:05
  • $\begingroup$ 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." $\endgroup$
    – nbbo2
    Feb 20 at 6:57

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Agreed with nbbo2. I did this exact thing a while back.

[So, for example If I want to compare 𝜎𝑏=10% with 𝜎𝑟=15%. That means I should calculate initial option value with 𝜎=10%, but then on each step: montecarlo simulation as well as 𝑑1(𝑓𝑜𝑟Δ) will use 𝜎=15%?]

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.

Monte Carlo Excel

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