I am struggling in interpreting results of my simulations. I use Monte Carlo algorithm to simulate stock paths and calculate option price. The notation: $r$ is a risk free interest rate, $T$ is time to maturity, $K$ - strike of an option, $S_T$ - stock price at time $T$, $\sigma$ - volatility of stock log-returns, $c$ - price of a call option, $t$ is a time step.
First I simulate stock paths assuming that stock price follows GBM process and calculate option price. Lets assume that $r = 0.05$, $K = 70$, $S_0 = 70$, $\sigma = 0.3$, $T = 0.28 $ (100 days). Number of sims $N = 1000$ and there will be 700 stock movements in each path (7 each day). I get $c = 5.008$.
BSM model with same parameters gives $c = 4.88$
Now I assume that log returns can be described by NIG disrtibution with parameters $\mu$, $\beta$, $\delta$, $\alpha$ and that log returns, $r_i$ can be calculated as (see this question): $$r_i = \hat\mu + \hat\beta\sigma_i^2 + \sigma_i\varepsilon_i$$ where $\sigma^2 \sim IG(\hat\delta/\hat\gamma, \hat\delta^2)$, $\varepsilon \sim N(0, 1)$.
I estimated parameters of NIG distribution using Google hourly returns for 2015 year ($\sigma = 0.29$). After running simulation I get $c = 8.77$ which is far bigger than BSM price and price obtained by GBM simulation.
The reason is that growth rate in the latter model bigger than in GBM. For the NIG model expected value for growth rate $R_T$ for 100 days (700 movements, 7 each day) is $$\mathbb{E}(R_T^{NIG}) = \left(\exp\left[\hat\mu + \hat\beta\frac{\hat\delta}{\hat\gamma}\right]\right)^{700} = 1.083$$
while GBM expected growth rate is $$\mathbb{E}(R_T^{GBM}) = \left[\exp(r - 0.5\sigma^2)t\right]^{700} = 1.00139$$
Finally, the question: if I want to sell an option which model should I use? Intuitively it should be GBM because it prices under $\mathbb{Q}$ measure. But why one want to construct different models that prices under real world $\mathbb{P}$ measure e.g. NIG model above? And what are applications of them?