As an entry level financial engineer, I'm trying to make sense of a practical case using the concepts I learned including local vol, monte carlo, so I really appreciate your advice if my understanding is correct:
Question:
Suppose we price a Down and In Barrier Call Option using local vol and monte carlo, I think we should implement it like this:
Firstly, we calibrate local vol surface using Dupire formula: $$ \sigma_{LV}^2(T,K) = \frac{\Sigma^2 + 2\Sigma T \left( \frac{\partial \Sigma}{\partial T} + \mu(T)K \frac{\partial \Sigma}{\partial K} \right)} {\left( 1-\frac{Ky}{\Sigma} \frac{\partial \Sigma}{\partial K} \right)^2 + K \Sigma T \left( \frac{\partial \Sigma}{\partial K} - \frac{1}{4} K \Sigma T \left( \frac{\partial \Sigma}{\partial K} \right) ^2 + K \frac{\partial^2 \Sigma}{\partial K^2} \right)} \tag{1} $$ where $y = \text{ln}(K/F(0,T))$ and $\Sigma = \Sigma(T,K)$
Secondly, we simulate several paths of the stock price evolution using $S_i = S_{i+1}*e^{(r-0.5*\sigma^2)*\Delta{}t+\sigma*sqrt(\Delta{}t)*\epsilon_i}, $ $\Delta{}t=(T-t)/N,$ $t_i = t+\Delta{}t*i, i = 0,1,2...N$, so here is my doubt and again really appreciate your advice: what is $\sigma$ in the evolution? is it $\sigma(K,T)$, $K$ is the strike of the barrier option and $T$ is the time at step $i$ and $T=t_i = t+\Delta{}t*i$?
Thirdly, after computing all the paths (ex. number of path is 100) of stock price evolution, for all the stock price (at maturity time T of course) that is below the barrier, the corresponding barrier option prices are zero, and for all the stock prices that are above the barrier, we use payoff formula $max(S_T-K,0)$ to compute option prices, then we add up all the computed option prices and divide it by the number of paths, and discounted the average price to current time.
I'm wondering if my understanding is correct?