I have encountered an interesting question. Is it better to simulate the geometric brownian motion process for call itself or GBM for the underlying.
My question is can we actually apply GBM to call? is it possible?
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1$\begingroup$ The call price does not follow a geometric Brownian motion when the stock price does. We have $\mathrm{d}C_t = \frac{\partial C}{\partial t} \mathrm{d}t + \frac{\partial C}{\partial S} \mathrm{d}S_t + \frac{1}{2} \frac{\partial^2 C}{\partial S^2} \mathrm{d} \langle S \rangle_t$. So simulating the callprice does not work if our aim is pricing as the dynamics depend on its unknown derivatives. Or did I misunderstand your question? $\endgroup$– LocalVolatilityDec 26, 2017 at 13:36
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LocalVolatility's comment is correct. Just to provide a reference, the below is from Monte Carlo Methods in Financial Engineering by Paul Glasserman:
"Valuing a derivative security by Monte Carlo typically involves simulating paths of stochastic processes used to describe the evolution of underlying asset prices, interest rates, model parameters, and other factors relevant to the security in question."
The general idea is to assume the underlying price follows GBM and then simulate many price paths.
