monte carlo method in finance

whats is monte carlo method in finance?

its a numerical method for calculating option’s price.
it consist in generating many possible scenarios and calculate average.

example – call option

suppose Black-Scholes (lognormal) dynamics for underlying stock S. suppose interest rate and dividends are 0:

$$dS=S \sigma dW_t $$
$$dW_t$$ is standard browinian motion

1) discretize the equation with just one time step:
$$ S_1=S_0+S_0\sigma \epsilon $$
where
$$\epsilon$$ is a standard gaussian

2) generate N random gaussian numbers $$ \epsilon_1,\epsilon_2,\epsilon_3,..\epsilon_N $$
3) calculate $$S_1$$ for every $$\epsilon$$
so we’ll have $$ S_1(\epsilon_1) , S_1(\epsilon_2) ,.., S_1(\epsilon_N) $$
4) calculate payoff of the option for every $$ S_i$$
call payoff is$$ (S_1(\epsilon_1)-K)^+ $$
5) calculate option’s price as average of these payoffs

if interest rates are not 0 we’d need to change S dynamics and discount payoff with corresponding discount factor