I am currently considering the price $C_0$ of a call option on a stock $S$ with $$ S_0 = 1 \\ K = 1.1 \\ r = 1\% \\ T = 1 $$
Based on the Black-Scholes formula, I have deduced that $C_0 = 0.356$.
However, I am currently trying to replicate this result in R. To do this, I have:
- Simulated 1000 Wiener processes (each with 1001 time steps between $t=0$ and $t=1$)
- Based on these processes, I have created 1000 models for the evolution of the stock price $S_t$, based on the formula $$ S_t = \exp(-W_t + t) $$
- Based on the 1000 obtained values for $S_1$ I have calculated the payoff of the call option $C$ in each case
- Discounting each of these values, by multiplying each by $e$, I have found 1000 possible values for $C_0$, which I have then taken the mean of
However, this method gives a result (of approx. 4) which varies significantly from the theoretical result obtained using the Black-Scholes formula. I am presuming that this is due to an error in my method used in R.
Can anyone help me to understand where I might be going wrong?
EDIT: The following image shows the exact question that I am attempting to answer.