Monte Carlo simulation estimates uncertain quantities by generating random samples or simulated paths. Its implementation pattern is to define a stochastic process, sample many trials, record outcomes, and summarize their variation. section 2.3 section 2.4
A practical simulation proceeds as follows:
- Specify the state process, transition probabilities, horizon, time step, and number of simulations. section 2.3 section 2.4
- Sample transitions in batches, compute each terminal state or path statistic, and retain the resulting outcomes. section 2.3 section 2.4
- Inspect dispersion across trials: longer stock-price horizons produce greater volatility, while constrained walks require importance weights for weighted mean-square distances. section 2.3 section 2.4
Binomial stock-price simulation
For a stock beginning at S0, calculate up and down multipliers u and d and an up-transition probability p. For each horizon, draw binomial up counts for N simulations, obtain down counts by complementation, and compute terminal prices as S0 multiplied by u raised to the up count and d raised to the down count. Comparing one-month and one-year results exposes horizon-dependent uncertainty. section 2.3
