Monte Carlo Pricing of European Call

This question focuses on using Monte Carlo simulation to value a European call option in a standard stochastic asset-price setting, given access only to uniform random number generation. The candidate is asked to outline how to construct simulated future prices under an appropriate pricing measure, how to obtain the option payoff from those simulated paths, and how to convert that into a present value estimate. The second part then shifts attention to the practical problem of generating normal random variables with a specified mean and variance starting solely from uniform draws, as would be the case with a basic pseudo-random number generator.

It leans on a mix of continuous-time asset pricing intuition, risk-neutral valuation, and numerical methods. On the probabilistic side, it requires understanding distributions, transformations, and properties of the normal distribution. On the computational side, it tests awareness of standard algorithms for mapping uniforms to normals and for structuring a Monte Carlo pricing routine. Interviewers look for clarity of the simulation pipeline, correct handling of discounting and parameters, and an appreciation of convergence, variance, and implementation trade-offs in practice.

What it tests

Monte Carlo simulation for option pricing leverages the law of large numbers and risk-neutral valuation. The core idea is that, under the risk-neutral measure, the expected discounted payoff of a derivative equals its fair price. By simulating many possible future states of the world (price paths) and averaging the discounted payoffs, we approximate this expectation. The accuracy improves as the number of simulations increases, converging to the true value due to statistical averaging. The transformation of random variables from one distribution to another (e.g., uniform to normal) relies on the fact that a continuous random variable's CDF maps outcomes to uniform [0,1], and inverting this process allows us to generate samples from any desired distribution.