Monte Carlo with Geometric Brownian Motion

1. Monte Carlo Simulation for Option Pricing

Monte Carlo simulation was first applied to option pricing by Phelim Boyle in 1977. The method is conceptually straightforward: simulate a large number of possible future stock price trajectories under the risk-neutral measure, compute the option payoff for each trajectory, and take the discounted average as the option price estimate. While less efficient than closed-form solutions like Black-Scholes for simple European options, Monte Carlo is extremely flexible and can handle path-dependent payoffs, multiple underlying assets, and complex models where analytical solutions do not exist.

Under risk-neutral pricing with geometric Brownian motion, the terminal stock price after time \(T\) can be generated directly in a single step:

Drift: \(\;(r - q - \tfrac{\sigma^2}{2})\,T\)

The drift is the deterministic component that governs the expected direction of the stock price over time. It has three parts:

• \(r\) (risk-free rate): under risk-neutral pricing, the stock must earn the risk-free return on average. This is not a forecast that the stock will rise; it is a no-arbitrage requirement ensuring that holding stock is equivalent, in expectation, to lending at the risk-free rate.
• \(-q\) (dividend yield): continuous dividends reduce the stock's capital appreciation. Cash paid out as dividends leaves the stock, lowering its growth rate by exactly \(q\).
• \(-\tfrac{\sigma^2}{2}\) (Itô correction): this convexity adjustment arises from Jensen's inequality applied to the exponential function. Subtracting \(\tfrac{\sigma^2}{2}\) corrects for this, ensuring that \(\mathbb{E}[S(T)] = S_0\, e^{(r-q)T}\) exactly. Without this correction, the simulation would systematically overestimate the expected stock price.
• \(T\) (time): the drift accumulates linearly with time. The longer the horizon, the greater the deterministic shift in the exponent.

Diffusion: \(\;\sigma\sqrt{T}\cdot Z\)

The diffusion is the stochastic component that introduces randomness into each simulated path:

• \(\sigma\) (volatility): scales the magnitude of random fluctuations. Higher volatility produces wider dispersion of terminal prices and increases the value of both calls and puts.
• \(\sqrt{T}\) (time scaling): uncertainty grows with the square root of time, not linearly. This is a fundamental property of Brownian motion: doubling the time horizon increases the standard deviation by a factor of \(\sqrt{2} \approx 1.41\), not 2.
• \(Z \sim \mathcal{N}(0,1)\) (random shock): each simulation \(i\) draws an independent standard normal random variable \(Z_i\). This single draw determines the entire terminal price for that path. A positive \(Z\) pushes the price up; a negative \(Z\) pushes it down. The collection of all \(N\) draws creates the distribution of terminal prices.

For each simulation \(i\), the payoff at maturity is computed and the option price is estimated as the discounted average:

The standard error of the estimate and the confidence interval quantify the precision of the simulation:

Key variables:

• \(S_0\), initial stock price
• \(K\), strike price
• \(T\), time to maturity (years)
• \(r\), risk-free interest rate (annualized)
• \(q\), continuous dividend yield (annualized)
• \(\sigma\), volatility (annualized)
• \(N\), number of simulated paths
• \(Z\), standard normal random variable
• \(\hat{C}\), Monte Carlo price estimator
• \(\text{SE}\), standard error of the estimate

2. Why ATM Calls Are More Expensive Than Puts When \(r > q\) and Volatility Implications

When the risk-free rate exceeds the dividend yield, ATM call options are always priced higher than ATM puts. The difference is driven entirely by the spread \(r - q\): the higher the interest rate relative to dividends, the larger the gap. The call holder defers paying \(K\) until expiration, saving interest on that cash, while the put holder defers receiving \(K\), losing the interest they could have earned. The chart below shows this relationship; it crosses zero exactly at \(r = q\).

However, a higher call premium does not mean more calls expire in the money. The option premium depends on the mean payoff (which is set by the forward price, always above \(K\) when \(r > q\)), but the ITM probability depends on the median of the terminal distribution. The Itô correction (\(-\sigma^2/2\)) in the GBM exponent drives a wedge between the two.

The terminal price distributions below illustrate this. With \(r = 4\%\) and \(q = 0\%\):

• At σ = 20%: σ²/2 = 2.0% < r − q = 4.0%. The exponent drift \((r - q - \sigma^2/2) = +2.0\%\) is positive, so the median sits above \(K\). The ordering is: \(K\) < Median < Mean, so more calls expire ITM.
• At σ = 30%: σ²/2 = 4.5% > r − q = 4.0%. The exponent drift \((r - q - \sigma^2/2) = -0.5\%\) turns negative, so the median drops below \(K\). The ordering becomes: Median < \(K\) < Mean, so more puts expire ITM, yet calls are still more expensive because the right tail pulls the mean (and the premium) above what the puts can match.

A positive spread \(r - q\) ensures that the expected terminal price (the forward price) always sits above the strike. Changing implied volatility does not affect this mean; it only increases the dispersion of terminal prices and widens the gap between the mean and the median. The call payoff histograms below illustrate this: with lower volatility, more calls finish in the money with frequent, smaller payoffs. With higher volatility, fewer calls finish in the money, but the right tail stretches further, producing larger individual payoffs that sustain the average premium.

3. Convergence and Accuracy

Despite its slower convergence compared to closed-form or lattice methods, Monte Carlo simulation is the method of choice for pricing exotic options with path-dependent features (Asian options, barrier options, lookback options), options on multiple assets, or instruments under models with stochastic volatility and jumps. Variance reduction techniques such as antithetic variates, control variates, and importance sampling can significantly improve convergence speed in practice.