Common Probability Distributions

The normal distribution is widely used, but it allows for negative values—problematic when modeling asset prices, which cannot fall below zero. The lognormal distribution resolves this by ensuring all values are positive.

Lognormal Distributions

• Definition: A random variable \( X \) is lognormally distributed if \( \ln(X) \) is normally distributed.
• Key Properties:
  • Cannot take negative values—ideal for modeling stock prices.
  • Positively skewed (right-tailed).
  • Determined by the mean \( \mu \) and variance \( \sigma^2 \) of \( \ln(X) \).
• Mean and Variance: The mean and variance of the lognormal variable \( X \) are:
  \( \text{Mean} = \exp(\mu + 0.5\sigma^2) \)
  \( \text{Variance} = \exp(2\mu + \sigma^2) \times \left[\exp(\sigma^2) - 1\right] \)

Continuous Compounding and Volatility

If continuously compounded returns are normally distributed, then future asset prices are lognormally distributed.

• The future price \( S_T \) relates to the current price \( S_0 \) via:
  \( S_T = S_0 \cdot \exp(r_{0,T}) \)
  where \( r_{0,T} \) is the continuously compounded return over time \( T \).
• Continuously compounded returns are always less than or equal to holding period returns.
• Volatility is the standard deviation of returns. It is typically estimated from historical daily returns.
• Annualized Volatility: Scales volatility to a yearly basis:
  \( \text{Annualized Volatility} = \sigma_{\text{period}} \times \sqrt{T} \)
  where \( T \) is the number of periods per year (e.g., 252 for daily, 12 for monthly).

Independently and Identically Distributed (IID) Returns

If single-period returns are IID (independent and drawn from the same distribution) with mean \( \mu \) and variance \( \sigma^2 \), then the \( T \)-period continuously compounded return has:

• Mean: \( \mu T \)
• Variance: \( \sigma^2 T \)

Monte Carlo simulation is a computational technique used to model uncertainty and estimate the probability of various outcomes by simulating thousands of random scenarios.

Steps in a Monte Carlo Simulation

1. Define the underlying variables (e.g., stock price, interest rate) and their initial values.
2. Specify the time horizon and the assumed probability distribution (e.g., normal, lognormal).
3. Model the evolution of these variables using random draws from the chosen distribution.
4. Calculate the outcome (e.g., option payoff) at the end of each simulated path.
5. Discount the final value back to present value.
6. Repeat the process for thousands of trials to build a distribution of possible outcomes.

The accuracy improves with more trials. Monte Carlo provides a statistical estimate, not an exact answer, and is especially useful for complex instruments where analytical solutions are unavailable.

Bootstrapping is a resampling technique used to estimate the sampling distribution of a statistic by repeatedly drawing samples from observed data.

Core Concepts

• Samples are drawn with replacement from the original dataset.
• Used when traditional parametric assumptions (e.g., normality) are questionable or inapplicable.
• Enables inference (e.g., confidence intervals for mean, median, or VaR) without assuming a specific distribution.
• Treats the observed sample as a proxy for the true population, allowing simulation of new samples.

Both are simulation-based, but they differ in data source and purpose.