Library/CFA (Chartered Financial Analyst)/JuiceNotes™ 2024: Quantitative Methods - CFA® Level I/Simulation Methods

Simulation Methods

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Lognormal Distribution and Asset Prices5 min

In financial modeling, a distinction is made between the behavior of asset returns and asset prices. Asset returns are typically modeled using a normal distribution. However, asset prices follow a log-normal distribution. This is because asset prices cannot fall below zero, creating a lower bound, and they theoretically have no upper bound, resulting in a distribution that is skewed to the right (positive skew). The Black-Scholes-Merton model, a cornerstone of option pricing, relies on the assumption that the underlying asset price is log-normally distributed.

Key Points

Normal distribution is used for asset returns.
Log-normal distribution is used for asset prices.
Log-normal distribution is bounded below by 0.
Log-normal distribution is positively skewed (long right tail).
Black-Scholes-Merton assumes log-normal asset prices.

Continuously Compounded Returns and Scaling7 min

To model returns effectively, analysts use continuously compounded returns, calculated as the natural logarithm of the ratio of the closing price to the opening price. A fundamental assumption in this analysis is that returns are independently and identically distributed (i.i.d.). 'Independent' means past returns do not predict future returns, and 'identically distributed' implies stationarity—the mean and variance remain constant. Based on these properties, returns and volatility can be scaled over time. The mean return scales linearly (multiplied by time T), while the standard deviation (volatility) scales with the square root of time (multiplied by the square root of T).

Key Points

Continuously compounded return = ln(Ending Price / Beginning Price).
Assumption: Returns are i.i.d. (Independently and Identically Distributed).
Stationarity implies constant mean and variance over time.
Annual Mean Return = Daily Mean Return * 250 (assuming 250 trading days).
Annual Standard Deviation = Daily Standard Deviation * sqrt(250).

Monte Carlo Simulation6 min

Monte Carlo simulation is a computational algorithm that relies on repeated random sampling to obtain numerical results. It is likened to rolling dice many times to understand the range of outcomes. In finance, it involves identifying variables, defining their ranges and distributions, running thousands of simulations, and analyzing the resulting probability distribution of outcomes. Its primary strength is the ability to price complex securities, such as American options, where analytic formulas fail. Its main weakness is that it provides approximations rather than exact solutions and is computationally intensive.

Key Points

Used to model probability of different outcomes.
Process: Identify variables -> Define Ranges -> Run Simulations -> Analyze Results.
Strength: Prices complex path-dependent securities (e.g., American options).
Weakness: Provides statistical estimates, not exact analytic results.
Weakness: Offers less insight into cause-and-effect mechanics than analytic formulas.

Bootstrapping5 min

Bootstrapping is a statistical method that relies on random sampling with replacement. Unlike Monte Carlo simulation, which generates data based on estimated parameters (mean, variance) of a distribution, Bootstrapping uses the actual historical data as the 'population'. It repeatedly draws samples of the same size as the original dataset, replacing each drawn item so it can be drawn again. This allows analysts to construct a sampling distribution and estimate standard errors without making strong assumptions about the shape of the underlying population distribution.

Key Points

Resamples from the original dataset with replacement.
Does not require knowledge of population distribution parameters.
Uses the sample as the proxy for the population.
Useful when analytic formulas are unavailable or population distribution is unknown.
Both Bootstrapping and Monte Carlo rely on repetitive sampling.

Questions

Question 1

Which probability distribution is most commonly used to describe asset prices in financial modeling?

Simulation Methods

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