Simulation Methods

Quantitative Methods

Lognormal Distribution and Continuous Compounding

Learning Outcome Statement:

Explain the relationship between normal and lognormal distributions and why the lognormal distribution is used to model asset prices when using continuously compounded asset returns.

Summary:

The lognormal distribution is used in financial modeling, particularly for asset prices, because it is derived from the exponential of a normally distributed variable, which ensures that prices remain non-negative. This distribution is right-skewed, reflecting the potential for asset prices to increase significantly, while being bounded by zero on the downside. Continuously compounded rates of return, if normally distributed, imply that the asset prices follow a lognormal distribution.

Key Concepts:

Lognormal Distribution

A variable is lognormally distributed if its natural logarithm is normally distributed. This distribution is appropriate for modeling asset prices because it is bounded below by zero and can model the asymmetric upside potential of asset prices.

Continuously Compounded Rates of Return

If the continuously compounded return of an asset is normally distributed, the future price of the asset will follow a lognormal distribution. This relationship is crucial for modeling stock prices over time in financial applications.

Relationship between Normal and Lognormal Distributions

The lognormal distribution is directly related to the normal distribution through the exponential function. If a variable's logarithm is normally distributed, the original variable is lognormally distributed.

Formulas:

Future Asset Price

P_T = P_0 \exp(r_{0,T})

This formula shows how the future price of an asset is determined by its current price and the continuously compounded return over the period.

Variables:

$P_T$ :: future stock price
$P_0$ :: current stock price
$r_{0,T}$ :: continuously compounded return from time 0 to T

Units: currency

Mean of Lognormal Distribution

\mu_L = \exp(\mu + 0.5 \sigma^2)

This formula calculates the mean of a lognormal distribution based on the mean and variance of the associated normal distribution.

Variables:

$\mu_L$ :: mean of the lognormal distribution
$\mu$ :: mean of the associated normal distribution
$\sigma^2$ :: variance of the associated normal distribution

Units: none

Variance of Lognormal Distribution

\sigma_L^2 = (\exp(\sigma^2) - 1) \exp(2\mu + \sigma^2)

This formula calculates the variance of a lognormal distribution based on the mean and variance of the associated normal distribution.

Variables:

$\sigma_L^2$ :: variance of the lognormal distribution
$\mu$ :: mean of the associated normal distribution
$\sigma^2$ :: variance of the associated normal distribution

Units: none

Monte Carlo Simulation

Learning Outcome Statement:

describe Monte Carlo simulation and explain how it can be used in investment applications

Summary:

Monte Carlo simulation is a computational technique that uses random sampling from specified probability distributions to model and analyze complex systems and processes. It is particularly useful in investment applications for estimating risk and return, valuing complex securities without analytic pricing formulas, and testing model sensitivity to changes in assumptions. The simulation involves generating a large number of trials to produce a distribution of possible outcomes, from which statistical estimates of performance and risk measures are derived.

Key Concepts:

Monte Carlo Simulation

A method that generates a large number of random samples from specified probability distributions to estimate the statistical properties of a system. It is used to model complex systems and processes where analytical solutions may not be feasible.

Application in Investment

In investments, Monte Carlo simulations are used to estimate risk and return, value complex securities, and analyze the impact of different assumptions on model outputs. It allows for the valuation of securities with embedded options or other complex features.

Process of Monte Carlo Simulation

The process involves specifying the model and distributions, generating random numbers, converting these into model outputs, and repeating the process to create a distribution of outcomes. This helps in understanding the range and likelihood of different results under varied scenarios.

Valuing Contingent Claims

Monte Carlo simulation can be used to value options and other contingent claims by simulating different scenarios of underlying asset prices and calculating the resulting payoffs. Examples include Asian options and lookback options.

Formulas:

Stock Price Change Model

\Delta \text{Stock price} = (\mu \times \text{Prior stock price} \times \Delta t) + (\sigma \times \text{Prior stock price} \times Z_k)

This formula is used within the Monte Carlo simulation to model changes in stock prices over time, incorporating both expected returns and random fluctuations.

Variables:

$\mu$ :: Expected return of the stock
$\sigma$ :: Standard deviation of the stock's returns
$\Delta t$ :: Time increment
$Z_k$ :: Standard normal random variable

Units: currency units

Bootstrapping

Learning Outcome Statement:

describe the use of bootstrap resampling in conducting a simulation based on observed data in investment applications

Summary:

Bootstrap resampling is a statistical technique used to estimate population parameters by repeatedly drawing samples from an observed dataset. This method treats the observed sample as the entire population and mimics the process of random sampling to construct a sampling distribution. It is particularly useful when the population distribution is unknown and only sample data is available. Bootstrapping is used to generate statistical estimates for parameters like mean, variance, skewness, and kurtosis, and is often compared to Monte Carlo simulation, which uses random data from a known distribution.

Key Concepts:

Bootstrap Resampling

Bootstrap resampling involves repeatedly drawing samples of the same size, with replacement, from the original sample. This process treats the sample as if it were the population, allowing for the estimation of statistical distributions and parameters.

Monte Carlo Simulation vs. Bootstrapping

While both methods involve repetitive sampling, Monte Carlo simulation generates random data based on a known distribution, whereas bootstrapping uses the empirical distribution from observed data. This makes bootstrapping particularly useful when the true population distribution is unknown.

Implementation of Bootstrap in Simulation

The bootstrap process involves specifying the quantity of interest, using the empirical distribution to generate data, and calculating the value of the contingent claim based on the simulated data. This process is repeated multiple times to produce statistical estimates.

Formulas:

Average Stock Price

\text{Average Stock Price} = \frac{\sum_{k=1}^K \text{Stock Price}_k}{K}

This formula calculates the average stock price over K subperiods, which is used in the valuation of contingent claims in bootstrap simulations.

Variables:

$K$ :: Total number of subperiods
$Stock Price_k$ :: Stock price at subperiod k

Units: currency

Present Value of Contingent Claim

C_{i0} = \frac{C_{iT}}{(1 + r)^T}

This formula calculates the present value of the contingent claim, discounting the future value of the claim at maturity by the appropriate interest rate over the period until maturity.

Variables:

$C_{iT}$ :: Value of the contingent claim at maturity
$r$ :: Appropriate interest rate
$T$ :: Time to maturity

Units: currency

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