QM6 Simulation methods

Question 1

What is the log normal distribution?

Answer 1

• Lower bound 0 so cannot take on negative values
• Used in the black scholes merton model
• Right-skewed, but gets more bell shaped when the variance increases, as the left tail cannot expand into the negative
• mean = e^(mean + 0.5variance)

Question 2

What is the lognormal distribution used for?

Answer 2

• Used to describe distribution of asset prices since asset prices cannot go negative (except for derivatives like futures)
• A variable Y follows a lognormal distribution if ln(Y) is normally distributed

Question 3

What is volatility?

Answer 3

The annualised standard deviation of the continuously compounded daily returns of the underlying asset

Question 4

How do you annualise volatility?

Answer 4

Since r ~ mu (muT, sigma^2 T),
standard deviation = mu x sqrt (T)

So both the mean and variance of r scale linearly with time, but the standard deviation scales linearly with the sqrt of time

E.g., if daily volatility = 0.01, annualised volatility = 0.1 x sqrt(250) = 15.81%, when there are 250 trading days in the year

Question 5

How does a monte carlo simulation generate data?

Answer 5

• Random number generator
• Use to select points on a distribution (i.e., percentiles on a cumulative distribution function)
• Compound these over multiple periods, i.e., every month for 10 years
• Do many runs of these i.e., 1000
• Eventually it will draw out a normal distribution of all possible values of the portfolio at year 10

Question 6

How can we use monte carlo simulation to specify a minimum growth rate?

Answer 6

• Get the MCS to generate a distributions through many runs
• Look at the bottom i.e., 95% percentile
• Number here = return you can expect to get with 95% confidence
• End amount required / (1+95% confidence GM annual return)^10 = beginning capital required

Question 7

What does Monte Carlo Simulation provide?

Answer 7

• Statistical estimates, not exact results
• Does not support cause and effect conclusions

Question 8

What does the bootstrap method do?

Answer 8

• Creates, rather than estimates the distribution
• Involves randomly selecting observations from a set
• Some may be drawn multiple times, meaning each bootstrap is different
• Doing this e.g. 1000 times creates the distribution
• Can also be used to find the standard error of a measure of central tendency like the median, for which there is no analytic method available to find SE