MF3052/MF6011 – Derivatives, Securities and Option Pricing Assignment 1 – Testing normality of asset returns (9 marks)
The Black-Scholes model for the price S(t) of an asset is
S(t) = S0e(μ−σ2/2)t+σB(t), t ∈ [0,T],
where S0 = S(0) and B is a standard Brownian motion. The log-return on the asset over a time period [t + δt] is
t
1. Yahoo finance (https://finance.yahoo.com) allows you to download historical data for pub- licly listed companies and stock market indices. Choose a stock or index and download closing prices for a period of 3 years. Then compute the daily log-returns of the asset, defined to be ri = log(Xi+1/Xi), where Xi is the closing price of the asset on the ith day.
(1 mark)
2. If the assumptions of the Black-Scholes framework are reasonable for your selection, the values ri should be samples from a normal distribution N((μ−σ2/2)δt,σ2δt), where δt = 1/252, since 252 is the approximate number of trading days in a year. Assuming we have a total of N returns, the sample mean is the maximum likelihood estimator (MLE) of the true mean:
St+δt 2
=(μ−σ /2)δt+σ(B(t+δt)−B(t)) which is a random variable with distribution N((μ−σ2/2)δt,σ2δt).
log S
2 1N
(μ−σ /2)δt≈ N
ri =:mˆ.
i=1 and the sample variance is the MLE of the true variance:
1 n
σ2δt ≈ n − 1
Hence compute the sample mean mˆ and standard deviation vˆ for the daily log-returns.
(2 marks)
3. Plot a histogram of your data, rescaled to provide an empirical probability distribution (this is referred to as a kernel density estimator and can be done in R using the density function, or you can also try the bkde function in the KernSmooth library). Overlay this plot with the true normal distribution curve with mean mˆ and standard deviation vˆ.
(3 marks)
4. Generate a Q-Q plot and hence comment on the normality of your asset price log-returns.
(3 marks)
Submit this assignment as a printed report to the School office by 4pm, Friday October 18, 2019. Late submissions without appropriate documentation will be awarded a mark of zero.
(ri − mˆ )2 =: vˆ2.
i=1