finance
Question 1
Investigate the accuracy and speed of convergence of different methods of quadrature.
Your investigation should include all the methods of quadrature we have included in the course and
should consider how effective they are on a range of different functions.
An appropriate way of assessing the efficacy of each method would be to assess how many
function calculations are required to achieve a given level of accuracy for each function
considered.
Question 2
Investigate the accuracy and efficiency of convergence of different branching methods of
calculating options prices.
Your investigation should include the CRR Binomial Model, the Tian Shift Model and the Trinomial
Model.
Your investigation should include an analysis of how may branches are needed to produce a given
level of accuracy in each method and an explanation of why you think this is the case.
Question 3
Investigate the efficacy of different option pricing models in explaining real world option prices
Your investigation should include the Black-Scholes, Gram-Charlier and Heston Stochastic
Volatility Models and should include a comparison of the implied volatilities under each model with
the implied volatility taken from real world option pricing data.
Question 4
Choose one of the following investigations:
Extend the Gram-Charlier Model to include more cumulants (very very hard);Build stochastic
volatility into the CRR Binomial Model;Investigate the efficacy of the assumptions underlying the
Black-Scholes, Gram-Charlier and Heston Stochastic Volatility Models, at explaining historic stock
prices;Investigate whether a quadranomial model for option pricing is feasible;Investigate the
impact of dividend default probabilities in pricing trees;Use a binomial tree to investigate the the
trade-off between raising capital through equities and bonds;Can you continue the sequence
Simpson’s, Simpson’s 3/8ths, Boole’s Rule?Can you extend the trinomial model to include
something equivalent to the Tian shift?Investigate the impact of frequency of testing for exotic
options such as knock-out options and lookback options;Can you develop an extension of n point
Gaussian quadrature to accurately handle Fourier transforms?Investigate the ability of different
option pricing models to accurately explain volatility smile