Simulation Techniques and Financial Modelling – Coursework
DEADLINE: MARCH 25th 2021 Students are required to solve ALL questions
This is a group coursework; rules as follow.
1. Group size: max 4 – No exceptions allowed.
2. Group composition: as given by the Course Director/Course Office – No exceptions allowed.
3. Extensions on deadline will only be granted by the course director.
4. It is the group members responsibility to deal with any problem arising within the group. No extensions will be granted for this kind of issues.
5. The coursework has to be submitted electronically via Moodle.
6. Please use suitable wordprocessing software (e.g. Microsoft Word with Equation Editor/MathType, or LaTex) to type your document, and once ready for submission, convert it into .pdf format using appropriate software.
No other format will be accepted.
7. Submit also your codes together with any functions you develop to support your code. Codes must be in either Matlab or Python. No other format will be accepted.
8. Label your files using your group number as ‘SMM281-Group N.-text’ for the main body and ‘SMM281-Group N.-code’ for your code. No other format will be accepted.
9. Usual rules apply in case of plagiarism.
10. THIS IS A FORMAL PIECE OF ASSESSMENT.
Any questions regarding the coursework must be asked in the Thursday live session in front of the whole class, at the beginning or at the end. I will not respond to e-mail or personal enquiries regarding the coursework
SMM281 – Coursework 2021
1
Question 1
Consider the VG process
Xt = μt + θGt + σWGt
with Gt denoting the Gamma subordinator – note we are using the same parametrization
as in the Lecture Notes and Week 1 slides.
a) Use the COS method to recover the probability density function of the process Xt att=1fortheparametersetμ=0.1,θ=−0.3,σ=0.2andκ=0.15. Verifythe accuracy by checking the mean, variance, skewness and excess kurtosis obtained from the two numerical schemes against the corresponding analytical expressions.
b) Perform a sensitivity analysis versus the model parameters. This task means to repeat the above exercise by changing one parameter at a time. As reference you can use the following values
−1, −0.5, 0, 0.5, 1 −0.5, 0, 0.5
0.1, 0.2, 0.3
0.01, 0.7, 2.
μ = θ = σ = κ =
c) Use the procedure of Eberlein et vanilla options (calls and puts) dence of the base parameter set.
d) Repeatthepreviousexerciseusingtherangesfortheparametersθ,σ,κgivenabove, and extract the corresponding implied volatility.
e) What conclusions can you draw concerning the role of these parameters in con- trolling the shape of the implied volatility, and the log-return distribution? Your analysis needs to be well justified. You can refer, for example, to Backus, D.K., Foresi, S., Wu, L., (2004), ‘Accounting for biases in Black-Scholes’, SSRN Elec- tronic Journal.
al. (2010) to price 6 months to maturity European for different strikes and maturities, in correspon-
SMM281 – Coursework 2021 2 Question 2
A general procedure to recover the Value at Risk with Monte Carlo simulation can be organized along the following steps.
1. Generate M i.i.d. random variables Rj , j = 1 . . . , M representing the holding period log-return
2. Sort {R1,R2,…,RM} in ascending order as {R1∗,R2∗,…,RM∗ }.
3. SetVaR=−Rk∗,fork=⌊αM⌋
then V aR is the (1 − α)% Value at Risk.
Assume 252 days in a year.
a) Assume the stock price is modelled as
St = S0eμt+σWt ,
with parameters μ = 0.1 and σ = 0.2.
Write a Monte Carlo code which computes the 1 day and 10 day 95% Value at Risk of the holding period log-returns using the Brownian bridge. Test the accuracy of your procedure against the closed form expression for the Value at Risk under the Gaussian distribution.
b) Consider now the case of a stock price of the form St = S0eμt+Xt ,
with Xt a VG process. Using the Variance Gamma bridge of Ribeiro and Webber (2003 – ‘Valuing path-dependent options in the Variance-Gamma model by Monte Carlo with a Gamma bridge’, Journal of Computational Finance, 7 (2)), write a Monte Carlo code to compute the 1 day and 10 day 95% Value at Risk of the holding period log-returns. Assume the following parameters
(μ, θ, σ, κ) = (0.11, −0.01, 0.2, 0.045).
Test the accuracy of your procedure against the exact method based on the Fourier
transform discussed in Week 5.
c) Under the assumptions in part (a) of this question, use the Brownian bridge to develop a Monte Carlo code which returns the 1 day and 10 day Value at Risk for different level of the confidence interval under the assumption that in one year the stock drops by 20% of its current value. Comment your results
d) Use the Variance Gamma bridge to repeat the same experiment as in part (c) under the assumptions of part (b). Comment your results.