Problem Sheet 1, 4CMP Spring Term 2018/19
PROBLEM SHEET 1
4CMP, SPRING TERM, PART 2: PROGRAMMING IN C++
Lecturer: Dr Fabian Spill ([email protected])
Due Date: Saturday 2
nd February 2019 at 11.59 pm
Weighting: This problem sheet counts 15% to your final mark for the spring part of 4CMP, but only
the best two out of problem sheets 1-3 count.
INSTRUCTIONS FOR THE SUBMISSION:
Submit your code by the deadline on Canvas. Each problem on this sheet should be implemented in
a single file named as follows: main_X.cpp. Here, X is the number of the problem. For example,
main_P1.cpp. Additionally, problem 2 requires some explanations that should be submitted as a pdf
file.
This file should compile without errors or warnings on Visual Studio 2015 as installed on the clusters
in the learning centre, and the executable should run without problems.
The source code should also be well formatted. This means that opening and closing brackets should
be aligned in a readable way, and we will discuss such styles briefly in the lectures. Also, it is crucial
that you comment your code well. For this first problem sheet, this means that every single nontrivial
statement you write should be explained with a comment. Moreover, at the beginning of each
file, give some brief description of what the code in this file is doing. You can follow the style I use
(which contains author, version, file, description etc, but you can also create your own style of
comments. The critical bit is that it contains the information required to follow what is going on in
the code). The programs you write should run in a self-explanatory way. If you are asked that the
program provides some input, it should first have some output on the command line telling the user
what kind of input is required. For instance, if you are supposed to read in the price of a share, do
the following:
double price;
std::cout << “ Please enter the price of share: “ << std::endl;
std::cin >> price;
Then, note that obviously group work or copying solutions from other students or sources
constitutes plagiarism and is not allowed.
Marking: The most important criteria for marking is the correctness of the code. It needs to compile
and run correctly, and do what you were asked to do. However, style, efficiency, readability and
formatting are also part of the evaluation criteria. In particular, if the code is so unreadable that one
cannot evaluate if it is working correctly, then you risk to loose many marks for that.
PROBLEM 1: MEAN, STANDARD DEVIATION AND CORRELATION
Write a function that calculates the sample mean and standard deviation for two vectors that are
input for these functions, and the correlation between the two vectors. If the length of the two
vectors is not the same, an error message should be given. The function should then “return” the
means, standard deviations and correlations. You should implement this “return” by using
references as arguments of the function as we discussed in the lectures.
The main program should declare two double-valued std::vector x,y that are populated with the
values
x = {1.2, 4.3, 2.3, 3.3}
y = {2.2, 3.1, 4.3, 2.1}.
Call the function that performs the calculations of means, standard deviation and correlation, and
then write the outputs (means, standard deviation and correlation) to the command line.
PROBLEM 2: A DOUBLE SUM
Write a function that has a single integer m as an input and returns the following sum as an integer:
In main, ask the user to input m and call the function that calculates the above sum. Then, compare
your result with the analytic result that is given by
(1 + m). Have the program check if the results of the direct calculation and the formula above
agree, and write the result of both the check and the actual numerical results on the command line.
If you encounter unexpected problems during the comparison, notice that the combined arithmetic
operations with multiplications and divisions are evaluate are evaluated left to right. This means for
three integers a, b, c, the product a * b * c is evaluated as (a * b ) * c. Likewise, division. So, can you
think of what could potentially go wrong when you rearrange the factors in a product?
Finally, check what the largest integer m is before your calculation exceeds the maximal integer,
which is 2
31 1. Is that maximum different for the direct calculation of the sum and for the analytic
formula 1
(2 + m)(1 + m )? Which m is the smallest for which the analytic
calculation matches the calculation through the sum. Can you explain your answer? Submit your
answer and explanations as a pdf file on canvas.
PROBLEM 3: BLACK-SCHOLES
First, write a function that returns the value of the cumulative distribution function of a Gaussian
distribution. For this purpose, you can use the complementary error function erfc, recalling that the
Gaussian CDF is obtained by CDF(x) = erfc(-x /sqrt(2)).
Both the function erfc and sqrt are in the library cmath. That means if at the beginning of your
source code, you write: #include <cmath>, then you can simply use erfc and sqrt in your code.
Then, write a function that calculates the price of a European Call option through the Black-Scholes
formula. The input for this function should be the current share price, the current time, the expiry
time, the strike price, the volatility, the interest rate and the dividend yield. Motivate your choice of
appropriate data types in your code comments. This function should call the Gaussian CDF function
that you created before. Next, write a function that calculates the value of a European Put option.
The function main should have the following functionality: It should declare the following nondimensional
parameters for testing purposes: share price: 50, strike: 50, Expiration: 1, Interest rates:
0.1, Volatitility 0.5, Divident: 0.2. The functions for put and call prices should be called with these
parameters, and the returned put and call prices written on the command line.
Problem Sheet 1, 4CMP Spring Term 2018/19
PROBLEM 4: SINE FUNCTION
Write a function that calculates the Taylor approximation
for the sine function to
arbitrary N. In main, call this function for inputs x and N that are obtained from the command line,
and then write the result of the Taylor approximation to the command line. Also write the output of
the function: “std::sin” which you can find in <cmath> on the command line, and calculate the error
between the Taylor expansion and the value obtained from the cmath function. When you have
solved this exercise, think also of the efficiency of your calculations. Each arithmetic operation costs
computational time. Thus, have you tried to keep your number of arithmetic operations low?
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