Question 1 [20 marks]

Please type your answers for this question into a text file named question01 .txt using a text editor such as Notepad.  (Do not use other file formats such as Microsoft Word.)  Please put your name and student ID number on the first line of this file.

Note:  You should answer  using your own words, to demonstrate that you fully  understand the concepts.

(a)  Explain briefly the difference between a vector and a set (in the C++ library).                               4

(b)  Give a precise definition of the following terms, and also a finance-related example of each:

1. A default constructor,

2. A parametrised constructor,

3. A copy constructor.

(c) This question concerns virtual functions in C++.

1.  Explain what is meant by a virtual function.

2. Write a short code sample to illustrate clearly the usage of a virtual function. Your code should be written in such a way that its behaviour will change if the keyword “virtual” is removed.

Question 2 [30 marks]

In this question, you are going to write a class to represent a 2-by-2 matrix of the form  ╱ c(a)   d(b) 、.

(a)  Create a new Visual Studio project with a single file, main02 .cpp.  You should implement everything for this question in this single file. On the first line of this file please type a C++ comment with your name and student ID number.

(b)  Copy-and-paste the following code into your file.  This will form the starting point for your solution.

(c) Add a constructor to the class so that a new matrix object can be created with a particular set       of values for the elements a, b, c and d. (This will enable lines 1 and 2 to compile.)                  4

(d)  Explain why you do not need to write either a copy constructor or an assignment operator for

the Matrix class in this simple example.

(Type your answer as a C++ comment at the end of the file main02 .cpp .)                                4

(e) Add two new member functions to the class so that two matrix objects can be either added or      

multiplied together, to form a new matrix. (This will enable lines 3, 4 and 5 to compile.)          8

(f) Write a new function so that line 6 will display the matrix S on the screen in the single-line

format  “(a,  b,  c,  d)” . What output does your program give (for the matrix S)?

(Type your answer as another C++ comment at the end of the file main02 .cpp .)                     4

(g)  Replace the empty function calculate() with the following code which creates a collection of four matrices. Add any other necessary code to your file so that the program can be built

and will run.                                                                                                                              2

(h)  In the space indicated, write some code that will use an iterator to loop over all the matrices in the collection, in order to determine the sum of all of them. The sum should be displayed on the screen. What output does your program give for the sum?

(Type your answer as another C++ comment at the end of the file main02 .cpp .)                     4

(i)  Finally, write some code to  “free” the memory allocated by new.                                                       4

Question 3 [50 marks]

In this question you will be pricing so-called “power options” using the Black-Scholes model. Please read the following two pages carefully, before attempting the assignment which starts on page 8.

Power options are vanilla European options whose payoffs at expiry time T depend on the share price ³T  to some fixed power. Specifically, the payoffs for call and put power options are given by:

Call power option:           令T(³T) = max ╱ ³T(n) - 云」0、

Put power option:           抄T(³T) = max ╱云 - ³T(n)」0、

where n S 0 is some fixed number, 云 S 0 is the strike price and ³T  is the price of the underlying share at expiry. Note that a power option with n = 1 is just the same as an ordinary option.

The price today (at time t = 0) of a vanilla European option expiring at future time T is given by the discounted expectation of the payoff VT(³T)

V0 = e_rT |0 X g(³T)VT(³T)d³T」

where g(³T)  is the probability density function of the share price at time T  (in the risk-neutral measure) as seen at time t = 0.  We will work within the framework of the Black-Scholes model where the share price follows geometric Brownian motion, and so g(³T) is the log-normal distribution

(

)   )   )

g(³T) = (

)   )   )

(

³Ts   2。1^  exp ┌ - ┐

0

for ³T  S 0」

otherwise」

with mean and standard deviation parameters

m = log ³0 + (r - #2 )T       and       s = #^T|

As usual, ³0  is the price of the share at time t = 0, # is the volatility of the share price, and r is the annualised continuously-compounded interest rate.  We are assuming here that the share pays no dividends during the lifetime of the option.

The integral above can actually be performed analytically for power options, giving exact formulas for the prices today:

令0 = ³0(n) exp ┌(n - 1)(r + n#2 /2)T┐Φ(h+) - 云e_rT Φ(h_ )」     抄0 = 云e_rT Φ(-h_ ) - ³0(n) exp ┌(n - 1)(r + n#2 /2)T┐ Φ(-h+)」

where

h+ = log(³0/云1/n )  )n - 、#2、T         and       h_ = h+ - n#^T」

and Φ(α)  is the cumulative distribution function of the standard  normal distribution.   You will implement these exact formulas for 令0  and 抄0  in part (b).

In part (c) you will evaluate the same integral numerically to determine approximate values for the prices which you will then compare with the exact values. You will need to replace the upper limit of the integral (infinity) with some suitable finite value ³T,max . You can then use Simpson’s rule for the integral using 之 steps, where 之 is even:

|ab y (α) dα ≈   !(┌)y (α0 ) + 2 1 y (α2j) + 4  y (α2j _1 ) + y (αN )[(2) 」

where h = (b - a)/之 and αj  = a + jh for j = 0」1」| | | 」之 .

The following C++ function implements the mathematical function Φ(α). You may incorporate this into your program for part (b) if you wish.

(a)  Create a new Visual Studio project, initially with a single file main03 .cpp. You may add other files to the project, as required.  On the first line of every file please type a C++ comment with your name and student ID number.

(b) Write a program to price a power option using the analytical (exact) formulas. Your program should behave as follows:

1. Ask the user which type of power option she wishes to price. She should be able to enter either “CALL” or “PUT” .

2. Then  prompt the user for the option  parameters K , n  and T , and the market data parameters ³0 , # and T , and check that the values entered are sensible.  (If she enters invalid values, then the program should simply display an appropriate error message, and terminate.)

3.  Calculate today’s price of the option using the appropriate analytical formula.

4.  Display the price on the screen.

Hint: For the highest marks, remember to use the techniques of object-oriented programming that we have covered in this module.

What are the exact prices of a power call option and a power put option, each with time to expiry T = 1|25 years, strike price K = 105 and n = 2, where ³0 = 10, T = 0|05 and # = 0|3?

(Type your answers as a C++ comment at the end of the file main03 .cpp.)

(c)  Now extend your program so that it can price a power option numerically, using Simpson’s rule for the integral.  The program should behave as follows (steps 1, 2 and 6 are the same as before):

1. Ask the user which type of power option she wishes to price. She should be able to enter either “CALL” or “PUT” .

2. Then  prompt the user for the option  parameters K , n  and T , and the market data parameters ³0 , # and T , and check that the values entered are sensible.  (If she enters invalid values, then the program should simply display an appropriate error message, and terminate.)

3. Ask the user which method she wishes to use.  She should be allowed to enter either “AN” for the analytical method, or “NUM” for the numerical method.

4.  If the user has chosen the numerical method, then prompt for ³T,max  (the upper limit of the integral) and N (the number of integration steps).

5.  Calculate today’s price of the option using the chosen method.

6.  Display the price on the screen.