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Bond options in the Vasiˇcek interest rate model

C++ Programming with Applications to Finance Spring 2020

The Vasiˇcek model for the short (interest) rate rt

in continuous time is formulated in terms of

the stochastic differential equation

drt = (θ − αrt)dt + σdWt,

equivalently,

where r0 is the current short rate and Wt

is a Brownian motion.

The aim of this project is to create a program in C++ that can be used to study the prices

of options on zero coupon bonds produced by means of an approximation method based on

the Vasiˇcek trinomial tree. Your program should be accompanied by end-user and developer

documentation.

This project description is accompanied by two files, namely a C++ header file Project.h

and two sample data files, rates.txt and prices.csv. The header file Project.h must be

included in your project without change.

General hints and tips

• This project description contains all the properties of the Vasiˇcek model that are needed

to complete this project. Further knowledge of the model (which is covered in the module

Modelling of Bonds, Term Structure and Interest Rate Derivatives) might make the project

more interesting, but is not required, and will not give any advantage when completing

the project.

• Read the submission instructions at the end of this document before starting work on the

project.

• Read through all the tasks before starting work on the project. Tasks do not have to be

completed in the order that they are listed here.

• You are free to recycle any code produced by yourself during the module. You are also

free to use any code provided in Moodle during the module, provided that such code is

acknowledged.

• Test your project with a freshly downloaded copy of Project.h just before submission.

This project contributes 30% to the mark for this module. The deadline for submission is

1pm on Monday 4 May 2020. The marks for each of the tasks below will be split into 60%

for coding style, clarity, accuracy and efficiency of computation, and 40% for documentation

(including comments within the code) and ease of use. Credit will be given for partial completion

of each task.

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Task 1: Data input, estimation and calibration (20%)

The Vasiˇcek model has three parameters: θ ∈ (−∞, ∞) is the long term mean parameter, α > 0

is the mean reversion parameter, and σ > 0 is called the volatility.

Your program should ask the user to enter the value of the parameter α, as well as the

current value of the short rate, denoted r0.

Your program should then estimate the value of θ, as follows. It should read a time series

of observed short rates ˆr1, . . . , rˆn from a file called rates.txt containing a series of past rates,

as in the sample rates.txt provided. The estimated value of θ is then calculated asθ = αr,

is the average of the observed short rates. Your program should display the value of θ once it

has been calculated.

Finally, your program should calibrate the value of σ by using the formula for the price of

a zero coupon bond in the continuous-time Vasiˇcek model. A zero coupon bond with maturity

date T is a financial contract guaranteeing a payoff of 1 at time T, and its price at time 0 in

the continuous-time Vasiˇcek model is

B(0, T) = e−r0A(0,T)+D(0,T),

where r0 is the current short rate.

Your program should ask the user to enter a maturity date T and the observed market price

Bquote of the zero coupon bond with maturity date T, and then solve the non-linear equation

Bquote = B(0, T)

for σ. When calculating B(0, T), your program should use the values of r0, θ and α that have

already been obtained. Your program should display the value of σ once it has been calculated.

The value of σ should be accurate to at least 5 decimal places.

Technical note for Task 1

The actual rates.txt file used for testing the program by the marker will contain a different

and much longer time series. Include the sample rates.txt file in your submission, placed in

the directory where your program will be looking for it, but do not hard wire the absolute path

to the file into your program as this path is likely to be different on the marker’s computer.

Explain in the end-user instructions how to propare and provide the file rates.txt so that your

program can read it.

Task 2: Vasiˇcek model (10%)

The header file Project.h contains a class VasicekModel that encapsulates a few of the essential

properties of the continuous-time Vasiˇcek model. Create a new source file VasicekModel.cpp

containing the implementations of the two member functions of VasicekModel that are incomplete.

The function VasicekModel::Long_term_average() should return the long term average

rate θα, and the function VasicekModel::Is_well_defined() should return true if α > 0 andσ > 0, otherwise it should return false.

Task 3: Vasiˇcek trinomial tree (20%)

The header file Project.h also contains a class VasicekTree encapsulating a trinomial tree that

can be used to compute approximate prices of financial derivatives, such as options, in the

Vasiˇcek model. Create a new source file VasicekTree.cpp containing the implementations of

the member functions of VasicekTree that are not already implemented in Project.h, using the

information provided below.

The starting point for the tree is the length ∆t > 0 of the small time interval between

successive steps. The value of the short rate r(n, k) at each time step n = 0, 1, . . . and node k

in the tree is

r(n, k) = e−αn∆tr0 +θα1 − e−αn∆t+ k∆r,

where∆r = σ√3∆t

is the vertical space between nodes. Here r0 is the initial short rate and θ, α and σ are the

parameters of the Vasiˇcek model.

The key idea behind the trinomial tree approximation is that the Vasiˇcek model is meanreverting,

in other words, whenever the short rate moves away from the long term mean θ/α,

it tends to move back closer to it over the long run. This means that it is less important to

model the short rate at values far away from the mean, and we incorporate this by limiting the

number of nodes in the tree at each time step to a fixed number. Thus the number of nodes in

the tree do not grow over time, which leads to very efficient pricing procedures.

Let kmax be the smallest integer that is larger than (or equal to) 0.184α

∆t

. At early time steps

n = 0, . . . , kmax − 1, the tree grows like a standard trinomial tree: it has 2n + 1 nodes at time

step n, numbered −n, . . . , n, and every node has three successors. At time step kmax, the tree

stops growing. At each time step n = kmax, kmax+1, . . ., the tree has 2kmax+1 nodes, numbered

−kmax, . . . , kmax. At such time steps, every node still has three successors, but the choice of

successors are now constrained by the fact that the tree does not grow any more. In all cases,

the three successors of each node k, called uk, mk, dk are always chosen so that mk is as close

to k as possible, and then uk = mk + 1 and dk = mk − 1.

Here is an illustration of the node structure on a tree with kmax = 3. Each node in the

diagram is labelled (n, k), where n denotes the time step and k the node number. The successors

of each node are indicated by arrows.

The probability of transitioning from any node k at any time step n = 0, 1, . . . to a node l at

time step n + 1 is denoted by pk,l. The successors and transition probabilities are summarised

3

in the following table:

Node k Successors Transition probabilities

Note that the successors and the transition probabilities associated with any node k do not

depend on the time step n at which the node appears.

Technical note for Task 3

You can use the ceil() function in the cmath library to determine kmax. The smallest integer

larger than a given number x is (int)ceil(x).

Task 4: Derivative pricing in the Vasiˇcek trinomial tree (20%)

The header file Project.h contains a pure virtual class EurOptionVasicekTree encapsulating a

European style derivative that offers a payoff at its maturity date N in the Vasiˇcek trinomial tree.

It also contains an inherited class ZeroCouponBondVasicekTree, which encapsulates a zero coupon

bond with payoff 1 at its maturity date N. Create a new source file EurOptionVasicekTree.cpp

containing the implementation of the function EurOptionVasicekTree::Price() using the information

below.

The payoff of a European style derivative with maturity date N can be represented by the

numbers H(−kN ), H(−kN + 1), . . . , H(kN ) where kN = min{kmax, N}, and where H(k) is the

payoff of the derivative at each node k at time step N. The price P(n, k) at each node k at

some earlier time step n < N is then calculated by means of the following procedure:

• At time step N, let

P(N, k) = H(k) for all k = −kN , . . . , kN .

• At every time step l = N −1, N −2, . . . , n, assume that kl+1 is known, as well as P(l+1, k)

for all k = −kl+1, . . . , kl+1. Then define

kl = min{kmax, l}

and

P(l, k) = e−r(l,k)∆t(pk,dk P(l + 1, dk) + pk,mk P(l + 1, mk) + pk,uk P(l + 1, uk))for all k = −kl, . . . , kl.

If n = 0, then the number P(0, 0) is referred to as the price of the derivative at time 0.

4

Task 5: Call options on zero coupon bonds (10%)

A European call option on a zero coupon bond with maturity date M is a financial asset with

maturity date N < M that gives its holder the right, without obligation, to buy the zero coupon

bond at a predetermined strike price K > 0 at time N. Its payoff at time N is therefore

max{B(N, M) − K, 0},

where B(N, M) is the price of the zero coupon bond at time N.

Create a subclass CallOption of EurOptionVasicekTree in a new header file CallOption.h to

represent European call options on zero coupon bonds in the Vasiˇcek tree model. The subclass

should use the class ZeroCouponBondVasicekTree that is defined in Project.h. This header file

may be accompanied by a code file CallOption.cpp if appropriate.

Task 6: Numerical study of call options (20%)

Your program should perform a numerical study of the prices of call options. In this task, your

program should use the values of α, r0 and T entered by the user, and the calculated values of

θ and σ, all from Task 1. It should ask the user to enter the strike K of a call option on a zero

coupon bond with maturity date T.

Your program should produce a file prices.csv using the comma separated values (csv)

format. It should contain a table of call option prices, together with appropriate row and

column headings. Every row m = 1, 2, . . . , 10 corresponds to the Vasiˇcek tree with∆t =TM ,

where M = 100m, and contains 10 column entries. The first entry is the price at time 0 of

a zero coupon bond in the Vasicek tree with maturity date M, and for n = 2, . . . , 10, the nth

entry in the row is the price at time 0 of a call option with strike K on a zero coupon bond in

the Vasiˇek trinomial tree, where the maturity date of the bond is M, and the maturity date of

the option is

N =n − 110× M = 10(n − 1)m.

Technical notes for Task 6

• A sample file prices.csv has been provided to illustrate the structure of the output. Your

output should contain numerical results in place of each of the entries marked “price”,

and you may vary the heading text if you wish.

• Spreadsheet applications (e.g. Microsoft Excel) can be used to open .csv files and create

graphs from the data they contain.

Submission instructions

Submit your work by uploading it in Moodle by 1pm on Monday 4 May 2020.

5

Format

Submit the code as a single compressed .zip file, including all Code::Blocks project (.cbp)

files, data files (.csv), source code and header (.cpp and .h) files and the executable (.exe)

file produced by compiling and linking the project, all residing in a single parent directory.

The .zip file should preserve the subdirectory structure of the parent directory (that is, the

subdirectory structure must be automatically recreated when unzipping the file). It must be

possible to open all project files and to compile, link and run the project using the

version of Code::Blocks running on computer lab machines.

The code should be accompanied by detailed documentation, split into two parts:

1. Code developer documentation containing information to help understand the code.

It should contain the following:

• Description of the file structure.

• Description of the available classes and functions.

• Instructions on how to extend the code by adding new options or derivatives.

• Test runs, including tables. Graphs are optional but encouraged.

The developer documentation should not include extensive extracts from the code (brief

snippets are perfectly fine, if typeset correctly—no screenshots!), and there is no need

to describe the mathematical methods in any detail. It is expected that the developer

documentation will be less than 10 pages in length.

2. The end user instructions should contain instructions on how to use the compiled .exe

program, how to input data and how the results are presented, and a brief description

of the methods implemented. The contents of this document should be appropriate for a

reader who is not familiar with C++. It is expected that the end user instructions will

be less than 5 pages in length.

The documentation files must be submitted in .pdf format (two separate .pdf files containing

developer documentation and user instructions) and uploaded in Moodle separately from the

code .zip file.

It is advisable to allow enough time (at least one hour) to upload your files to avoid possible

congestion in Moodle before the deadline. In the unlikely event of technical problems in Moodle

please email your .zip files to alet.roux@york.ac.uk before the deadline.

Code usage permissions and academic integrity

You may use and adapt any code submitted by yourself as part of this module, including your

own solutions to the exercises. You may use and adapt all C++ code provided in Moodle as

part of this module, including code from Capi´nski & Zastawniak (2012) and the solutions to

exercises. Any code not written by yourself must be acknowledged and referenced

both in your source files and developer documentation.

This is an individual project. You may not collaborate with anybody else or use code that

has not been provided in Moodle. You may not use code written by other students. Collusion

and plagiarism detection software will be used to detect academic misconduct, and suspected

offences will be investiaged.

If things go wrong

Late submissions will incur penalties according to the standard University rules for assessed

work.

It must be possible to open all project files and to compile, link and run the

project using the version of Code::Blocks running on computer lab machines. It

may prove impossible to examine projects that cannot be unzipped and opened, compiled and

run on computer lab machines directly from the directories created by unzipping the submitted

.zip files. In such cases a mark of 0 will be recorded. It is therefore essential that all

project files and the directory structure are tested thoroughly on computer lab machines before

being submitted in Moodle. It is advisable to run all such tests starting from the .zip files

about to be submitted, and using a different lab computer to that on which the files have been

created.

All files must be submitted inside the zipped project directory, and all the files

in the project directory should be connected to the project. A common error is to

place some files on a network drive rather than in the submitted directory. Please bear in mind

that testing on a lab computer may not catch this error if the machine has access to the network

drive. However, the markers would have no access to the file (since they have no access to your

part of the network drive) and would be unable to compile the project.

As part of the marking process, the file Project.h will be replaced. If the copy of Project.h

used in a project has been modified in such a way that the code does not compile

with the original file, then a mark of 0 will be recorded.

References

Capi´nski, M. J. & Zastawniak, T. (2012), Numerical Methods in Finance with C++, Mastering

Mathematical Finance, Cambridge University Press.

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