HONMD1Y/101/0/2012
Tutorial letter 101/0/2012
Introduction to the mathematical
modelling of derivatives 1
(HONMD1Y)
Year module
Department of
Decision Sciences
This tutorial letter contains important
information about your module
1
CONTENTS
Page
1. Introduction and welcome . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2. Purpose and outcomes of the module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3. Lecturer and contact details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
4. Module related resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
5. Student support services . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
6. Module specific study plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
7. Module practical work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
8. Assessments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
9. Examinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
10. Other assessment methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
11. Frequently asked questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7
12. The syllabus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
13. Appendix A - Evaluation of South African stocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
14. Appendix B - Swap analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
15. Appendix C - Assignments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
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1.
HONMD1Y/101/0
INTRODUCTION AND WELCOME
We welcome you as an honours student in the Department of Decision Sciences, and especially to the module
HONMD1Y (Introduction to the mathematical modelling of derivatives 1).
The tutorial matter for this module consists of Tutorial Letters 101 and 301. Some of this tutorial matter
may not be available when you register. Tutorial matter that is not available when you register will be
posted to you as soon as possible, but is also available on myUnisa. Further material for this module is
appended to this tutorial letter. Other tutorial letters will be sent to you by the Department of Decision
Sciences. 200-series tutorial letters will only be sent to you once you have submitted your assignments.
Note that you must be registered on myUnisa in order to submit assignments, to access library functions,
download study materials, contact or chat with the lecturer or fellow students and access additional resources.
Mathematical techniques form the basis of any attempt to model economic and financial reality in a quantitative way. This module is designed to introduce you to some of the most useful of these techniques, and
also to give you some idea of the environment in which these techniques are applied.
In this module we shall endeavour to equip you with a ready knowledge of the operation of models used
in pricing instruments in the economic and financial markets, as well as the assumptions underlying these
models. Our approach is largely international, although in the first part — bond pricing — we look mainly
at South African examples.
There is some confusion of terminology. The British usage of “shares and stocks,” to refer to what in America
are called “stocks and bonds”, was originally adopted in South Africa, but the all-pervading American
influence is taking its toll. One now often sees the phrases “equities and stocks” or even “equities and
bonds”. We shall be using “shares” and “stocks” and “stocks” and “bonds” interchangeably since the
textbook is of American origin. It will be clear from the context what is meant by “stock”.
In the English-speaking world there is further disagreement as to what the word “billion” means. In the
United States it means 109 = 1 000 000 000; in British (and South African) English — and all other main
European languages — it means 1012 = 1 000 000 000 000. We and the British add six zeroes per step up
where the Americans add three. So a British “trillion” is 1018 , whereas an American “trillion” is only 1012 ,
ie the same as a British or South African “billion.” The non-US system is more logical but the general
usage in the financial world tends to be American. It is, however, important to keep this distinction in
mind, especially when referring to non-English language sources. In Afrikaans and French, for example,
a thousand million is a milliard or miljard, so speakers of these languages can play it safe by speaking in
multiples of “milliard”, as one can in English by speaking of a “thousand million”. In languages other than
English, it should, however, still be considered a mistake to use “billion” to refer to anything other than
1012 .
We hope that you will enjoy this module. We are here to help (and also learn from you!), so please do not
hesitate to contact us.
3
2.
PURPOSE AND OUTCOMES OF THIS MODULE
The purpose of the module is to introduce the learner to the derivative investment environment. After
completion of the module the learner should:
• understand the characteristics of derivatives
• understand how futures and options can be used to increase profits and reduce risks
Learner outcome 1
Learners are able to understand, construct and evaluate different derivatives such as forward contracts,
futures contracts and options.
Assessment criteria
• Characterise the different examples of derivatives:
– forward contracts;
– futures contracts;
– options.
• Distinguish between exchange traded and over-the-counter derivatives markets.
• Determine payoff diagrams of forward contracts, futures contracts and options.
• Apply hedging, speculation and arbitrage principles to specific stock information.
• Determine and adjust margin accounts to reflect daily settlements.
• Compare forward contracts with futures contracts.
• Apply different interest rate measures.
• Understand the principle of short selling.
• Determine the forward/futures price of an asset.
• Determine the value of forward/futures contracts.
• Characterise forward and futures foreign currency contracts.
• Apply index arbitrage.
• Apply hedging strategies using futures.
• Calculate basis risk in hedging.
• Determine the optimal hedging ratio.
• Calculate zero rates, par yields and forward rates.
• Calculate the value of a forward rate agreement.
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HONMD1Y/101/0
Learning outcome 2
Learners are able to evaluate and construct bonds, swaps and trading strategies involving options.
Assessment criteria
• Evaluate South African bonds (see Tutorial Letter 101).
• Apply the duration measure.
• Analyse, evaluate and represent interest rate swaps using a diagram.
• Analyse, evaluate and represent currency swaps using a diagram.
• Analyse and evaluate South African bond swaps.
• Understand the terminology and mechanics of option markets.
• Distinguish the factors affecting option prices.
• Apply the no-arbitrage principle to determine upper and lower bounds for options.
• Represent different options strategies and spreads using payoff tables and diagrams.
Learning outcome 3
Learners are able to model the behaviour of share and option prices using binomial trees and the BlackScholes model.
Assessment criteria
• Apply no-arbitrage valuation to determine option prices.
• Apply the risk-neutral valuation approach to value option prices.
• Apply the one- or two-step binomial model to determine option prices.
• Apply the Markov property to stock prices.
• Model stock prices using the Wiener process.
• Understand how Ito’s lemma can be applied to the behaviour of stock prices.
• Apply the lognormal property to stock prices.
• Estimate the volatility of stock prices using historical data.
• Understand the Black-Scholes-Merton differential equation.
• Apply Black-Scholes pricing formulas to determine option prices.
5
3.
LECTURER AND CONTACT DETAILS
Please re-read the information in Tutorial Letter 301 thoroughly!
3.1 Lecturer
You can find the name, telephone and facsimile number for the lecturer of this module in Tutorial
Letter 301. Please write down the name and numbers in the space just below this sentence, so that
the information is always at hand.
3.2 Department
You may contact the Department of Decision Sciences by email at qm@unisa.ac.za. Alternately, you
may call 012.429.4564 or send a fax to 012.429.4898.
3.3 University
Contact addresses of the various administrative departments are included in “my Studies @ Unisa”,
which you received with your tutorial matter. Always have your student number at hand when you
call the University, and always include it in the subject line of your emails. The Unisa postal address
is
PO Box 392
Unisa
0003
To discuss administrative matters, you may contact the student advisor
Miss KM KEKANA
TEL: 012 441-5819
tkekankm@unisa.ac.za
4.
MODULE RELATED RESOURCES
4.1 Prescribed books
The prescribed book for this module is:
Options, Futures and Other Derivatives (Seventh edition) by John Hull. Prentice Hall, New Jersey.
2009.
or
Options, Futures and Other Derivatives (Eighth edition) by John Hull. Prentice Hall, New Jersey.
2011.
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HONMD1Y/101/0
Prescribed books can be obtained from the University’s official booksellers. If you have difficulty in
locating your book(s) at these booksellers, please contact the Prescribed Book Section at Tel: 012 4294152 or e-mail vospresc@unisa.ac.za. Please consult the list of official booksellers and their addresses
in “Your Service Guide @ Unisa”
Note that it is often cheaper and faster to order books online than through bookshops. Browse some
of the online stores to compare prices. Also compare shipping prices, since books from England or
Europe may sometimes work out cheaper once shipping is included.
4.2 Recommended books
There are no recommended books for this module
4.3 Electronic Reserves (e-Reserves)
There are no e-Reserves for this module
5.
STUDENT SUPPORT SERVICES
Please refer to the my studies @ Unisa brochure.
6.
MODULE SPECIFIC STUDY PLAN
Please refer to the my Studies @ Unisa brochure for general time management and planning skills. You
should finish the study material (including the assignments) for this module by the due date for the third
assignment. This will allow you ample time for revision before the exam. Note that simply completing the
assignments by the due dates will not be sufficient to pass the examination. You will need to use the time
after completing the third assignment to study further exercises and examples in the prescribed book.
7.
MODULE PRACTICAL WORK AND WORK INTEGRATED LEARNING
There are no practicals for this module.
8.
ASSESSMENT
8.1 Assessment plan
You are required to submit two compulsory assignments (Assignments 1 & 2) to obtain admission to
the examination. Admission will be obtained by submitting the compulsory assignments
and not by the marks you obtain for it.
Please ensure that these assignments reach the University before the due date. Late submission
of the assignments will result in your not being admitted to the examination. Note that neither
the Department nor the School of Economic Sciences will be able to confirm whether or not the
University has received your assignments. For any queries not strictly concerning the study material
(eg whether your assignment has been received, registered, processed or returned) please contact the
Assignment Section (on 012-429-2959, 011-670-9000 or by e-mail to assign@unisa.ac.za) or consult
https://my.unisa.ac.za/.
7
The submission dates for compulsory Assignments 1 and 2 are 25 June and 8 October,
respectively.
Although only submission of the compulsory assignments is necessary for admission to the exam, note
that the mark you receive for these assignments will count 10% towards your final mark (a maximum
of 5% of your final mark can therefore be contributed by each assignment).
Plagiarism is the act of taking words, ideas and thoughts of others and passing them off as your own.
It is a form of theft which involves a number of dishonest academic activities.
The Disciplinary Code for Students (2004) is given to all students at registration. Students are advised
to study the Code, especially sections 2.1.13 and 2.1.4 (2004:3-4). Kindly read the University’s Policy
on Copyright Infringement and Plagiarism as well.
8.2 Assignment due dates
You are expected to complete three assignments for this module (see Appendix C on page 24). The
due dates are as follows:
9.
Assignment
Work covered
1
2
3
Section 1
Section 2
Section 3
Due date
Unique number
25 June 2012
28 October 2012
7 November 2012
826773
703121
788796
EXAMINATIONS
All registered students who submitted Assignments 1 and 2 will be admitted to the examination.
One three-hour examination will be written for this module during January/February 2013. The examination
for HONMD1Y is a restricted open-book examination where you may refer to your textbook only; notes
and other material will not be allowed. You are also allowed to use a programmable or financial calculator.
To pass you need to obtain a total mark (Assignment 1 and 2 + Exam) of at least 50%. Thus, if you
obtained a total mark of 0% for the assignments, you would need 56% in the exam in order to pass.
10.
OTHER ASSESSMENT METHODS
There are no other assessment methods for this module.
11. FREQUENTLY ASKED QUESTIONS
Please refer to the brochure my Studies @ Unisa for more study information.
12. THE SYLLABUS
12.1 Section 1: Futures, forward contracts and yield curves
1. Chapters 1–6 of Hull (7th or 8th ed)
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HONMD1Y/101/0
12.2 Section 2: Evaluation of bonds, swaps and options
1. Appendix A Evaluation of stocks (bonds) (Section 13 on page 9)
2. Appendix B Swap analysis (Section 14 on page 18)
3. Chapters 7–8 of Hull (7th ed) or Chapters 7 and 9 of Hull (8th ed)
12.3 Section 3: The behaviour of share prices and the Black-Scholes model
1. Chapters 9–13 of Hull (7th ed) or Chapters 10–14 of Hull (8th ed)
2. The appendices to Chapters 12/13 and 13/14 of Hull (7/8th ed) are not included in the syllabus.
9
13. Appendix A — Evaluation of South African stocks
Authors: Prof P Salemink & Prof PH Potgieter
13.1 Introduction
The capitalisation of the world bond1 market was $21,1 trillion at the end of 1994, slightly more than
that of the world stock market.2 In contrast to equities, which are real assets and whose capitalisation
represents real wealth, the size of the bond market does not represent real wealth, apart from that
part of the bond market (around $5,4 trillion) that represents corporate leverage. (Since corporate
bonds are part of the aggregate capitalisation of businesses, they represent real corporate assets over
and above that which is measured by stock market capitalisation.)
Bonds are issued by governments, state-owned firms and large companies for which the risk of default
is considered to be very small. It is not only national governments that may issue bonds, but also
municipalities and various local governments. In the mid-80s, however, a major international banking
crisis was triggered by the imminent default of several Latin American nations on their debt. The
Brady plan allowed these countries — under certain conditions — to buy back their own debt at the
deeply discounted prices of the secondary market.
A short survey of the current state of the world’s bond markets may be found on the Research
section of Merrill Lynch’s website at http://www.ml.com/. It should be noted that many countries with relatively small stock markets may have large bond markets (for example, Italy) or even
relatively small economies and fairly large bond markets (Denmark, for instance). Other useful Internet references for the US and international bond markets are http://www.bonds-online.com/ and
http://www.ibbotson.com/. A site devoted to South African bonds can be found at
http://www.bondexchange.co.za/. Their introduction to bonds is recommended reading.
Bonds are sold and redeemed (by the issuing authority) at their face value — known as par. The
holder receives a fixed rate of interest (say 15%, paid semi-annually) while holding the bond. This
interest rate is sometimes called the “coupon”. Each bond has a fixed redemption date (when the
money will be repaid), interest rate and face value.
In South Africa bonds are issued by government (gilts) and parastatals (semi-gilts), Eskom, Telkom
and Transnet (Transport utility), et cetera. Some bonds amortise over three periods (R153 and R157),
rather than in a single payment but these are nevertheless quoted as if they matured at the middle
of their amortisation schedule. South African institutions have also recently started raising off-shore
rand denominated funds, leading to the establishment of the so-called Eurorand market. (More about
this at http://www.imf.org/external/pubs/ft/icm/97icm/pdf/file17.pdf — you will need the
free Acrobat viewer from http://www.adobe.com/.)
1
Please refer to the note on the confusion of terminology on page 2.
THE $40 TRILLION MARKET: GLOBAL STOCK AND BOND CAPITALIZATIONS AND RETURNS, Laurence B.
Siegel, in Quantitative Investing for the Global Markets: Strategies, Tactics, and Advanced Analytical Techniques, Peter Carman
(ed), Glenlake Publishing Co., Chicago, 1997.
2
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HONMD1Y/101/0
The price of a bond is determined by the current level of interest rates. A bond that pays a fixed
interest rate of 16% becomes more valuable when interest rates fall, because institutional buyers cannot
get such a high interest rate in the market. They are thus prepared to pay more than the face value
for that bond. In the bond market, a round lot is R1 million worth of bond (stock) at face value,
which puts this market way outside the range of most private investors, except by way of the newly
introduced (in South Africa) money market funds. Options on bond market futures contracts are,
however, heavily traded on the South African Futures Exchange.
In summary, a stock (bond) is characterised by the following features:
1. A redemption value (face value/nominal value).
2. A maturity date on which the redemption is payable.
3. A coupon rate expressed as a percentage (in terms of the redemption value).
4. A schedule of coupon payments.
Unlike most other countries, stocks (bonds) in South Africa have their price quoted in terms of their
yield to maturity. The price paid for the stock is the sum of the discounted cash flows, discounted
at the quoted yield, on the settlement date. In the United States, for example, the price of Treasury
bonds is quoted as the equivalent cash price for a face value of $100, expressed in integer dollars and
32nds of a dollar. (Here also, reform is being discussed — in the United Kingdom quotes went decimal
some time ago, by way of comparison.)
13.2 Pricing of stock
In South Africa the settlement date for a bond transaction, since October 1997, has been three trading
days after the trade (‘T+3’). In the past, trades were settled the second Thursday after trade date.
Therefore, the actual number of days between trade date and settlement date varied from 14 (for a
trade executed on a Thursday) to 8 (for a trade executed on a Wednesday). When a holiday occurred
on a Friday, the settlement date for that week used to move from Thursday to Wednesday. This was an
anachronism in our markets, removed by the reform of 1997. Settlement conventions in other markets
are different (http://www.jpmorgan.com is a good source).
In South Africa there is one more peculiarity: stocks used to go ex interest one calendar month before
coupon payment is due. This has however been changed and the period is now 10 days for
most bonds. In this course, we will assume the 10-day rule to hold for all South African bonds.
The coupon payment, if the settlement occurs in the ex interest period, remains with the seller (ie the
buyer does not receive the next coupon). In some markets such as Germany, the trade date rather
than settlement date determines custody of the coupon. If the stock is trading outside the ex interest
period, it is said to be trading cum interest. In some foreign markets no ex interest (or, ex dividend )
period exists.
We need the following formulas to determine the price of stocks:
An annuity is a sequence of equal payments at equal intervals of time. An annuity is termed an
ordinary annuity when payments are made at the same time that interest is credited, that is, at the
end of the payment intervals. If the payments in rands, made at each interval in respect of an ordinary
11
annuity at interest rate i per payment interval, are denoted by R, then the amount or future value of
the annuity after n intervals is
µ
¶
(1 + i)n − 1
FV = R
i
The present value of the annuity is
µ
PV = R
(1 + i)n − 1
i(1 + i)n
¶
The method to determine the price of South African stocks will be explained in the following examples.
Example 1 In this example the price of a stock (or bond) is determined on an interest date. The
following is the relevant data for the stock:
Nominal value
Coupon rate (half-yearly)
Interest dates
Maturity date
Yield to maturity
Settlement date
R1 000 000
9,250% per annum
30 June and 31 December
31 December 2019
15,9% per annum
30 June 2003
By convention the coupon payment on 30 June 2003 belongs to the seller. The coupon payments will
be paid out every six months from 31/12/2003 to 31/12/2019. The coupon payments to be received
at n = 33 half years are
1
× 9,25% × 1 000 000 = R46 250
2
This stream of coupon payments constitutes an ordinary annuity. The present value of this stream
must be evaluated using the yield to maturity of 15,9% per annum or 15,9
2 per half year. The present
value of the coupon payment is
µ
P Vc = 46 250
(1 + 0,0795)33 − 1
0,0795(1 + 0,0795)33
¶
= 535 159,79
The present value of the nominal value is
P Vf = 1 000 000(1 + 0,0795)−33 = 80 103,72
The value of the stock on 30 June is
P = P Vc + P Vf = 615 263,51
Thus the price of the stock is R615 263,51.
Unless otherwise stated, it may be assumed that coupons are paid half yearly at the maturity date
(here 30 June) and six months later (here 31 December).
The R% convention is used to represent nominal values in units of R100 and to express the price as
the number of rands per R100 unit. We could write the price of the stock as R61,52635%. Note that
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HONMD1Y/101/0
the convention requires five decimal places. Then to obtain the actual price of this stock we multiply
the R% price by 10 000.
The MS-Excel function PRICE may be used to determine the price if the settlement date occurs on a
coupon date. You should use the following input if your system is set up to use decimal points.
settlement
maturity
rate
yld
redemption
frequency
basis
“2003/06/30”
“2019/12/31”
0.09250
0.159
100
2
3
The result should be 61,52635036. Multiply this value by 10 000 to determine the price. This function
may only be used to calculate South African stocks if the settlement date occurs on a coupon date.
In South Africa a stock is sold cum interest if the settlement date is 10 days or more from the next
coupon date. That means the coupon must be added to the price of the stock. A stock is sold ex
interest if the settlement date is strictly less than ten days from the next coupon date. Obviously
whether it is sold “ex” or “cum” affects the price of the stock. The following examples illustrate the
relevant principles.
Example 2 The relevant data for stock BBB is:
Nominal value
Coupon (half yearly)
Interest dates
Maturity date
Settlement date
Yield to maturity
100%
11% per annum
1 June and 1 December
1 June 2025
31 March 2007
15,61% per annum
Notice that we use the R% convention. This is a cum interest case. Verify that the value of the stock
on the next coupon date 1 June 2007 is
P (1/6/2007) = 72,44142%
This is the value on 1 June 2007, which excludes the coupon due on that date. Since 31 March is more
than 10 days before 1 June, the coupon payment of 5,5% should be included in the determination of
the all-in price. We add the coupon value to the previous value:
72,44142 + 5,5 = 77,94142%
13
This value should be discounted to 31 March 2007. The number of days from 31/3/2007 to 1/6/2007
is R = 62. The number of days in the half year from 1/12/2006 to 1/6/2007 is H = 182. This value,
often referred to as the all-in price, is
µ
¶
0,1561 −62/182
P = 77,94142 1 +
= 75,97130%
2
In the case of cum interest a coupon has been added to the present value of the stock. Since the
settlement date falls between two coupon dates, a part of the coupon added does not belong there.
We determine this part, referred to as accrued interest, by using the the formula H−R
365 × c, with c the
coupon rate per annum. Hence the accrued interest is
accrued interest =
182 − 62
× 11 = 3,61644%
365
Next we can find the clean price the price of the stock:
clean price = all-in price − accrued interest = 75,97130 − 3,61644 = 72,35486%
The price the buyer pays for the stock BBB on 31 March 2007 is the all-in price R75,97130% or
R759 713,00 (ignoring transaction fees). The clean price is used for accounting purposes. From the
viewpoint of buyers and sellers of stock, the important parameter is the all-in price.
Example 3 If the settlement date in the previous example is 25 May 2007, it will be an ex interest
case. The present value of the stock on 1 June 2007 is again 72,44142%. The coupon should not be
added. The number of days between 25 May 2007 and 1 June 2007 is R = 7. The number of days
in the half year from 1/12/2006 to 1/6/2007 is H = 182. We discounted the present value back to
determine the all-in-price. The all-in price is
µ
¶
0,1561 −7/182
P = 72,44142 1 +
= 72,23233%
2
Since it is an ex interest case we calculated the accrued interest using the formula −R
365 × c, with c the
yearly coupon rate. Thus,
accrued interest =
−7
× 11 = −0,21096%
365
Hence the clean price is
clean price = 72,23233 − (−0,21096) = 72,44329%
Example 4 Consider the following RSA124 stock, bought on Tuesday 19 May 1998. The following is
the relevant data for the stock.
Nominal value:
Semi-annual coupon
Maturity date:
R1 000 000
13% per annum
15/7/2005
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HONMD1Y/101/0
The ruling yield, at purchase, is 13,9%. What is the settlement price?
Settlement is scheduled for Friday 22 May 1998 according to the T+3 rule. We shall start by calculating
the price (fair value) at the next interest/coupon date, ie on 15/7/1998. This is a simple annuity
calculation with n = 7 × 2 payments (at the end of each six month period — therefore excluding
the coupon of 15/7/1998). The coupon payment is p = 13%×12000 000 = 65 000 and the future value is
F V = 1 000 000. The interest rate per period used is 13,9
2 %.
This yields a present value P V = 960 527,12 which implies a fair price, on 15/7/1998, for the benefit
of receiving the future cash flows of R960 527,12. However, the buyer will also receive the coupon on
that date, so the price on 15 July 1998 should be 960527,12 + 65 000 = R1 025 527,12.
This amount must now be discounted back to the settlement date of 22 May 1998. The remaining
days from 22/5 to 15/7 is 54 days. The fraction of a half year to be discounted back using the US
Street method 3 and an actual/actual day count is then
n=
54
= 0,29834
181
since the number of days in the half-year between 15/1/1998 and 15/7/1998 is 181. (Note that five
decimal places were used — this is the convention for settlement pricing.) The all-in price is
¶
µ
0,139 −0,29834
1 025 527,12 × 1 +
≈ 1 005 174,25 rand.
2
The accrued interest, the pro-rata part of the next coupon with respect to the time elapsed since the
last coupon, can be calculated as follows:
accrued interest =
=
days since last coupon
× 130 000
365
127
× 130 000
365
= 45 232,88.
The clean price is then
clean price = all-in price − accrued interest
= R959 941,37.
Summary of basic bond pricing
1. Calculate price on next interest/coupon date.
2. If cum interest, add coupon payment.
3. Calculate the fraction of the half-year from settlement to date of next coupon.
3
The standard yield to maturity calculation used in the United States by market participants other than the US Treasury.
Yield is compounded semi-annually. If the value date does not fall on a coupon date, the present value of the bond on the next
coupon date is discounted over the fractional period with compound interest.
15
4. Discount back to settlement date, giving the all-in price. This is the price also known
as the total consideration on settlement date.
5. Deduct accrued interest to get the clean price.
NOTE If you encounter difficulties with the pricing of SA bonds you may ask the lecturer to email/send
you notes from the first year course Introductory Financial Mathematics.
13.3 Yield to maturity
In South Africa, stocks are quoted as a yield-to-maturity, and from this the cash price is calculated.
By definition we have:
The yield-to-maturity (YTM) is the interest rate at which the present value of the future
cash flows of the stock is equal to the market price.
From this it is clear that there is an inverse relationship between the YTM and the settlement price
— the higher the YTM, the lower the settlement price and vice versa.
Consider the following three stocks with different coupon rates, all with 10 years to maturity and all
trading at 15% YTM.
Nominal value
Coupon (semi-annual)
YTM
All-in price
A
R1 000 000
13%
15%
R898 055,09
B
R1 000 000
15%
15%
R1 000 000
C
R1 000 000
17%
15%
R1 101 944,91
We call A a discount stock, B a par stock and C a premium stock. It is clear that if the YTM equals
the coupon rate, the stock will be a par stock. If the YTM is larger (respectively, less than) then the
stock will be a discount (respectively, premium) stock.
An enlightening view on these three stocks is exposed by a breakdown of the attributable income
expressed as a percentage of the nominal value (negative values represent outflows).
Income
A
B
C
Coupon
Interest on coupon
Price discount
Interest on discount
Nominal value
130
151,48
10,19
33,11
100
150
174,79
0
0
100
170
198,09
-10,19
-33,11
100
Total future value
424,79
424,79
424,79
Under the assumption that all interest is earned at the YTM, all stocks perform the same. However,
we can make the following interesting income comparison:
16
Total interest
A
184,59
B
174,79
HONMD1Y/101/0
C
164,98
From this it is clear that C would be the least sensitive to interest rate fluctuations, whereas A is the
most sensitive.
13.4 Running yield
The running yield of a stock is simply the coupon received from the stock, expressed as a percentage
of the current clean price. For the RSA124 stock purchased on 19 May 1998 at a YTM of 13,9% we
had a clean price of R959 941,37 and an annual coupon payment of R130 000. The running yield is
then
running yield =
annual coupon
× 100
price
130 000
× 100
959 941,37
≈ 13,54%.
=
This is roughly similar to the dividend yield of a share, although the two should really not be compared
at all.
13.5 Realised compound yield
With the YTM it is assumed that all cash flows received from the stock (ie the coupons) are re-invested
at the YTM rate. This may not be a realistic assumption and, in order to overcome this assumption,
the concept of realised compound yield (RCY) was introduced. The RCY is simply the modified rate
of return on your investment in the stocks.
The realised compound yield is the return obtained on a stock if it is assumed that all
future cash flows are re-invested at a specified rate, r.
Consider, once more, the RSA124 of the first example, bought for settlement on Friday 22 May 1998
with a YTM of 13,9%. Let us assume a semi-annually compounded re-investment rate of 10%. The
all-in price was R1 005 174,25.
The future value (FV) of future coupon payments, with a re-investment rate of r = 10%, is given by
an annuity calculation using n = 7 × 2 = 14 periods, a coupon payment of p = 13%×12000 000 = 65 000
and an interest rate per period of 10
2 %, and present value P V = 65 000. (The stock is bought cum
interest.) The future value obtained in this way4 is F V = 1 402 606,63 which can be added to the
face value of the stock to give a total future value of all inflows of R2 402 606,63. The RCY is now
the semi-annually compounded rate i that will grow the all-in price to R2 402 606,63 over 14 half-year
periods. We solve the equation
µ
¶
i 14
1 005 174,25 1 +
= 2 402 606,63
2
4
In MS-Excel, use -FV(0,05;14;65000;65000;0) or - FV(0.05,14,65000,65 000,0) if your system uses the point as decimal
separator.
17
to give
õ
i=2
2 402 606,63
1 005 174,25
!
¶1
14
−1
≈ 0,1284 ≈ 12,84%.
Note that
(i) the RCY is expressed as a nominal per annum rate, compounded semi-annually;
(ii) as the re-investment rate drops, so will the RCY;
(iii) the term is actually somewhat more than the 14 half-year periods. Round the term to the nearest
integer value.
13.6 The effective par rate
The effective par rate (EPR) is that coupon of par stock which, under the assumed investment rate,
will give the same RCY as the given stock. To obtain the effective par rate, we proceed as follows:
1. Calculate the RCY.
2. Calculate the future value of R1 million with this RCY.
3. Subtract R1 million (the final payment) from this future value.
4. Calculate the semi-annual payments necessary to achieve this value if the interest is at the assumed
re-investment rate.
5. Take the computed payment, express as a percentage of R1 million and multiply by 2 to give the
EPR.
18
HONMD1Y/101/0
14. Appendix B — Swap analysis
A swap (or switch) of stocks is the selling of one stock with the simultaneous buying of another stock. The
purpose of a swap is typically to increase or reduce the sensitivity of a stock position with respect to an
expected market change.
When analysing swaps, one should consider a number of factors, including
• the stock’s future value;
• realised compound yield;
• tax liability;
• cash flow and
• statutory requirements.
The technicalities of a swap may require a
• nominal for nominal ;
• cash for cash;
• or clean for clean swap.
14.1 Future values in swap considerations
Let us compare the value of two different stocks at a common point in time, assuming an anticipated
re-investment rate. Note that we are sometimes evaluating older stocks, which still use the 30-day
convention to determine whether it is cum interest. However, all recent stocks and bonds (in the
assignments or exam) should use the 10-day convention. For stocks older than 2004, either may be
used, as long as it is specified.
Example 5 Consider the following:
A. Sell R1 million of 14,5% 15/10/2006 RSA stock for settlement on 25/10/1990 at YTM of 15,3%.
B. Use the proceeds to buy 10,5% 15/04/2000 RSA stock for settlement on 25/10/1990 at YTM of
15,4%.
Should the swap be made for the following assumptions on the investment rate (semi-annually compounded)?
(i) 12%
(ii) 14%
(iii) 16%
(i) First we calculate the all-in price (as a percentage of nominal value) for the stocks in question,
and get
19
A. R95,652164%
B. R76,264844%
where the handy practice of expressing prices in R%, or percentage of face value, has been used.
The nominal amount of stock B that can be purchased with R1 million nominal value of A is
95,652164
× 1 000 000 ≈ R1,254210m nominal.
76,264844
We now compare the future values of each of these scenarios. The future value has two components,
– the coupon stream; and
– the nominal value of the stock.
For the coupon stream of A we have n = 16 × 2 payments of p = 14,5%×12 000 000 and the present
value is P V = 0. The interest rate per period used is 12
2 %. This gives a future value of F V =
6 589 508,91 to which must be added the R1 million nominal, to give a total future value (on
15/10/2006) of R7 589 508,91.
For stock B we first calculate the future value on the maturity date of the stock, 15/4/2000. The
1
interest rate per period used is 12
2 %. For the coupon stream there are n = 9 2 × 2 payments of
p = 10,5%×12 000 000 and the present value is P V = 0. This gives a future value of F V = 1 772 399,56
to which must be added the R1 million nominal, to give a total future value (on 15/4/2000) of
R2 772 399,56. The future value on 15/10/2006 of this amount, using a re-investment rate of 12%
is
µ
¶
0,12 6,5×2
2 772 399,56 1 +
≈ R5 913 329,38.
2
However, we need the value of R1 254 210 nominal of stock B, which is
R7 416 556,84.
In this case it is therefore clearly not desirable to swap A for B.
(ii) Going through the same calculation for a re-investment rate of 14% gives future values, on
15/10/2006, of
A. R8 990 816; and
B. R8 953 698.
Again it would (only just) not be worth swapping A for B.
(iii) Going through the same calculation once more for a re-investment rate of 16% gives future values,
on 15/10/2006, of
A. R10 730 481; and
B. R10 833 027.
This time it is indeed profitable to swap A for B.
20
HONMD1Y/101/0
14.2 Realised compound yield considerations
Basis: Compare the RCY of the two different stocks at a common point in time, assuming an anticipated re-investment rate.
Example 6 For the swap under consideration in Example 5 we shall calculate the RCY in each case.
The present value (on 25/10/1990) is the same (R956 521,64) for the corresponding nominal amount
of each stock and each re-investment rate scenario.
We find the RCY for R1 million nominal of stock A by computing the semi-annual rate i that grows
R956 521,64 over 16 years to R7 589 508,91,
µ
¶
i 16×2
956 521,64 1 +
= 7 589 508,91
2
which can be solved to find
i ≈ 0,1337 = 13,37%.
In the same way we can compute the other RCYs. In the following table the RCYs for the two stocks
using the three different re-investment rates are listed.
A
B
12%
14%
16%
13,37%
13,22%
14,51%
14,48%
15,70%
15,76%
Comparing the RCYs, we reach the same conclusion as in Example 5. Why is this not very surprising?
14.3 Other considerations
Tax considerations Tax considerations are of importance in swap operations. This depends on the
tax status of the investor, of course. If the investor is taxed on both capital and interest it may
be in the investor’s interest to sell just before an ex-interest date.
Cash flow To generate cash flow sooner, a stock with a coupon payment five months hence may be
sold in order to purchase a stock with a coupon payment one month away.
Statutory requirements To comply with prescribed asset requirements, for example, one might
swap into a discount stock in order to “create” investments by amortising a bond to par.
14.4 Stock values used in swaps
The value used in a swap can differ, depending on the basis for the swap:
– nominal for nominal ;
– cash for cash; or
– clean for clean.
Consider a swap of the following two stocks:
21
Selling stock (A)
Buying stock (B)
15,00%
16,05%
26/02/1990
15/09/2007
R93,91719%
R-0,69863%
(17 days ex)
R93,21856%
14,25%
16,16%
26/02/1990
01/11/2008
R88,80381%
R4,56781%
(117 days cum)
R93,37162%
Coupon (semi-annual)
YTM
Settlement date
Maturity date
Clean price
Accrued interest
All-in price
Nominal for nominal swap
Example 7 Suppose an investor wants to sell R50 million nominal of stock A and purchase the
same nominal amount of stock B. What are the consequences for the investor, ie will he/she have
to pay in, or not?
For R1 million nominal of each stock,
proceeds from sale of A
= R932 185,60
cost of purchase of B
= R933 716,20
additional amount to be paid =
R1 530,60.
Therefore, each million in face value of the nominal for nominal swap will cost the investor
R1 530,60. To swap R50 million from A to B, the investor will have to put up an additional
50 × 1 530,60 = R76 530,00.
Example 8 Suppose now the investor were to contemplate this swap (at the same rates) with
a settlement date of 12/04/1990, when stock A is cum interest and stock B is ex interest. What
would the consequences then be?
The all-in prices are now
All-in price
Selling stock A
Buying stock B
R95,00629%
R88,12580%
and in this case the investor will receive an additional amount of R68 804,94 for each R1 million
nominal of the swap. In total, therefore, he/she will receive R3 440 247.
Why this huge difference from the swap less than two months earlier? The answer is that the
investor has forfeited the coupon payment to the value of
0,1425
× R50m = R3,5625m
2
which he/she would have received on 01/05/1990.
Evidently, whether a swap will suit the investor, and on what date, will depend on his cash flow
and tax considerations. (Whether the swap is actually profitable is another question, as discussed
in Section 8.)
Timing is essential when evaluating a swap!
22
HONMD1Y/101/0
Cash for cash swap
Example 9 Suppose the settlement date is 12/04/1990 and all proceeds of stock A will be used
to purchase stock B. How much nominal of stock B will be purchased?
Since each R1m nominal of stock A is worth R950 062,94 and each R1m nominal of stock B costs
only R-881 258,00 the sale of R50m of stock A will purchase
950 062,94
× R50m ≈ R53,903791m
881 258,00
of stock B.
In order to avoid working with “odd lot” quantities of stock, a cash for cash swap is usually
rounded off to the nearest million. Thus, in this case the investor would purchase R54m of stock
B, leaving him/her with a slightly impaired cash position.
NB Once again, timing can be critical. Had the swap been done with a settlement date of
26/02/1990, the investor would have had to pay in a small amount (about R80 000) to obtain
exactly R50m nominal of stock B.
Clean for clean swap
Example 10 Consider the swap mentioned in the last example, on the settlement date of
26/02/1990, by re-investing the proceeds in terms of the clean price of stock A fully in terms
of the clean price of stock B.
For each R1m of stock A, a clean price of R939 171,90 is obtained. However, R1m nominal of
stock B costs, in terms of clean price, R888 038,12. In other words, in a clean for clean swap the
sale of R1m of stock A leads to the purchase of
939 171,90
= R1,0575806m
888 038,12
nominal of stock B.
In order to avoid odd lots, the investor purchases R53m nominal of stock B. This costs
R53m × 0,9337162 ≈ R49,486959m
and the sale of A yields
R50m × 0,9321856 ≈ R46,609280m.
The investor therefore has to make an additional capital investment of R2,8776786m for which
he/she obtains an additional R3m nominal of stock B.
14.5 Categories of swaps
Stocks that are swapped may differ in terms of
– coupon rate;
– time to maturity; and
– market segment (RSA, Eskom, Transnet etc).
23
Broadly speaking, there are three kinds of swaps:
– substitution swaps;
– differential swaps; and
– strategic swaps.
Substitution swaps Also known as a yield pick-up swap. One swaps to stock that is a close
substitute but gives greater RCY (or future value). See Example 6 (in the case of a re-investment
rate of 16%). Take care, though, that stock B must be equally tradeable in the secondary market,
since unmarketable stocks always trade at relatively higher yields.
Differential swaps Also known as an anomaly swap. It can occur either on an intracategory basis,
or intercategory basis.
Intracategory differential swap Say the differential between the YTM of two long-dated RSA
stocks of similar maturity widened from the usual 10 to 30 basis points. Then, swap from the
“expensive” stock into the “cheap” stock (if positioned correctly). When the yield differential
returns to normal, swap back.
Intercategory differential swap Similar to an intracategory differential swap, but involves
stock from two different categories (RSA to Eskom, say).
Strategic swaps This depends on the investor’s long-term view on interest rates. The investor will
try to position him/herself so as to take advantage of anticipated changes in interest rates.
Suppose the investor feels rates will rise. Then it will be advisable to move into stocks with
either a higher coupon or shorter maturity or both. However, the advantage could be negated by
unfavourable adjustments of the yields of shorter-dated stocks relative to longer-dated stocks.
Usually strategic swaps are done on a nominal for nominal basis.
24
HONMD1Y/101/0
15. Appendix C — Assignments
Assignment 1
The exercises refer to those in Hull (7th or 8th ed). Please take note of the discrepancies between the
editions, as indicated below, and answer the questions appropriate to your edition.
1. Problem 2.11
[3]
2. Problem 2.16
[5]
3. Problem 2.17
[5]
4. Problem 3.18
[2]
5. Problem 3.23 (3.26 in 8th ed)
[6]
6. Problem 4.4
[3]
7. Problem 4.22
[3]
8. Problem 4.28 (4.33 in 8th ed)
[8]
9. Problem 5.9
[8]
10. Problem 5.10
[2]
11. Problem 5.15 (If using the 7th ed, assume the spot price of silver is $15 per ounce.)
[2]
12. Problem 6.7
[8]
13. Problem 6.25 (6.28 in 8th ed)
[4]
25
Assignment 2
Question 1
Consider a stock with nominal value R1000 000, maturity date 1 June 2020 and two coupon payment dates
of 1 June and 1 December with a coupon rate of 12,55% per annum. The stock goes ex-dividend from
21 May and 20 November. The re-investment rate is 13,6%. The treasurer of a large company decides to
purchase the stock at a yield of 18,5% on 20 May 2011 (with settlement on 23 May 2011).
1. What is the total consideration (all-in price)?
2. Calculate
(a) the clean price
(b) the realised compound yield
(c) the effective par rate.
3. The yield decreases to 14% by 16 November 2012. The treasurer decides to sell the stock on that date,
with settlement on 19 November.
(a) What is the price that he receives?
(b) What was the effective yield on his investment for the period that he held the stock, expressed
as a continuously compounded rate?
Question 2
A bank enters on the first of January 2012 into a 10-year currency swap with an export company X. The
principal amounts are $9 000 000 and R63 000 000. Under the terms of the swap the bank pays interest at
6% per annum in US dollars, and receives interest at 13% in SA Rand, every year on 31 December. Suppose
that company X defaults during the year 2014 when the exchange rate is R8,90 per dollar. (No payment
is made at the end of 2014 and thereafter.) Assume that in 2014 the interest rate is 6% per annum in
US dollars and 10% per annum in SA Rand for all maturities. All interest rates are quoted with annual
compounding.
1. Use the exchange and interest rates prevailing during 2014 to calculate one-year forward exchange
rates for years 2015–2018.
2. What is the cost of the default to the bank (calculated at the end of 2014)?
26
HONMD1Y/101/0
Question 3
Problem 7.2
Question 4
Consider the following transaction:
A. Sell R1 million of 7,5% 15/10/2020 RSA stock for settlement on 25/01/2012 at YTM of 6,3%.
B. Use the proceeds to buy 5,75% 15/04/2021 RSA stock for settlement on 25/01/2012 at YTM of 6,4%.
Should the swap be made for the following assumptions on the investment rate (semi-annually compounded)?
(i) 7,5%
(ii) 8,5%
Question 5
Problem 7.3
Question 6
Problem 7.9
Question 7
Problem 8.17 (7th ed) or 9.17 (8th ed)
Question 8
Problem 9.3 (7th ed) or 10.3 (8th ed)
Question 9
Problem 9.12 (7th ed) or 10.12 (8th ed)
Question 10
Consider a non-dividend paying share whose current price is R150. The risk-free interest rate is 15% per
annum (continuously compounded). Find a lower bound for the price of a six-month
(a) call option with strike price R90; and
(b) put option with strike price R105.
Is there a theoretical upper bound to the option prices?
Question 11
Consider Example 4, Appendix A. Suppose now that settlement was on 22/7/1998. What would the all-in
price be in this case?
Question 12
Problem 10.19 (7th ed) or 11.20 (8th ed)
Question 13
Problem 10.20 (7th ed) or 11.21 (8th ed)
27
Assignment 3
Question 1
How can a forward contract on a share with a certain delivery price and delivery date be created from
options?
Question 2
Consider a call and put on the same underlying asset. The call has an exercise price of R100 and costs R20.
The put has an exercise price of R90 and costs R12.
(a) Graph a short position in a strangle based on these two options.
(b) What is the worst outcome from selling the strangle?
(c) At what price of the asset does the strangle have a zero profit?
Question 3
Suppose that put options on a stock with strike price $30 and $35 cost $4 and $7, respectively. How can
the options be used to create
(a) a bull spread; and
(b) a bear spread?
Question 4
Use put-call parity to show that the cost of a butterfly spread created from European puts is identical to
the cost of a butterfly spread created from European calls.
Question 5
A covered call is constructed using a particular futures contract and options on the futures contract. Each
futures contract controls 5000 units of the spot asset. Each option contract controls one futures contract.
The call option has exercise price of R150 per unit, and is sold at R6,75 per unit. If the futures price when
the covered call is initiated is R155 per unit and the futures price at the expiration of the option is R152,
find the total profit per contract at expiry.
Question 6
(a) A share price is currently R120. Over each of the following three-month periods it is expected to go up
by 10% or down by 10%. The risk-free interest rate is 16,5% per annum with continuous compounding.
Calculate the value of a six-month European put option with a strike price of R126.
(b) An American put option on a non-dividend-paying asset with strike price $45 matures in one year.
Divide the one-year interval into two six-month intervals. The continuously compounded risk-free rate
of interest is 4,5% and the volatility is 20% per annum.
(i) Determine the up and down factors for a two-period binomial tree.
28
HONMD1Y/101/0
(ii) Determine the risk-neutral probability of an up movement of the price during one period.
(iii) If the current asset price is $35, determine the value of the option using the risk-neutral probability.
(iv) At the left-most (initial) node in the binomial tree, describe the replicating portfolio based on an
investment in the asset and the risk-free security.
Question 7
A company’s cash position, measured in millions of rand, follows a generalised Wiener process with a drift
rate of 1,5 per quarter and a variance of 4,0 per quarter. How high does the company’s initial cash position
have to be for the company to have a less than 5% chance of a negative cash position by the end of one
year?
Question 8
Shares A and B both follow geometric Brownian motion. Changes in any short interval of time are uncorrelated with each other. Does the value of a portfolio consisting of one share A and one share B follow
geometric Brownian motion? Explain your answer.
Question 9
Suppose that a share has an expected return of 18% per year and a volatility of 30% per year. The price at
the close of trading today was R60. Calculate the following:
(a) The expected share price at the close of trading tomorrow.
(b) The standard deviation of the share price at the close of trading tomorrow. Explain what the number
means to a non-technical friend.
(c) The 95% confidence limits for the share at the end of trading tomorrow.
Question 10
The following table shows the daily closing prices and daily returns for a non-dividend-paying share for
consecutive trading days.
³
´
i
i
Day (i) Price (Si ) SSi−1
ln SSi−1
1
2
3
4
5
6
7
8
9
10
R30,00
R31,00
R31,50
R30,00
R32,00
R34,00
R32,00
R32,50
R32,50
R31,25
1,033
1,016
0,952
1,066
1,062
0,941
1,015
1,000
0,962
0,0328
0,0159
-0,0492
0,0645
0,0606
-0,0606
0,0155
0
-0,0392
29
(a) Estimate the current annualised volatility of the share price, assuming a 252 day trading year.
(b) State at least two assumptions regarding the stochastic process followed by the share price that you
have used to compute the volatility above.
Question 11
What is the price of an European put option on a non-dividend-paying share if the share price is £23, the
strike price is £24, the risk-free interest rate is 5% per annum (continuously compounded), the volatility is
35% per annum, and the time to maturity is six months?
Question 12
Consider an American call option on a share. The share price is $70, the time to maturity is eight months,
the risk-free interest rate is 10% per annum, the exercise price is $65, and the volatility is 32%. Dividends
of $1 are expected after three months and six months. Show that it can never be optimal to exercise the
option on either of the two dividend dates. Calculate the price of the option.
Question 13
Consider an option on a share that is expected to go ex-dividend in 2 months. The expected dividend is
R1. What is the price of an European call option if the share price is R30, the exercise price is R29, the
risk-free interest rate is 15% (continuously compounded), the volatility is 25% per annum, and the time to
maturity is 4 months?
Question 14
If S folllows the geometric Brownian motion process given by the equation
dS = a(x,t)dt + b(x,t)dz
(as in equation 13.11), what is the process followed by
a. y = 2S?
b. y = S 2 ?
c. y = eS ?
d. y = er(T −t) /S?