The document discusses various financial instruments and valuation methodologies. It provides background on Nobel prize winners such as Scholes, Merton, and Black who developed the Black-Scholes options pricing model. It then discusses various derivatives like options, forwards, futures, and swaps. It explains how these instruments can be used for hedging, speculation, and gaining exposure to markets. It also discusses how concepts like volatility and yield curves are important for pricing derivatives.

1. FINANCIAL INSTRUMENTS
valuation methodologies
Dr A. A. Kotzé
Financial Chaos Theory
April 2007
Saggitarius A*: supermassive black
hole at the Milky Way’s center

2. Before I came here I was confused
about the subject. Having listened to
your lecture I am still confused. But on
a higher level.
Enrico Fermi (1901-1954)
Niels Bohr
and
Albert Einstein

3. What’s in the 1997 Nobel prize?
Myron Scholes (1941 - )
Robert Merton (1944 - )
Fischer Black (1938 - 1995)
Along the way, it changed the way investors
and others place a value on risk, giving rise to
the field of risk management, the increased
marketing of derivatives, and widespread
changes in the valuation of corporate
liabilities.
The theory "is absolutely
crucial to the valuation of
anything from a company
to property rights“. In my
view, financial
economics deals with
four main phenomena:
time, uncertainty, options and
information.
William F. Sharp

4. Drivers of the markets
• Only two emotions drive the market – GREED
and FEAR
• Two types of traders – SPECULATORS and
HEDGERS

5. DERIVATIVES: why?
Derivatives expanded the
universe of instruments
available for trading and
hedging
• Speculators use derivatives to
expose portfolios to some
market risk
• Hedgers use derivatives to
reduce the market risk they
are exposed to

6. Benefits of derivatives
• part of universe of instruments – bigger choice
• basic instruments – used to construct others
• participation in price moves – gearing or leverage
• used as hedges
• can buy or sell underlying at more favourable prices
• can earn premium income
• derivatives give options
• tailor your position to your situation or risk tolerance
• position yourself to your market expectation
• limited influence on underlying market

7. Hedging
• Transfer risk
• Risks more precisely tailored to risk preferences and
tolerances
• Those willing to bear risk must be compensated
• Most transactions still for speculating and not hedging
• Speculators provide liquidity

8. Rapid growth
• Rapid growth and development
• Emergence of international financial institutions who
intermediate bulk of international capital flows in the
financial markets
• Increase capacity of the financial system to bear and
price risk and allocated capital
•

9. Historical Developments

10. The Financial Market
• Financial markets are mechanisms that link surplus and
deficit investors
• Participants: borrowers, lenders, financial intermediaries
and brokers
• Financial Market: interaction between the players in
setting a price on financial instruments fulfilling demands
and requirements

11. Time value of money
• The notion that money has a time value is one of
the basic concepts in the analysis of any
financial instrument
• Money has time value because of the
opportunities for investing money at some
interest rate
• Interest is a fee for having the use of money

12. Zero coupon bond
• A zero-coupon-bond is a bond that makes no
interest payment over its lifetime
• The face value is the amount paid at maturity
• Most money market instrument are zero-coupon
• Value
Cash invested
Cash + interest

13. Coupon bond
These are bonds with regular interest payments
e.g. R153, R157
The value of the bond is given by its net present
value (NPV)

14. Pricing SA bonds
Use Makeham’s formula: basically the NPV with a
few twists to make it practical

15. Pricing SA bonds: the bond pricing formula

16. The bond pricing formula
Settlement on coupon date; more than 6 months
Settlement on coupon: less than 6 months
Ex-coupon: more than 6 months
Ex-coupon: less than 6 months
Cum-interest: more than 6 months
Cum-interest: less than 6 months

17. Bond Trading
• Debt probably oldest financial product stemming from
origins of mankind
• Governments have been issuing public debt since 16th,
17th century?
• First majour default: British Crown (Edward III) in 1345
• Bond trading also quite old – 1800’s
• Derivatives on interest rate instruments are quite new

18. The zero coupon yield curve
• There is no single interest rate for an economy
• Factors: term of loan; credit
• Government best credit – can borrow at best
rates
• A yield curve is a graphical representation of the
term structure of interest rates for instruments of
a similar credit rating

19. Yield curve is dynamic
South African Yield Curves
5.500%
6.500%
7.500%
8.500%
9.500%
10.500%
11.500%
12.500%
13.500%
14.500%
0.01
0.08
0.25
0.75
1.25
1.75
3.00
5.00
7.00
9.00
11.00
13.00
15.00
17.00
19.00
21.00
23.01
25.00
27.00
29.00
Years
YTM
06-Jan-03
30-Jun-03
31-Mar-04
30-Sep-04
05/05/2005
17/02/2006
28/02/2005
02-Apr-07

20. The forward rate
• Can invest money today for a certain date
• Question: if we receive money in the future, can
we invest it at that point in time but fix the
interest rate today?
• Yes, by using forward rates
• Forward rate determined by arbitrage argument

21. The forward rate: arbitrage

22. The Interest Rate Swap
• A swap is an agreement between two parties to
exchange cash flows in the future according to a
prearranged formula
• There is a fixed rate payer and a floating rate
payer
• Only the net interest differential is exchanged
First swap done in 1981 between the World Bank and IBM
Size of market: more than $150 trillion notional outstanding

23. The swap: mechanics

24. Swaps: an example
• Two companies agree the following:
• company B lends company A R100 million at the 3 month Jibar rate;
• Company A lends company B R100 million at a fixed rate of 9.5%

25. Value of a swap
Value of a swap using yield curve
• Fixed leg: NPV
• Floating leg:

26. Forwards/Futures
A person holding a futures contract has the right to
buy a certain underlying asset at a future date at
a pre-determined price
Payoff

27. What is an option?
• Call option: gives the holder the right, BUT NOT
the obligation to buy the underlying asset on a
future date at a pre-determined price
• Put option: gives the holder the right, BUT NOT
the obligation to sell the underlying asset on a
future date at a pre-determined price
Payoff

28. The Black-Scholes formula
If you buy an option, you buy a RIGHT without any
OBLIGATION – when the option expires, you can
decide to exercise or not. This costs money – the
premium of the option

29. The Black-Scholes world
• Stock prices follow a continuous random walk – Brownian motion
• The efficient market hypothesis holds
• Investors live in a risk-neutral world
• Delta-hedging is done continuously
In general, Black & Scholes assumed that
the financial market is a system that is in
equilibrium – without outside or exogenous
influences, the system is at rest; everything
balances out and supply equals demand.
Any distortion or perturbation is thus
quickly handled by the market players and
equilibrium restored

30. Misconception
A common misconception about option pricing models is that
their typical use by option traders is to indicate the “right”
price of an option.
Options pricing models do not necessarily give the “right”
price of an option. The “right” price is what someone is
willing to pay for a particular option.
An efficient market will give the best and truest prices for
options. Many individuals trade warrants and some of
them do not know what an option is nor do they have a
pricing model.

31. Benefits of having a model
• the concepts behind the formula provided the framework
for thinking about option valuation and dynamics;
• Due to research within the Black-Scholes framework many
new insights into the market dynamics have come to light
• such research led to an expanded universe of financial
instruments encompassing more complex option
structures;
• quantitative risk management, stress testing and scenario
analysis became possible;
• hedging of derivatives were easier;
• The market has progressed and expanded.

32. Types of options
• European – can only exercise on expiry date;
• American – can exercise any time on or before expiry
• Exotics;
– Barrier option: option either disappears or is created when certain
market levels are breached
– Asian: use averages to determine the final payoff value
– Exchange one asset for another option
– Cliquet
– Ladder
– Reset
– Lookback
New types of options are created nearly every day

33. Value of an option: risk management
• The concept of delta-hedging:
If we purchase an option we can trade in the cash instrument (called
``trading spot" or ``trading the cash") to hedge the option. By buying
the option we pay the premium upfront. As the underlying's share
price changes with time, the Delta will show us that we have to buy
the cash at the lows and sell at the highs - we thus make money by
delta hedging. In theory, if we hedge continuously we should make
exactly what we paid for the option.
On the other hand, if we sell the option, we earn the premium. The
Delta will then show us to buy the cash at the highs and sell it at the
lows - we thus loose money by delta hedging. By doing this
continuously we should loose exactly what we earned with the
premium.
Hedging costs = value of derivative

34. Valuation: American options
• No closed-form formula
• Use numerical procedures like binomial trees
• The discrete version of normal distribution is the binomial
distribution
• The binomial model models the time to expiration as a very large
number of time intervals, or steps. Using probability theory a
tree of stock prices is produced.
• At each step it is assumed that the stock price will move up or
down by an amount calculated using probabilities. This produces
a binomial distribution, or recombining tree, of underlying stock
prices. The tree represents all the possible paths that the stock
price could take during the life of the option

35. The binomial distribution

36. Factors affecting option prices
First 3 are known. Last 3 needs to be estimated
• The spot price
• The strike price
• The time to expiry
• The risk-free interest rate
• The dividends/cash flows expected during the life of
the option
• The Volatility during the life of the option

37. Volatility
• The most important of the three uncertain parameters, is
the volatility.
• The importance of the volatility parameter was highlighted
by Black & Scholes through their model. Practitioners now
needed to estimate only one parameter, the volatility, and
input it into a relative simple formula to find the price of an
option. Of the three uncertain parameters, changing
volatility has the biggest impact on the price of an option.
• Volatility measures variability, or dispersion about a central
tendency — it is simply a measure of the degree of price
movement in a stock, futures contract or any other market.
• Standard Deviation

38. Volatility is not constant

39. New listed option
• Safex listed a lookback option
• Investors know it is near impossible to get the timing
right. The floating strike lookback option comes to the
rescue.
• This path-dependent option is described as the portfolio
manager’s “best friend”. This option will always let the
investor have the best possible strike price thus
maximising his payoff!
• Very expensive!
• To cheapen a lookback, one can limit the time in which
the strike is set. One can even discretise the lookback
feature to only include certain pre-determined dates.

40. New listed option
Price
• Continuous lookback: Black-Scholes type closed-
form equation
• Discrete lookback: value using numerical procedures
like Monte Carlo simulation

41. Monte Carlo simulation
• This is a difference equation
• Through simulation of this
equation we can obtain share
prices at expiry and thus
option prices by their payoff
functions












  ttdSS ttt 


2
exp
2
1
• If we assume stock prices follow Brownian motion we
can describe their behaviour with

42. Derivative Strategies
• If a fund manager wants to hedge a portfolio against
market moves, is buying a put the only option?
• NO
• Options are very versatile and, used in combination
with the underlying instruments give powerful
strategies that can help you to hedge a portfolio, get
synthetic exposure to the market or gear a position.

43. Payoff profiles
• A payoff profile shows the payoff that would be received if
the underlying is at its current level when the option
expires
• It highlights the risks associated with the strategy in a
simple diagram: a future has unlimited profit potential, but
such a diagram also shows the potential losses
• It is easy to work with payoff profiles - they are additive
meaning that we can add or subtract them from one
another ---
• useful in constructing more complex financial instruments
or strategies

44. Derivatives as simple diagrams
a future has unlimited profit potential, but such a diagram also
shows the potential losses
P/L
Short
Future
K K’
Long
Future

45. Payoff profiles: options
K
Long
Put
Payoff
K
Short
Put
Payoff
K
Short
Call
K
Long
Call

46. The put strategy
• Buying a put is also a strategy
• In general investors buy a put as a hedge
when they are long the underlying stock
• A bearish strategy

47. Put-call-parity
SK
Long
put
Long
stock

48. Valuation of optionality in transactions
• Most BEE deals have inherent optionality
• SBSA: partial American option. Black shareholders
obtain a put option on SBK to protect them against any
downside. The European put turns into an American put
on 1 January 2015. The strike is R40.50
put + long shares = synthetic call
• Absa: variable strike partial American option where the
American date is 2 July 2007 and the strike is
  






00.10000.69
00.10000.70,0maxint70.000.48
RSifR
RSifRSRR
K
A
AA
Binomial Model

49. The zero cost collar strategy
• Fund manager has a portfolio of shares
he/she needs to hedge
• Fund manager buys an ATM put from risk
taker – this costs money (premium)
• Risk taker buys an OTM call from the fund
manager
• Strike of OTM call is determined by ensuring
that the call premium is equal to the premium
of the ATM put
Value = Put - Call

50. Advantages
• Hedge down side market moves with zero
upfront premium
• Participates in upward moves
• Tailor-made in terms of strike and expiry e.g.
important for unit trust’s quarterly reports
• Generally OTC and not Safex
• Mostly done on the Alsi Top 40 index
• Also suited to single stocks, Indi, Fini, Resi
or a tailor- made index

51. The zero cost collar: graphically
S
Long
stock
K
Short
call
Floor
Ceiling
Long
put

52. Employee stock options
• Companies issue these to employees
• A particular structure:
– Options are granted on certain dates.
– 20% of the granted options vests 2 years after the grant date
and 20% vests every year thereafter for the next 4 years.
– No rights can be taken up until the vesting date of a
particular tranche.
– Employees have until 1 year after the last vesting date to take
up the vested option rights. Vested options can be taken up
on any date from the vesting date.
– The option is a delayed delivery option.
– Option strikes are set on the grant date.

53. Employee stock options
Experts believe Black-Scholes is not appropriate:
• There is usually a vesting period - options cannot be exercised.
This vesting period can be as long as four to five years.
• Options are very long dated – up to 10 years.
• When employees leave their jobs during the vesting period they
forfeit unvested options.
• When employees leave after the vesting period they forfeit
options that are out of the money and they have to exercise
vested options that are in the money immediately.
• Employees are not permitted to sell their employee stock options
in the open market – this means there is no efficient discovery of
prices. They must exercise the options and sell the underlying
shares in order to realize a cash benefit or diversify their
portfolios. This tends to lead to employee stock options being
exercised earlier than similar regular options.

54. Employee stock options
• A lattice structure, such as the binomial or
trinomial model, incorporates assumptions about
employee exercise behaviour over the life of each
option grant. This results in more-accurate option
values and compensation expense.
• This is set out in IFRS 2

55. An equity linked note: guarantees
• The fund manager now has cash and not a
portfolio of shares
• An interest rate play
• Invested funds are guaranteed at expiry +
there is potential upside if the share market
performs well

56. Example
• Investor invests R100 million with risk taker for 1
year
• Risk taker guarantees R100 million after 1 year –
credit risk
• Risk taker invests R90,497,737.56 in money
market – 10.5% NACA
• This leaves R9,502,262.44 to buy exposure to
market

57. Equity linked note structure
Buy ATM
call
Cost = R13,040,447.90
Need R3,538,185.46 to
make zero-cost
Sell OTM
call
Value = Call1 – Call2 + money market

58. Contact
Dr Antonie Kotzé
Email: consultant@quantonline.co.za
Phone: 082 924-7162
Disclaimer
This article is published for general information and is not intended
as advice of any nature. The viewpoints expressed are not
necessarily that of Financial Chaos Theory Pty Ltd. As every
situation depends on its own facts and circumstances, only specific
advice should be relied upon.