Syllabus Trading Financial Derivatives

1
Syllabus Trading Financial Derivatives
Course Fin 3600AW/6600AW Trading Derivatives -
How to make money and not lose it with Futures,
Swaps and Options
ProfessorDr. Gunter Meissner Web
www.dersoft.com E-mail gmeissne_at_aol.com Office
FHT 5th floor, office 1, Tel [phone]
Description The course will cover the theory and
application of Futures, Swaps and Options. It
will analyze the valuation and risk of
derivatives as well as focus on the practical
application of derivatives in debt and portfolio
management.
Goals To obtain a complete overview of the
existing derivatives To understand the
valuation and risk calculation of
derivatives. To develop the ability to use
derivatives as a risk and cost reduction tool
2
Syllabus Trading Financial Derivatives, cont.
Goals To compare derivative and standard
instruments and to choose the better financial
alternative. To be able to relate derivatives to
one another. To be able to trade derivatives.
Literature 1) Slides on www.dersoft.com/hpu
2) Gunter Meissner Trading Financial
Derivatives - Futures, Swaps and
Options in Theory and Application
Voluntary 3) Gunter Meissner Credit
Derivatives - Application, Pricing and Risk
Management 4) Gunter Meissner
Outperform the Dow - Using Options, Futures and
Portfolio Strategies to Beat
the Market 5) RISK Magazine Financial
Engineering News 6) John Hull Options,
Futures and other Derivative Securities
Cool Websites www.cboe.com, www.eurexchange.com,
www.cme.com, www.cbot.com,
www.nymex.com, www.marketwatch.com,
www.yahoo.com
3
Syllabus Trading Financial Derivatives, cont.
Grading Participation/Homework 10 Black-Sch
oles Project 30 Presentation of the
Model 10 Mid term 25 Final 25
Point System
95.00 lt A lt 100
90.00 lt A- lt 95.00
86.66 lt B lt 90.00
83.33 lt B lt 86.66
80.00 lt B- lt 83.33
76.66 lt C lt 80.00
73.33 lt C lt 76.66
70.00 lt C- lt 73.33
65.00 lt D lt 70.00
60.00 lt D lt 65.00
F lt 60.00
Black-Scholes Project The students will, in
groups of 3, program the Black-Scholes
pricing model, including risk-parameters for
different markets.
4
Topics for the financial paper
1) Trading in the 90s Fundamental analysis
versus chart technology (chapter 2) 2) Interest
rate futures (chapter 4) 3) Swaps, usage and
pricing (chapter 5) 4) Building an option
pricing model based on Black-Scholes
including hedge parameters (chapter 7,8) 5)
Building the binomial option pricing model
including hedge parameters 6) Building a model to
price an exotic options (parts of chapter 11)
including hedge parameters 7) Monto Carlo
Simulation - Structure and Application 8)
Disasters in Derivatives (parts of chapter 2)
Proctor Gamble, Gibson Greetings Nick
Leeson, Metallgesellschaft, Sumitomo, Daiwa,
State of being today, Lessons to learn 9) VAR
The Value at risk concept, Features and
Limits 10) Term structure based models Types,
How do they work? 11) Credit Derivatives Types
and Usage 12) Weather Derivatives Will They be
popular?
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Jobs in the Derivatives Area www.pacifica.co.jp
numa.com financewise.com topmoneyjobs.com
www.gloriamundi.com www.garp.com
Of the SP 500 companies, how many use
Derivatives???

• Trader

• Marketer

• Risk Management (fast growing area) Fin3801/6801

• Product Developer (well paid)

• Structured Products

• Derivatives Controller

• Programmer

• Settlement

• Broker (Private Investor, Interbank)

6
Trading according to Seasons
Since 1950, 86.97 of the Dow gain occurred in
the month from
November to April !!!
?
Sell in May and go away
(Data since 1968)
7
What is a derivative in finance ?
A financial derivative (Call on Google) is a
security, whose value is at least in part derived
from the value of an underlying security (Price
of Google)
Derivatives, also called contingent claims, can
be divided in three main categories
(Book p. 3
8
Derivatives Industries
Derivatives Investment Banking
Hedge Funds
Risk Management
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Myths about Derivatives
Warren Buffet Derivatives are weapons of mass
destruction
Some CEO and CFOs We dont use Derivatives
because they are too dangerous
Alan Greenspan Derivatives are a useful tools
to reduce risk
Truth about Derivatives
Derivatives are powerful instruments
In the hands of maniacs, they can do a lot of
damage as Nick Leeson with Barings Bank 1996
Loss 1.24 billion Robert Citron with Orange
County in 1997 Loss 2 billion Nobel Prize
Laureates with LTCM in 1998 Loss 4 billion
Derivatives are very useful instruments to
Reduce Risk in an Economy and a Company and for
an Investor!!! more uses of
Derivatives
10
Why are derivatives so popular?
? Speculation
? Hedging
? Cost reduction
? Arbitrage
Formerly Off-balance sheet feature but not
true anymore Financial Accounting Standard
(FAS) statement 133 requires since 1999 that
all derivatives are included in the balance
sheet at their fair market value
(Book p. 5 - 8)
11
Types of Derivatives in different Markets
12
Options
What is an option?
A Call option gives the buyer (holder) the right
to buy the underlying asset at a predetermined
price (strike), at a predetermined date (European
style) or during a predetermined period
(American style).
A Put option gives the buyer (holder) the right
to sell the underlying asset at a predetermined
price (strike), at a predetermined date (European
style) or during a predetermined period
(American style).
(Book p. 164f)
13
Main Features of Options
Strike K Price at which the option buyer can
buy (in case of a Call) or sell (in case of a
Put) the underlying asset.
Premium C for a Call, and P for a Put
Option price, paid from the option buyer to
the option seller
Maturity date T Last date to exercise the option
(if American style, only
date to exercise the option (if European style)
Exercising Buying the underlying asset (in case
of a call), selling the
underlying asset (in case of a put)
Trading Buying and selling options at the market
price. (Trading options is usually
better than exercising pre-maturely)
Right Only the option has a
right. The right to buy or sell the
underlying asset. The option seller has the
to fulfill the right of
the option buyer.
(Book p. 164 - 169)
14
Cash Flows of long Call
t
Tm
Option period
Cash Flows of long Put
t
Tm
Option period
15
Options on Commodities Futures Contract
Specifications
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Options on Commodities Futures Contract
Specifications cont.
17
Options on Financial Futures Contract
Specifications
Interest Rates
Eurodollar
CME
720 - 200 (5AM OR 2PM)
1,000,000 ED
ALL
1/2 PT 12.50
Every 25 PTS
LIBOR
CME
720 - 200 (5AM OR 2PM)
3,000,000
ALL
1/2 PT 12.50
Every 25 PTS
Municipal Bond
CBOT
720 - 200
1000 X INDEX
HMUZ
1/64 PT 15.625
Every 1 BASIS PT
Treasury Bill
CME
720 - 200 (1000)
1,000,000
ALL
1 PT 25.00
Every 25 PTS
Treasury Bond
CBOT
720 - 200 (1200)
100,000
ALL
1/64 PT 15.625
Every 1 BASIS PT
520PM - 805PM CST
620PM - 905PM CDT
1000PM - 600AM PROJECT A
Treasury Note
CBOT
720 - 200 (1200)
100,000
ALL
1/64 PT 15.625
Every 1 BASIS PT
520PM - 805PM CST
620PM - 905PM CDT
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Options on Financial Futures Contract
Specifications cont.
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Why are Options so popular?
Speculation The option holder participates in
the leverage effect
Leverage means Little money as input, high
potential profits as output. An example
Price of call on Yahoo 5 7 9 10 100
Price of Yahoo 100 105 111 113 13
Day 1 Day2 Day 3 Day 4 change
(Book p.169,170)
20
Why are Options so popular? continued
Hedging E.g Covered put buying Buy a put
toeliminate the downside risk
Arbitrage E.g put-call parity C K e-rt P
S C Call price K Strike e-rt Discount
factor P Put price S Price of underlying
Details later (Book p. 172 - 174)
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The Hockey sticks Payoff at option maturity
Buying the stock at 100
(Short) Selling the stock at 100
Profit
Profit
100
Spot price of stock
Spot price of stock
100
100
100
Loss
Loss
((Book p.180))
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More Hockey sticks payoff at maturity
Long (long) call with strike 100, premium 10
Short (sold) call with strike 100, premium 10
Profit
Profit
Spot price of stock
10
Spot price of stock
100
110
110
100
10
Loss
Loss
(Book p.176,177)
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More Hockey sticks payoff at maturity
Long (bought) put with strike 100, premium 10
Short (sold) put with strike 100, premium 10
Profit
Profit
Spot price of stock
Spot price of stock
10
90
90
10
100
100
Loss
Loss
(Book p.177,179)
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Option strategies
Covered call writing to enhance profits Stock
purchase at 100 Short Call, Strike 100,
premium 10
Profit
covered call writing
Spot price of stock
short call
long stock
Loss
See www.dersoft.com/consecutive.doc
(Book p.179,180)
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Option strategies, continued
Covered put buying to hedge a stock Stock
purchase at 100 Long Put, Strike 100,
Premium 10
Profit
covered put buying
Spot price of stock
long put
long stock
(Book p.180)
Loss
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Option strategies, continued
The Straddle Long Call, Strike 100, Premium 10
Long Put, Strike 100, Premium 10
Profit
straddle
Spot price of stock
long call
long put
(Book p.181)
Loss
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Option strategies, continued
The Strangle
Profit
strangle
Spot price of stock
long call
long put
(Book p.181,182)
Loss
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Option strategies, continued
Vertical bull spread
Profit
vertical bull spread
Spot price of stock
short call, strike 120
long call, strike 80
(Book p.182,183)
Loss
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Option strategies, continued
Vertical bear spread
Profit
long call, strike 120
short call, strike 80
Spot price of stock
vertical bear spread
Loss
(Book p.183,184)
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Put-call parity C K e-rt P S cont.
Portfolio A at t0 C Ke-rT
Portfolio B at t0 P S
At option maturity T (T t0), if S gt K, we have
for portfolio A and B, expressed via S and K
Portfolio AT
Portfolio BT Call value CT is Put
value PT is Ke-rT value becomes
Stock value is Sum
Sum
At option maturity T, if K gt S, we have
Portfolio AT
Portfolio BT Call Value CT is Put
value PT is Ke-rT value becomes
Stock value is Sum
Sum
Therefor, for any ST ,
Therefor, if the put-call parity equation is
violated, arbitrage exists!!
(Book p. 172 - 174)
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Put-call parity Arbitrage
To do an Arbitrage you
Example A call and a put on Msft with a 1-year
maturity both trade at 3. The strike is 50, the
price of Msft is 45 and the 1-year risk-free
interest rate is 3. Is arbitrage possible?
C Ke-rT
P S
Now what???
Our Arbitrage at t0 is
At T we have no obligation for any value of S.
For example, if S 60, it follows that
the value of our portfolio is P S - C
K
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When to buy call options on stocks, when to buy
stocks
Call Premium 10 for 1 call on IBM, strike 100
Money invested in call is 10 money invested in
stock in 100!
50.00
120
10
100
20
20
1
130
20
200
66.67
30
30
1
140
30
300
75.00
40
40
1
Result
(Book p. 174,175)
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The time value of a call
Value of
the call C
C
Time value
Intrinsic value
max (S K, 0)
Asset
price S
K
Result
(Book p.204 - 207)
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The time value of a put
Result
(Book p. 204 -207)
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Do you have a good intuition for options???
Lets sum up, before we discuss the pricing of
options
YES!!!
Are options risky???
Why are they so popular??
What is leverage?? Is it always good??
When should you buy options, when the underlying??
When should you buy a straddle??
Should you buy out of the money options??
What does the time value tell us??
(joke)
(Book p. 1- 397)
36
Volatility
Volatility is a measure of the fluctuation-intensi
ty of a security
Volatility of asset a is than
volatility of asset b
(Book p. 43)
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Volatility cont.
Types of volatility
There are basically 3 types of volatility
implied
volatility
actual volatility
historical volatility
t1
t-1
t0
t0
(Book p. 44)
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Volatility cont.
1) Historical volatility is the volatility
measured from a point in time in the past until
today.
2) Implied volatility is the volatility that a
trader assumes to occur from today until the
maturity of the option. It is an estimation of
the actual volatility. Implied volatility is the
figure that is input in the option pricing
formula (Black-Scholes formula) to value an
option.
3) Actual volatility is the volatility of the
security that will actually occur in the future.
It is an ex ante (an expected) figure. If an
option trader believes that implied volatility is
smaller than actual volatility he should buy
options, vice versa.
(Book p. 44)
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Calculation of Volatility
The standard deviation of a price S is
The Volatility is the standard deviation of
logarithmic price differences
(Book p. 45)
40
Calculation of Volatility
where n number of prices i points in time ln
natural logarithm Si price at time i
(Book p. 45, 46)
41
Calculation of Standard deviation
Let's assume a stock has the prices Si on day 1
to day 5
1 2 3 4 5
Thus, the standard deviation for the 5-day stock
price movement is
This means
(Book p. 45, 46)
42
Calculation of Volatility
Let's assume the same stock has the prices Si on
day 1 to day 5
Taking 0.0026 and multiplying it with 1/(n-1) (n
4) gives.
Taking the square root of 0.0009 results in a
volatility of
(Book p. 47)
43
Shortcut to the Calculating of Volatility
1) Calculate all
2) Use EXCELs Stdev function on all
The result will be the same as the procedure used
in the previous slides
Homework Calculate the 30 day volatility of your
favorite stock
(New book p. 160 to 163)
44
Interpretation of Volatility
The 5-day volatility of the stock price is 3.00.
That means that there is 68.27 probability (the
68.27 represents 2 standard deviations of a
normal distribution) that the stock price for the
next day will be within
Furthermore there is a 95.45 probability (95.45
represents two standard deviations of the normal
distribution) that the stock price for the next
day will be within
(Book p. 47)
45
Annualization of Volatility
To annualize the volatility, the calculated
volatility has to be multiplied with the square
root of the observation frequency.
In the above example, we have daily data. There
are roughly 252 trading days in a year. Thus, the
annualized volatility is
So there is a 68.27 probability that the price
of the stock within the next year will be
(Book p. 47)
46
Features of the Volatility Concept
? The fact, that the financial markets use
volatility and not the standard deviation to
determine and compare the fluctuation intensity
of securities, is sensible
Stock prices moving 1, 2, 3 and 1001, 1002, 1003
lead to the same standard deviation of 1 ,
although the relative price fluctuation of 1, 2,
3 is much higher than a stock price movement of
1001, 1002, 1003.
The volatility concept reflects this relative
difference in the fluctuation. The volatility of
a bond moving 1, 2, 3 is 20.34, the volatility
of a stock moving 1001, 1002, 1003 is close to
zero.
(Book p. 48)
47
Features of the Volatility Concept
? The volatility concept is a trend concept
If the price of a security increases with the
same constant rate e.g. 10 100, 110, 121,
133.1, 146.41 and so on, it follows, that the
volatility is
The volatility is for any constant rate of
growth of a security.
The reason for the zero volatility is the fact,
that in an arbitrage free market, all securities
have to grow with the risk-free interest rate.
Arbitrage occurs, if an asset grows by more or
less than the risk-free interest rate.
An investor would buy the security with the
higher return and sell the security with the
lower return.
(Book p. 48)
48
The Volatility Smile
In trading practice, traders use a higher implied
volatility for out of the money puts and,
depending on the market, also for out of the
money calls
Why???
49
Typical Volatility Grimace in the Stock Market
50
Typical Volatility Smile in the Currency Market
51
Typical Volatility Smile in the Commodity Market
52
Pricing options
There are principally three ways to price options

• on a binomial model

• with the Black-Scholes formula

• on a term-structure based model

(Book chapter 7, p. 185 - 207)
53
The CRR model
The Cox, Ross, Rubinstein model is the simplest
binomial model. It is the discrete Black-Scholes
model.
In the CRR model the option price is derived in
two steps
a) Building the (stock) price tree b) Deriving
the option price on the (stock) price tree
(Book p.185 -190)
54
a) Building the (stock) price tree
The (stock) price tree is built starting with the
spot price S and assuming S can go two (bi)
ways, up and down
uS
110
S
100
90
dS
p (e(r ?t) - d) / (u - d)
e Eulers number 2.718283...
implied volatility
length of time step option maturity in years
/ number of time steps
(Book p. 186 -190)
55
A two step binomial (stock) price tree
(Book p. 188)
56
An example of two step stock price tree
Lets assume, the price of the security today is
100, the annual implied volatility is 10, the
annual risk-free interest rate is 5, and the
length of each time step is 1/12. What are the
values of the stock price in a two-step, binomial
tree?
u d 1/u
The probabilities p, (which are not
necessary for deriving the values of S), are p
Using these factors and rounding to four
decimals we get uS , dS ,
u2S d2S and udS .
Thus
t
t
t
o
1
2
100
(Book p. 189)
57
b) Deriving the option price on the (stock) price
tree
To determine the call price C as of today, we
have to roll backwards through the binomial
tree, starting at the last node t2, then derive
the option price at t1 and finally at t0
(Book p. 189)
58
b) Deriving the option price on the (stock) price
tree, cont.
At the last node of the tree, we know that the
option value is the intrinsic value. For a call,
the intrinsic value is the spot price S minus the
strike K or 0, whatever is bigger. Formally,
C max (S - K, 0)
The call prices for the two step binomial tree at
the last node, that is in two months (t2) are
therefore
u2C max (u2S - K, 0), udC max (udS - K, 0)
and d2C max (d2S - K, 0)
(Book p. 189, 190)
59
b) Deriving the option price on the (stock) price
tree, cont.
Next, we calculate the option prices at t1. We
discount the probability-weighted call prices
from t2 to t1
For the node uC and dC, the call prices are uC
u2C p udC (1-p) e-r?t and dC d2C
(1-p) udC p e-r?t
Finally, for the call price C today, we take the
results of uC and dC and discount them with the
relevant probabilities to today
C uC p dC (1-p) e-r?t
(Book p.190)
60
b) Deriving the option price on the (stock) price
tree, an example
Lets assume the security price S is 100 today,
the annual volatility of the security is 10,
and the annual risk-free interest rate is 5 and
each time step is 1/12. What is the price of a
2-months call with a strike of 98 on the basis of
the two-step binomial model ?
a) Firstly, we have to calculate the intrinsic
value at the last node t2. Using the stock price
tree created earlier, we get
u2C
max (u2S - K, 0)

udC
max (udS - K, 0)

d2C
max (d2S - K, 0)

(Book p.188,189)
61
b) Deriving the option price on the (stock) price
tree, an example, continued
To derive the call values at time t1, we discount
the probability weighted prices at t2 back to t1
uC
u2C p udC (1-p) e-r?t


dC
d2C (1-p) udC p e-r?t


(Book p.190)
62
b) Deriving the option price on the (stock) price
tree, an example, continued
Finally we can derive the call price C as of
today, by discounting the probability weighted
prices uC and dC back to today
C
uC p dC (1-p) e-r?t

Result The 2-month call with a strike of 98 has
a price of .
For comparison reasons, the price derived on a
ten step binomial tree is 3.40, a 20 step
binomial tree results in 3.39. The Black-Scholes
price, being equivalent to an indefinite time
step binomial model is also 3.39.
(Book p.190)
63
Early Exercise of American Style Options
An American Style Call or Put should be exercised
before option maturity, if the intrinsic value
(also called dead value) is higher than the value
of the option (also called live value). Formally
For a call
(9.13) If IVC max(S - K, 0) gt C ? Exercise
the Call now
The value of a call can be divided into intrinsic
value IV and time value TV
(7.37) C IVC TVC
The time value of a call can be at least
represented by the fact that the call holder can
by the stock later at K, formally
TVC K - K e-rt gt0
Conclusion Since IVC ? 0 and TVC gt0 ? from
equation (7.37) that
Therefore, Equation (9.13)
(Book p.242,243)
64
Early Exercise of American Style Options
It follows, that
In the absence of returns (dividends, coupons),
it is never rational to exercise an American
Style call pre-maturely !
However, credit risk (default risk) weakens that
conclusion! as in a Vulnerable Option, where
the option seller can default
(Book p.242,243)
65
Early Exercise of American Style Options
If you exercise early, you get
If you sell the American Style Call, you get
(Book p.242,243)
66
Early Exercise of American Style Options
For a put, the early exercise condition is
(9.17) If IVP max(K - S, 0) gt P ? Exercise
the Put now
The value of a put can be divided into intrinsic
value IV and time value TV
(7.38) P IVP TVP
The time value of a put TVP can be negative!
TVP TVMP OVP K e-rt - K OVP
TVMP Time value of money When you exercise
early, you can invest the money from the sell
earlier at the interest rate r TVMP lt 0
OVP Opportunity value of a put. It reflects the
fact that buying a put has an advantage over
shorting the asset, because of the leverage
effect and the fact that the maximum loss of the
put is the premium OVP gt 0
Conclusion If Abs(TVMP)gt Abs(OVP )? TVP
Therefore, if TVPlt 0 ? from equation (7.38)
(Book p.245 - 248)
67
Early Exercise of American Style Options
Value of the put P
b
c
P
a
Asset price S
K
S
Interpretation
If S gt S ?
IVP lt P ?
If S lt S ?
IVP gt P ?
(Book p.245 - 248)
68
Pricing options with the Black-Scholes formula
The Black-Scholes formula was founded 1973 and is
until today the dominating formula for valuing
European style options.
The assumptions of Black and Scholes are
? the underlying security is log-normally
distributed !!! ? the risk-free interest rate and
the volatility are constant until option
maturity ? the market participants are
risk-neutral ? short selling is possible ? there
is market transparency (every market participant
has all market information)
(Book p.194)
69
Normal versus log-normal distribution
Result
(Book p.197)
70
Normal versus log-normal distribution, cont.
Result
(Book p.197)
71
Normal versus log-normal distribution, cont.
The log-normal distribution reflects the path a
stock price takes over time!
((Book p.197))
72
Deriving the Black-Scholes formula for a call
The Black-Scholes formula can be derived by
firstly looking at the expected value at
expiration T. For a call,
where C call price, European style E
expectation value ST price of underlying
security at option maturity T K strike price
(Book p.194)
73
Deriving the Black-Scholes formula for a call,
cont.
Discounting the expectation value back to today
gives
where r annual risk free interest rate
Expressing the expectation value with an integral
gives
(Book p.194)
74
Deriving the Black-Scholes formula for a call,
cont.
From -? to K the value of the call at maturity is
zero
Separating the integrals gives
After transforming the integrals into cumulative
standard normal distributions, the original
Black-Scholes formula follows
(Book p.194)
75
Finally, the Black-Scholes formula for a call
where N cumulative normal distribution. N(d) is
the surface of the standardized normal
distribution (see table 12.1), and
C f (S, K, r, T, ?)
?
?
?
?
?
(Book p.195,199)
76
The Black-Scholes formula for a put
where N cumulative normal distribution. N(d) is
the surface of the standardized normal
distribution (see table 12.1), and
P f (S, K, r, T, ?)
?
?
?
?
?
(Book p.195,196)
77
Advantages and disadvantages of Black-Scholes
The reasons for the popularity of the
Black-Scholes formula are
? the high intuition of the formula. It is fairly
easy to understand and easy to program and apply
in practice
? modifications by Merton 1973, Black 1976 and
Garman/Kohlhagen 1983 allow to value options with
a known return, such as a known dividend yield of
a stock or a coupon of a bond as well as currency
options (compare chapter 9).
? modifications by Rubinstein/Reiner 1991,
Rubinstein 1992, Reiner 1992, and others allow
the valuation of exotics options such as barriers
options, compound options or quanto options and
others (compare chapter 11).
(Book p.202)
78
Advantages and disadvantages of Black-Scholes,
cont
The disadvantages of the Black-Scholes approach
lies in the fact, that
? the valuation of path dependent options (i.e.
American style options or lookback options) is
not possible with Black-Scholes
? the valuation of certain interest rate options
is problematic, because interest rate revert to
their mean. I.e a bond at maturity will have a
value of par, usually 100. Therefore the
distribution function of a bond looks as follows,
sometimes also referred to as the fish-effect.
(Book p.202,203)
79
The RISK of options
Stocks, bonds and futures (and most other basic
instruments) principally have only one form of
risk, the price risk If the price of a stock,
bond or future decreases and the investor is long
(has bought) the asset, he loses money
The risk of stocks, bonds and futures are largely
linear If the stock or bond, or the underlying
drops by one dollar, you lose one dollar.
Options though behave non-linear with respect to
the underlying instrument (see the leverage
effect), therefor the risk of options is
non-linear!
(Book p.208)
80
The Delta
The delta, ?, is an important risk-parameter of
an option.
The delta of an option, ?, measures how much the
option price changes, if the price of the
underlying instrument changes. The delta is
calculated assuming the other variables
influencing the option price implied volatility,
time, strike, and interest rate are constant
(ceteris paribus). Therefore the delta is the
first partial derivative of the option price with
respect to the price of the underlying asset.
(Book p.209)
81
The Delta of a call
For a call the delta is
delta of a call partial derivative
coefficient C call price S price of the
underlying asset
The delta expresses
(Book p.209)
82
The Delta of a call, cont.
Graphically the delta is the slope of the
call-function at a certain point
(Book p.210)
83
The Delta of a put
For a put the delta is
delta of a put partial derivative
coefficient P put price S price of the
underlying asset
The delta expresses
(Book p.213,214)
84
The Delta of a put, cont.
Graphically the delta is the slope of the
put-function at a certain point
(Book p.213,214)
85
The Delta determining the hedge amount
Example Dynamic delta hedging
Lets assume, on day 1 a trader buys an at the
money call on a non-dividend paying stock. The
call has a delta of 0.51 and an option premium of
10,000. The stock trades at 100. The investor
decides to delta hedge the price risk of the
option
He stocks for a value of

During the next 5 days, the stock moves as
follows
(Book p.215)
86
The Delta determining the hedge amount, cont.
It is important to note, that the sum of the
hedging activities is positive (162). The payoff
of hedging a long option position is always
positive (for standard options), because a trader
buys on declines and sells on increases of the
stock in order to stay .
(Book p.215,216)
87
The delta neutral position No price risk?
Bank A sells a call on 100 shares of IBM Call
price per share 5, delta 0.5, IBM price
100 Bank A receives 5 100 500
To delta hedge the price risk Bank A buys
shares.
If the price of IBM increases from 100 to 101,
Bank A loses, because the call price increases by
cents.
5.5 100 550. .
But Bank A makes money on the hedge 50 shares
101 5,050. .
If the price of IBM decreases from 100 to 98,
Bank A makes money, because the call decreases
from 5 to
Thus, Bank A makes on the call
.
Bank A loses money on the hedge.
88
The delta neutral position No price risk? cont.
Result A delta neutral position
price risk.
Unfortunately, the delta changes, when the stock
price changes (as seen in the graph earlier).
Thus, the price risk of a delta hedging investor
is eliminated, only for very small changes of the
underlying asset. If the underlying changes by a
large amount, the delta hedge becomes imprecise.
The changes in the delta, which occurs when the
underlying asset changes, is called
(Book p.215,216)
89
The Gamma
The gamma of an option, ?, measures how much the
delta changes, if the price of the underlying
instrument changes.
As the delta, the gamma is calculated assuming
the other variables influencing the option price,
implied volatility, time, strike, and interest
rate are constant (ceteris paribus).
Therefore, mathematically the gamma is the second
partial derivative of the option price with
respect to the price of the underlying asset.
Therefore the gamma measures
the of the option function.
The gamma can be viewed as the level expressing
the necessary delta hedge adjustment or the
inaccuracy of the delta hedge.
(Book p.216)
90
Deriving the delta and gamma of a call graphically
C
(Book p.206)
S
1
(Book p.212)
S
(Book p.217)
S
91
The Gamma, a free lunch?
A long call C and short underlying position g as
a hedge
Result
(Book p.220,221)
92
The Gamma, a free lunch?, cont.
If the gamma of a long call (and long put) is
always positive, why doesnt every trader buy
options and hedges them to achieve a risk-less
profit???
And the trader loses every day time value,
measured by the theta.
Before we discuss the important gamma - theta
relationship, lets look at the theta.
(Book p.221)
93
The Theta
The theta of an option ?, measures how much the
option price changes, when time passes on
As with the delta and gamma, the theta is
calculated assuming the other variables
influencing the option price, implied
volatility, asset price, strike, and interest
rate are constant (ceteris paribus).
Therefore the theta is
As already discussed an option price can be
comprised of the intrinsic value and the time
value
The theta measures the amount of change of the
time value with respect to an infinitesimal
small change in time.
(Book p.221)
94
The Theta, cont.
The following graph shows that an at the money
option loses much time value if the option is
close to maturity.
This makes sense from an intuitive point of view,
because an at the money option close to maturity
changes its value Therefore the profit
potential is high, which
(Book p.222,223)
95
The important Theta - Gamma relationship
The delta-hedging option buyer makes money on the
gamma and loses money on the theta. The
delta-hedging option seller makes money on the
theta and loses money on the gamma.
Thus, the crucial question for the delta-hedging
option trader is
? (known) Theta lt ? (expected) Gamma ? ?
(known) Theta gt ? (expected) Gamma ?
Since the (known) implied volatility largely
determines the theta, and the (expected) actual
volatility largely determines the gamma, the
above relationship can also be stated as
(known) implied volatility lt (expected) actual
volatility ? (known) implied volatility gt
(expected) actual volatility ?
(Book p.225,226)
96
The important Theta - Gamma relationship cont.
Naturally, if an investor does not delta hedge
her option, the relevant question to buy an
option is whether
the premium paid is
vice versa
Buy the call if
vice versa
Buy the put if
(Book p.226)
97
The Vega
Besides delta and gamma, the vega is the third
crucial risk-parameter of an option.
The vega of an option measures
As with delta and gamma, the vega is calculated
assuming the other variables influencing the
option price underlying asset price, time,
strike, and interest rate are constant (ceteris
paribus).
Therefore the vega is
(Book p.227)
98
The Vega cont.
The vega as a function of the underlying price S
looks as follows
Result
(Book p.227)
99
The Vega cont.
Just like delta and gamma, the vega bears
opportunity and risk.
Option traders deliberately play the vega.
This means they take a view on the future
development of the implied volatility.
If a trader believes, the implied volatility will
increase, she will
If a trader believes the implied volatility will
decrease, she will
If the implied volatility moves in an unexpected
direction, the option trader will
(Book p.227)
100
The Rho
A further risk parameter of an option, which
however does not have the significance of delta,
gamma and vega, is rho.
The rho of an option measures how much the option
price changes, if the risk-free interest rate r
changes.
As with delta, gamma and vega, rho is calculated
assuming the other variables influencing the
option price underlying asset price, implied
volatility, time, strike, and are constant
(ceteris paribus).
Therefore rho is the first partial derivative of
the option price with respect to the risk-free
interest rate.
(Book p.230)
101
The Rho
The rho of a call is positive the rho of put is
negative.
Explanation
A higher interest rate r
For the put, a higher interest rate r
(Book p.231)
102
The Leverage effect or omega
The leverage of an option, also called omega,
gearing or elasticity, is not a classical
option risk parameter, like delta, gamma, theta,
vega or rho, which option traders hedge.
However, especially for the small investor, it is
an important criteria for the potential gain of
the investment.
The leverage effect measures the relative change
of the option price with respect to a relative
change in the price of the underlying asset.
(Book p.232)
103
The Leverage effect or omega cont.
The leverage effect of a call is the higher, the
more the option is out of the money
Thus, to participate strongly in the leverage of
an option, buy
(Book p.233)
104
Futures
A Future is the agreement between two parties
to trade an asset at a certain date in the
future, at a price, which is determined
today. (Book p.52)
105
Futures
Buying a Future is agreeing to buy the underlying
asset at the Future maturity date, at today's
Future price
Selling a Future is agreeing to sell the
underlying asset at the Future maturity date, at
today's Future price
(Book p.52f)
106
(Book p. 53)
107
Dates of a Future trade
t0
tM
Last future sell possibility
Future purchase date
Possible future sell dates
108
A Future is a bet. The bet is characterized by

• Delivery price Price of the future today!

• Expiration dates (last trading day, delivery day)

• Other Specifications (Physical versus cash
  delivery,
• quality of the product for commodity futures,
  composition of the index for index
  futures, etc)

(Book p.53)
109
An Example of a Future Trade
The Dow-Future contract trades at 10,450. You buy
one contract. One point in the Dow represents 10.
Scenario 1 The Dow-Future has increased to
10,550. You sell the Dow-future at that price.
What is your profit?
Scenario 2 The Dow-Future has decreased to
10,250. You decide to sell the Dow-future at
that price. What is your loss?
110
Difference Option - Future
The difference between a future and an option
lies in the fact
So an option will lead to a
(Book p.165)
111
Commodity Futures Contract Specifications
112
COMMODITY FUTURES CONTRACT SPECIFICATIONS cont.
113
Financial Futures Contract Specifications cont.
114
Why are Futures so popular ?
(Book p. 63f, 79f)
115
Are Futures free? Kind of
When you trade futures, the only thing you pay is
the margin
Margin is insurance money, to make sure, you can
pay your debt, if you loose money on the future
trade. There is
Initial Margin Maintenance margin Variation margin
(Book p.56 -58)
116
Types of orders
? Market order
? Limit order
? Fill or Kill
? Order cancels order (oco)
? Stop order (stop loss, create profit order)
(Book p.61,62)
117

• The Fair Futures Price
• The fair futures price can be derived by cost of
  carry considerations
• Lets assume the future of Gold expires in one
  month (31 days)
• and Gold trades at a spot price (also called cash
  price) of S 320.

The spot Gold buyer who wants buy Gold and hold
it for one month, will have to borrow cash for
one month. If the one month interest rate is r
5, the cash buyer has to pay
Therefor the fair futures price F This is the
same as F
(Book p. 72-74)
118
Arbitrage
If the Futures price is not equal to the Fair
Futures price, we can do arbitrage.
Example
The Futures price is 325. How can we do
arbitrage???
119
The difference between the futures price and the
spot price is called the Basis. The basis is
converges to zero in time
F Fair futures price, dotted line actual future
price
(Book p.67,68)
120
If the underlying asset pays a return in form of
a
dividend
or
coupon
,
the cash buyer benefits relative to the future
buyer,
because
Let's assume the one month interest rate is 5.
Therefor the spot
treasury buyer will have to pay
((Book p. 83))
121
Arbitrage with Futures
The most popular type of arbitrage with Futures
is the Cash and Carry Arbitrage
Example 4.16 Today is June 1st. The CTD
(Cheapest to deliver) is priced 99.50 and has a
conversion factor of 1.052. The CTD also has an
annual coupon of 7 and pays coupons semiannually
on May 1st and November 1st. The future has a
price of 96.00 and expires on June 23rd. The
interest rate from June 1st to June 23rd is 6.
Does cash and carry arbitrage exist?
(Book p. 114,115)
122
Arbitrage with Futures cont.
On June 1st the potential cash and carry
arbitrageur buys the bond. Including accrued
interest she pays 0.995 100,000 31/365
0.07 100,000 100,094.52. This amount has to
be financed from June 1st to June
23rd 100,094.52 0.06 23/360 383.70. On
June 1st the arbitrageur sells the future at
96.00.
On June 23rd the arbitrageur will deliver the
bond under the terms of the futures contract.
Including the accrued interest from May 1st to
June 23rd, so for 53 days, the arbitrageur will
receive 0.96 100,000 1.052 53/365 0.07
100,000 102,008.44
Therefore the cash and carry arbitrage is
102,008.44 - (100,094.52 383,70) 1,530.24
per one future contract.
(Book p. 114,115)
123
The Currency Forward
The /DM currency spot exchange rate is 0.5556.
Considering the US one month interest rate is 5
and the Deutsch Mark one month interest rate is
7, the fair forward exchange rate in one month
is 0.5556 (1 0.05 31/365) / (1 0.07
31/365) 0.5547
If an American tourist wants to buy DM in one
month from Bank A, Bank A borrows and changes
the into DM today. For borrowing US Bank A
pays 5. Bank A invests the DM at 7
Bottom Line Buying the higher yielding currency
forward is advantages
((Book p. 82,83))
124
Swaps
A swap is an agreement between two parties to
exchange a series of cash flows.
(Book p.233)
125
Swaps - An overview
Most actively traded swaps in the different
markets
(Book p.118)
126
Interest rate swaps
An interest rate swap is an agreement between two
parties to exchange a series of cash flows based
on different interest rate indices, on the same
principal amount, for a given period.
The most common type of swap is the plain vanilla
fixed-floating interest rate swap
In a fixed-floating interest rate swap, one party
pays a fixed interest rate. This fixed rate, i.e.
7, is paid on the principal amount, annually or
sub-annually, during the life of the swap.
The other party pays a floating interest rate.
This floating rate is more interesting. It
depends on the future process of interest rates
and is therefore stochastic (unknown).
(Book p.117)
127
Interest rate swaps cont.
7
A
B
6ML
For a 2-year swap, annual on fixed side ,
semiannual on the floating side the cash flows
are
(Book p.119)
128
Example of an interest rate swap
Lets assume, we have a 2-year swap against 6ML,
7 annually on the fixed side, semiannually on
the floating side, and 1,000,000 principal
amount.
Lets assume the fixings of the 6ML are 6 at t0,
6,5 at t0.5, 4 at t1 and 6 at t1.5.
Lets also assume, t0.5 - t0 and t2 - t1.5 are
182 days and t1 - t0.5 and t1.5 - t1 are 183 days.
Thus, the fixed rate payer pays, at t1 and
at t2.
The floating rate is usually paid in
arrears.This means it is paid at the following
fixing date.
Thus, the floating rate payer pays,
(Book p.119,120)
129
Example of an interest rate swap cont.
Thus, the cash flows are as follows
t
t
t
t
t
0.5
1
1.5
2
0
Result In retrospective, the swap created a loss
for the fixed rate payer. He paid and received
only
(Book p.119,120)
130
Why interest rate swaps ?
There are 5 main reasons for the success of
interest rate derivatives
? Speculation
? Hedging
? Cost reduction
? Arbitrage
? Off balance sheet feature
The main motives for entering into swaps are
and .
(Book p. 120)
131
Reducing cost with an interest rate swaps
Read p. 6 to 8
Thank you
OK heres the example
Company AAA and company BBB both want to invest
50 mio in the USA and 50 mio pound in GB. For
simplicity purpose, the exchange rate is 1.
Company AAA can borrow money in the USA and GB
cheaper than BBB
As seen, BBB has a relative advantage over AAA in
GB, where is only pays 1.5 more interest than
AAA, compared with paying 2 more when borrowing
in .
(Book p. 6)
132
Reducing cost with an interest rate swaps cont.
From the table it follows If AAA would borrow
50 million in the USA and 50 million pound in GB
it would pay
50 mio 7 50 mio 8 7,500,000 in
interest.
If BBB does the same, it would pay
50 mio 9 50 mio 9.5 9,250,000 in
interest.
If AAA however uses its comparative advantage and
only borrows in the USA, AAA would pay
100 mio 7 7,000,000.
If BBB uses its comparative advantage and only
borrows in GB, it would pay
100 mio 9.5 9,500,000
Since both companies want to invest 50 million in
each country, they can now enter into a
fixed-fixed currency swap
(Book p. 6,7)
133
Reducing cost with an interest rate swaps cont.
If AAA gives 50 million to BBB for 8, and BBB
gives 50 million pound to AAA at 8.75, it
follows
Fig Companies AAA and BBB each using their
comparative borrowing advantage and entering into
a fixed-fixed currency swap
(Book p. 6,7)
134
Reducing cost with an interest rate swaps cont.
In the swap, AAA pays 50 mio 8.75
4,375,000, BBB pays 50 mio pound 8
4,000,000. Netted, AAA pays 375,000 to BBB in the
swap (the exchange rate is assumed to be 1)
The overall borrowing costs for AAA are therefore
7,000,000 (from borrowing in ) 375,000 from
the swap 7,375,000
The overall borrowing cost for BBB is 9,500,000
(from borrowing in pound) - 375,000 from the
swap 9,125,000
Bottom line therefore is
Using their comparative borrowing advantages and
entering into a swap,
The interest rates paid in the swap, 8.75 and
8, were chosen so that both companies reduce
their financing costs by 125,000. In reality the
exact levels are determined by negotiation.
(Book p. 8)
135
Hedging interest rate risk with an interest rate
swap
Example
Company A has a floating rate loan on 1 million.
The company is afraid that interest rates will
rise and wants to eliminate this interest rate
risk.
The company therefore enters into a swap, where
it pays a fixed rate (7) and receives a floating
rate (6ML)
Swap
Company
counterpart
A
6ML
1 mio
Loan
provider
Result
(Book p. 121)
136
The cross-currency swap
An US company A wants to invest in Germany,
therefore needs Deutsch Mark (DM).
Since the company is better known in America, it
can borrow money cheaper in the US.
Furthermore the company wants to pay a floating
rate on its loan. The loan amounts to 1.5 million
DM, the spot exchange rate is 1.50 DM/.
In this case the following cross-currency swap is
advantageous for company A
(Book p. 122)
137
The cross-currency swap cont.
At the beginning of the swap the principal
amounts are exchanged
(Book p. 122)
138
The cross-currency swap cont.
During the investment period during the swap
period, the interest rates are exchanged
(Book p. 122)
139
The cross-currency swap cont.
At the maturity date of the swap, the principal
amount is re-exchanged
Result
With this cross-currency swap the US company
achieves the goal of borrowing in the favorable
US market and at the same time paying a Deutsch
Mark floating interest rate.
(Book p. 122)
140
The equity swap
With an equity swap, a floating rate return (or
liability) can be swapped into an equity return
(or liability).
E.g., an investor who has originally bought a
floating rate bond to receive Libor, however now
believes the IBM stock will increase, can enter
into the following equity swap
(Book p. 122)
141
Practice Mid-term exam Task 1 (25 points) 1)
In 1997 Robert Merton got the Nobel-prize,
principally for adding dividends to the
Black-Scholes formula. True
False ? 2) A put seller has to buy
the underlying asset, if the put buyer decides to
sell it. True False 3)
The potential loss of a long put is unlimited
True False ? ? 4) The potential
profit of a long call is unlimited True
False ? ? ?
142

• 6) In the absence of returns, it is never
  rational to exercise a call prematurely
• True False
• ? ?
• 7) In the absence of returns, it is never
  rational to exercise a put prematurely
• True False
• 
  ?
• 8) The time value of an option is the difference
  between the option price
• and the intrinsic value
• True False
• ?
• 9) If the implied volatility increases, the
  option holder makes money.
• True False
• ? ?
• 10) If the put-call parity equation is violated,
  arbitrage opportunities exist

143
Task 2 (25 Points) a)An investor sells a call
with a strike K2 and sells a put with a strike
K1. K2 gt K1. Draw the payoff diagrams of both
options in one diagram. Add the payoffs. Show
the profit and the loss.
b) What is this strategy called?
c) What views about the future volatility does
the investor have?
144
Task 3 (25 points) a) It is better to sell a
call than to exercise prematurely. Do you
agree? Give your answer graphically and
explain the graph!
b) It is better to sell a put than to exercise
prematurely. Do you agree? Give your answer
graphically and explain the graph!
c) A call on a stock with a dividend should be
exercise early. Do you agree? Give your
answer verbally!
145
Task 4 (25 points) a) Show the leverage effect
of an option in a numerical example.
b) The option buyer benefits from the leverage
effect. Do you agree with that statement?
146
Practice Final Trading Derivatives
1) On average Bonds are less risky financial
instruments than stocks.   True
False   2) In the long run, bonds
tend to outperform stocks  
True False     3)      The vega of an
option measures how much the option price
changes, if the underlying instrument changes by
an infinitesimal small amount   True
False     4) The binomial model can value
American style options.   True
False     5) The Black-Scholes model can
value American style options   True
False   6) The maximum profit of
buying a put option is unlimited.
True False     7) The maximum loss of
selling a call is unlimited.   True
False  
147
8) Except for the margin, buying and selling
futures is free
True False     9) A straddle
seller believes in high future volatility
True
False   10) Barrier
options are more expensive than equivalent
standard options.  
True False  
148
Task 2 (25 Points)   a) An investor has bought
one share of IBM at 100. Unfortunately, IBM is
falling. The investor decides to hedge his
position with buying a put. The put has strike of
100 and a premium of 10. Draw the IBM
position and the put in one pay-off diagram. Add
the two!                                     b)
Which option position does the overall position
represent?           c) Is the investor still
bullish? Do you agree with the investors decision
to hedge?                        
149
Task 3 (25 points)   Draw the put function of a
live put with respect to underlying asset price
S. Differentiate the put function (graphically)
to derive the delta in a second graph below.
Differentiate the delta (graphically) to derive
the gamma in a further graph below.
                                     a)
What values can the put delta and put gamma take?
150
Task 4 (25 points)   a)      Show graphically,
that a future pr