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C) Option markets
and contracts
identify the basic elements and describe the
characteristics of option contracts;

An option is a contract.
 It gives one party (the holder of the option) the right
to choose, during a specified period of time, to buy
(or sell, respectively) a specified quantity of a
specified asset (for instance a stock) at a given
price.
 The other contract party (the writer of the option)
has the obligation to fulfill the holder's right.
Put Option
LongBuyer of
PutHolder
Short-
Call Option
Long-
Buyer of
Seller of
PutPutHolder
Writer
ShortSeller of
Put-
Writer
define European option, American option,
money ness, payoff, intrinsic value, and
time value



American option: may be exercised at
any time up to and including the
contracts expiration date
European option: can be exercised only
on the contract’s expiration period.
At expiration an American and an
European on the same asset and the
same strike price are equal
define European option, American option,
money ness, payoff, intrinsic value, and
time value.. Contd ..

If 2 options are identical in all
respects, except that one is
American and the other is European,
tha value of an American option will
equal or exceed the value of
European option
Options Terminologies.
In-the-money:
 At-the money
 Out of Money

CALL OPTION
160 ITM
162 ITM
164
166 OTM
168 OTM
170 OTM
BUY
SP
SP
MP
SP
SP
SP
PUT OPTION SELL
160 OTM
162 OTM
164 MP
166 ITM
168 ITM
170 ITM
Intrinsic Value of an
Option..
An option’s intrinsic value is the
amount by which the option is in-the
money. It is the amount that an
option owner would receive if the
option were exercised.
 An option has a zero intrinsic value
if it is it is at the money or out of
money, regardless of whether it is a
call or put

Intrinsic Value of an
Option..contd

Lets look at the value of a call option
at expiration. If the expiration date
date price of the stock exceeds the
strike price of the option. The call
owner will exercise the option and
receive S-X. if the price of the stock
is less that the strike price, the call
holder will let the option expire and
get nothing.
Intrinsic Value of an
Option..contd

The intrinsic value of the call option
at expiration is the greater of (S-X)
or 0. That is C= max(0, S-X)
Value
Put Option pay off
Strike Price of 50
Long Call
5
0
-5
Short
Call
X=50
X=55
Stock price at
expiration
Value
Call Option pay off
Strike Price of 50
Long put
10
0
-10
Short
Put
X=40
X=50
Stock price at
expiration
Time Value

The time value of an option is the
amount by which the option
premium exceeds the intrinsic value
and is sometimes called the
speculative value of the option.
Option value= intrinsic value+time
value.
As discussed earlier, the intrinsic value of an option is
the amount by which the option is in the money
At any point during the life of an option its value will be
typically greater than its intrinsic value. This is because
there is some probability that the stock price will change
in an amount that gives the option a positive pay off at
expiration greater that the current (intrinsic) value.
Recall that an option’s intrinsic (to a buyer) is the
amount of payoff at expiration and is bounded by zero.
When an option reaches expiration there is no time
remaining and the time value is zero.
For American options and most cases for European
options the longer the time to expiration, the greater the
time value and other things being equal, the greater the
options premium
Identify different types of
options
Financial Options
 Options on futures
 Commodity Options

Financial Options
Bond Options
 Index Options*cash settled
 Stock Options

Option on futures


Sometimes called futures options, give the holder the
right to buy or sell a specified futures contract on or
before a given date enter into a long side of a
futures contract at a given futures price. Assume that
you hold a call option on a bond future @ 98 % of
the face value and at the expiration the future price
of the bond is 99. By exercising the call, you can take
a long position in the futures contract, and the
account is immediately marked to market on the
settlement price. Your account will be credited with
an cash amount of 1% of the face value of the face
value of the bond.
The seller of the exercised call will take a short
position in the futures contract and the mark to
market value of this position will generate the cash
position deposited in your account a
Compare IRO & FRA.
IRO are similar to stock options
except that the exercise price is an
interest rate and the underlying
asset is a reference rate
 IRO are also similar to FRA because
there is no delivery asset, instead
they are settled in cash




Consider a long position in a LIBOR based
Interest rate call option with a notional
amount of $ 1000000 and a strike rate of
5%.
If at expiration libor is greater than 5%
the option can be exercised and the
owner will receive 1000000*(LIBOR-5%).
IF Libor is less than 5%, the option
expires worthless
Interest rate cap and floor


An interest rate cap is a series of interest rate call
option, having expiry dates that corresponds to the
reset dates on the floating rate loan. Caps are often
used to protect a floating rate borrower from an
increase in the interest rates. Caps places a maximum
limit on the interest rates on the floating rate
Caps pay when rates rise above the cap rate. In this
regard, a cap can be viewed as a series of interest rate
call options which strike equals to the cap rate. Each
option in a cap is called a caplet.
Interest rate cap and floor


An interest rate floor is a series of interest rate put
option, having expiry dates that corresponds to the
reset dates on the floating rate loan. Floors are often
used to protect a floating rate lender from an decline
in the interest rates. Floors places a minimum limit on
the interest rates on the floating rate
Floors pay when rates fall below the floor rate. In
this regard, a floor can be viewed as a series of
interest rate put options which strike equals to the
floor rate. Each option in a floor is called a floor let
Example
Reset
Reference Rate
Cap
Floor

90 days
LIBOR
10%
5%
In the event that LIBOR rises above
10%, the cap will make a payment
to the cap buyer to offset any
interest expenses in excess of an
annual rate of 10%
Loan
Rate
Loan rate without
caps or Floors
Received by
cap owner
10%
10% cap
5%
5% floor
Received
by floor
owner
5%
10%
LIBOR
Identify the minimum and maximum
values of European and American
Options
St=price of underlying stock at time t
x = exercise price of the option
T = time to expiry
ct= price of european call at time t, at any time t prior to expiration
c
t=price of Auropean call at time t, at any time t prior to expiration
pt=price of european put at time t, at any time t prior to expiration
Pt=price of American pur at time t, at any time t prior to expiration
RFR
Lower case letters are used to denote
European style options
Lower bounds for Options (Call & Put)
for both American and European options.
Theoretically no option will sell for
less than its intrinsic value and no
option can take a negative value.
 This means that the lower bound for
any any option is zero for both
Americans and European options

Upper bounds for call Options
(American and European )


The maximum value of either an American or
European call option at any time t is the time t
share price of the underlying stock. This makes
sense because no one would pay a price for a
right to buy an asset that exceeds the assets
value. It would be cheaper to simply buy the
underlying stock
At time t=0, the upper boundary condition can
be expressed respectively for American and
European call option is
American Option
C0 <= S0
European Option C0 <= S0
Upper bounds for PUT Options
(American and European )

The price for an American put option
cannot be more than its strike price.
This is the exercise value in the
event the underlying stock price
goes below zero. However, since the
European puts cannot be exercised
prior to expiration, the maximum
value is the PV of the exercise price
discounted at the RFR.
Upper bounds for PUT Options
(American and European )
Even if the exercise price goes to
zero and is expected to stay zero,
the intrinsic value X, will not be
received until expiration date. At
time t=0, the upper boundary
condition can be expressed for
American and European option can
be expressed as
 P0<=X and p0 <=X/1+( RFR) ^ t

Option Strategies.
Basic Options Strategy:
-Long on call
-Short on call
-Long on Put
-Short on put

Spread Strategies
-Bull spread.(Buy call-Sell call @ Higher Exercise
Price)-Market view Bullish
-Bear spread. (Buy call-Sell call @ Lower Exercise
Price).-market view Bears

Option Strategies.
Straddle-market View Mixed(volatile)
-Long Straddle
-Short Straddle
 Strangle -market View Mixed(Range
bound)
-Long Straddle
-Short Straddle

Spread Strategy
Bull Spread
Market View
Bullish
Action
Buy Call, Sell call
@ higher strike
Profit potential
Limited
Loss Potentail
Limited
Bull Call Spread
This involves purchase and sale of Call options at different exercise prices but
with the same exercise date. The purchase call shud have a Lower Exercise
340
360
Bull Call Spread
-16
9
Buy call 340;
Sell call 360; Prem 9
Net P/L
Prem16
Spot
300
-7
-16
9
310
-7
-16
9
320
-7
-16
9
330
-7
-16
9
340
-7
-16
9
350
3
-6
9
360
13
4
9
370
13
14
-1
380
13
24
-11
390
13
34
-21
400
13
44
-31
Bull Spread
Payoff diagram of bull spread
15
5
-5
30
0
31
0
32
0
33
0
34
0
35
0
36
0
37
0
38
0
39
0
0
Sp
ot
Profit/Loss
10
-10
Reliance price
Net P/L
Spread Strategy
Bear Spread
Market View
Bearish
Action
Buy Call, Sell call
@ Lower strike
Profit potential
Limited
Loss Potential
Limited
Bear Call Spread
Bear Call Spread
This involves purchase and sale of Call options at different exercise prices but
with the same exercise date. The purchase call shud have a Higher Exercise
360
340
Bear Call Spread
-9
16
Buy call 360; Prem
Sell call 340; Prem 16
Net P/L
9
Spot
300
7
-9
16
310
7
-9
16
320
7
-9
16
330
7
-9
16
340
7
-9
16
350
-3
-9
6
360
-13
-9
-4
370
-13
1
-14
380
-13
11
-24
390
-13
21
-34
400
-13
31
-44
Bear Spread
10
Profit/Loss
5
0
Spot 300 310 320 330 340 350 360 370 380 390
-5
-10
-15
Reliance stock price
Net P/L
Straddle-market View Mixed LONG Straddle
Long Straddle
Market View
Mixed
Buy Call, Buy Put
@ same Strike
Spot
290
300
310
320
330
340
350
360
370
380
390
Pay Prm
Action Profit Potential
Buy call Buy Put Unlimited
@ same Exercise
Loss Potential
limited
Strike: 340
1 Call buy : Prem:
Net P/L
16 1 Put buy;
21
-16
11
-16
1
-16
-9
-16
-19
-16
-29
-16
-19
-6
-9
4
1
14
11
24
21
34
340
Prem: 13
37
27
17
7
-3
-13
-13
-13
-13
-13
-13
Profit/Loss
Payoff structure for Long Straddle
40
20
0
-20 290 300 310 320 330 340 350 360 370 380 390
-40
Reliance stock price
Short Straddle
Short Straddel
Market View
Mixed
Action Profit Potential
Sell call Sell Put limited
@ same Exercise
Loss Potential
Unlimited
Short straddle I M receiving Prm
sell Call, sell Put
@ same Strike
Sold/short on call
Strike: 340
Sold/short on put
340
Spot
290
300
310
320
330
340
350
360
370
380
390
Net P/L Call : Prem: 16
Put Prem: 13
-21
16
-11
16
-1
16
9
16
19
16
29
16
19
6
9
-4
-1
-14
-11
-24
-21
-34
-37
-27
-17
-7
3
13
13
13
13
13
13
Payoff structure for short straddle
40
Profit/Loss
30
20
10
0
-10 290 300 310 320 330 340 350 360 370 380 390
-20
-30
Reliance stock price
Long Strangle
Market View
I PAY PRM
Action
Profit Potential Loss Potential
Buy Call, Buy Put -Buy
Call at higher SP
Volatile
Spot
280
290
300
310
320
330
340
350
360
370
380
390
400
Buy Call
Premium: 12
Strike 350
Call:Prem 12
-12
350
Net P/L
28
18
8
-2
-12
-22
-22
-22
-12
-2
8
18
28
-12
-12
-12
-12
-12
-12
-12
-12
-2
8
18
28
38
Buy PUT
Premium: 10
Strike 330
Put: Prem 10
-10
330
40
30
20
10
0
-10
-10
-10
-10
-10
-10
-10
-10
Payoff structure for Long Strangle
40
20
10
0
Reliance
400
-30
380
-20
360
340
320
300
-10
280
Profit/Loss
30
Short Strangle
Market View
I Recv PRM
Action
Profit Potential Loss Potential
Sell Call, Sell Put Sell Call at higher SP
Volatile
Spot
280
290
300
310
320
330
340
350
360
370
380
390
400
Sell Call
Premium: 12
Strike 350
Call:Prem 12
12
350
Net P/L
-28
-18
-8
2
12
22
22
22
12
2
-8
-18
-28
12
12
12
12
12
12
12
12
2
-8
-18
-28
-38
Sell Put
Premium: 10
Strike 330
Put: Prem 10
10
330
-40
-30
-20
-10
0
10
10
10
10
10
10
10
10
Short strangle payout structure
30
-30
-40
Reliance stock price
40
0
38
0
36
0
34
0
-20
32
0
0
-10
30
0
10
28
0
Profit/Loss
20
INTRINSIC VALUE
 For a call option:
Intrinsic value = Price of the underlying - Exercise price
 For a put option:
Intrinsic value = Exercise price - Price of the underlying
FACTORS AFFECTING
PREMIA

There are five major factors affecting
the Option premium:
 Price of Underlying
 Exercise Price Time to Maturity
 Volatility of the Underlying

And two less important factors:
 Short-Term Interest Rates
 Dividends

Intuition would tell us that the spot price of
the underlying, exercise price, risk-free
interest rate, volatility of the underlying, time
to expiration and dividends on the
underlying(stock or index) should affect the
option price.
OPTION Pricing

Black and Scholes start by specifying a simple
and well–known equation that models the way
in which stock prices fluctuate. This equation
called Geometric Brownian Motion, implies
that stock returns will have a lognormal
distribution, meaning that the logarithm of the
stock’s return will follow the normal (bell
shaped) distribution.

The Black-Scholes (1973) option pricing
formula prices European put or call options
on a stock that does not pay a dividend or
make other distributions
Part I of the model

In order to understand the model
itself, we divide it into two parts.
The first part, SN(d1), derives the
expected benefit from acquiring a
stock outright. This is found by
multiplying stock price [S] by the
change in the call premium with
respect to a change in the
underlying stock price [N(d1)].
Part II of the model

The second part of the model, Ke(- t)N(d2),
gives the present value of paying the exercise
price on the expiration day.
 The
fair market value of the call
option is then calculated by
taking the difference between
these two parts.
Assumptions of the Black and
Scholes Model:






The stock pays no dividends during
the option's life
European exercise terms are used
Markets are efficient
No commissions are charged
Interest rates remain constant and
known
Returns are log normally distributed
Value of PUT Option

P=Ke^-rt N(D2) – S N (D1)
Binomial tree

A useful and very popular technique for
pricing an option or other derivatives
involves constructing what is know as
Binomial tree
One step Binomial tree
We start by considering a very simple
situation where a stock price is currently
$ 20 and it is known that at the end of 3
months the stock price will be either $22
or $ 18
 We suppose that the stock pays no
dividend and that we are interested in
valuing a European call option to buy the
stock @ 21 in 3 months

Simple binomial tree
Stock Price $22
Option Price $1
Stock Price $20
Stock Price $18
Option Price $0
One step Binomial tree


We set up a portfolio of the stock and the option
in such a way that there is no uncertainty about
the value of the portfolio at the end of 3 months.
We also argue that since the portfolio has no
risk, the return must be equal to the RFR. This
enables to work out the cost of setting up the
portfolio and therefore the option price. Since
there are 2 securities ( the stock and stock
option) and only 2 possible outcomes, it is
always possible to set a riskless portfolio
One step Binomial tree




Consider a portfolio consisting of a long position in
delta shares of the stock and a short position in one
call option.
We will calculate the value of delta that makes the
portfolio risk less.
IF the stock price moves from 20 to 22, the value of
the shares is Delta 22 and the value of the option is 1,
so that the value of the portfolio is Delta 22-1
If the stock price moves down from 20 to 18, the value
of the shares is Delta 18 and the value of the option is
0, so that the total value of the portfolio is delta 18
One step Binomial tree
The portfolio is risk less if the value of delta is
chosen so that the final value of the portfolio is
the same for both the alternative stock prices.
 This means:
22Delta –1 = 18 delta
Delta = 0.25
A risk less portfolio is therefore
Long 0.25 share
Short = 1 option

One step Binomial tree





If the stock price moves up to 22, the value of
the portfolio is
22*0.25-1=4.5
If the value of the stock moves to 18, the value
of the portfolio is
18*0.25 = 4.5
Note: regardless of whether the stock price
moves up or down the value of the portfolio is
always 4.5 at the end of the life of the option
One step Binomial tree
Risk less portfolio must, in the absence of arbitrage
opportunity earn a RFR. Suppose in this case the RFR
is 12%. It follows that the value of the portfolio today
must be the PV of 4.5 or
4.5 e ^ -0.12*0.25= 4.367
The value of the stock price today is known to be 20.
The value of the portfolio today is therefore ,
20*0.25-F = 5 – F
Therefore F = 0.633

One step Binomial tree

This shows that in the absence of
arbitrage opportunity the value of the
option is 0.633
Interpreting volume and
open interest


While volume alone is not a useful determinant of
market direction, used in conjunction with other
data it can be very beneficial - especially to longer
term traders - in identifying whether a
continuation of or reversal in the prevailing trend is
likely.
Most traders incorporate Open Interest data with
their volume analysis. Open Interest is the net
number of open bullish positions in a futures
market.
Interpreting volume and open
interest


Volume at low levels reflects uncertainty regarding the
future direction of the market in question. Conversely, high
volume suggests a high level of confidence in the future
direction
Low levels of Open Interest reflect a market lacking in
liquidity and, therefore, one which will be relatively more
susceptible to being moved by a trade than a more liquid
counterpart. When there are high levels of Open Interest,
deals are likely to be rapidly swallowed up by the market due to the fact that there will be a vast array of participants
eager to open new positions or take profits - and
consequently have far less impact on the current price.
Interpreting volume and
open interest



If volume is relatively high while the market is going up and
remains relatively low during corrections, the inference is
that the market is in a strong uptrend, which should
continue.
both open interest and prices are increasing, then new buyers
are being brought into the market with a strong technical
picture unfolding. Expect the uptrend to continue
In the event of open interest declining while prices are also
slipping, liquidation by long positions is the implication,
therefore suggesting a technically strong market overall. In
other words, the market is strong as open interest declining
suggests no new aggressive shorts, as this would entail an
increase in open interest.
Risk Measures- Delta



Delta Hedging
Delta is defined as the rate of change of its price with respect
to the price of the underlying asset.It is the slope of the curve
that relates the derivatives price to the underlying asset
price.
For example, the price of a call option with a hedge ratio of
40 will rise 40% (of the stock-price move) if the price of the
underlying stock increases. Typically, options with high
hedge ratios are usually more profitable to buy rather than
write since the greater the percentage movement - relative to
the underlying's price and the corresponding little timevalue erosion - the greater the leverage. The opposite is true
for options with a low hedge ratio.
Theta


The theta of a portfolio of a derivative, is the
rate of change of the value of the portfolio
with respect to time with all else remaining
the same.
Some times referred to as time decay
Gamma

The gamma of the portfolio of derivatives on
an underlying asset is the rate of change of
the portfolio;s delta with respect to the price
of the underlying asset
Vega


The vega of a portfolio of derivatives is the
rate of change of the value of the portfolio
with respect to the volatility of the
underlying asset
If vega is high in relative terms, the
portfolio’s value is very sensitivity to small
changes and vice versa
Rho


The rho of a portfolio of derivative is the rate
of change of the value of the portfolio with
respect to interest rates.
It measures the sensitivity of the value of the
portfolio with respect to interest rates
Exotic options


Derivatives with more complex pay offs
than the standard American or European
options are sometimes referred to as exotic
options
Most exotics are traded over the counter and
are structured by Financial institutions
Types of Exotics








Packages
Non standard American option s
Forward Start Options
Compound options
As you like it option
Barrier options
Look back options
Asian options
Package

A package is portfolio consisting of standard
European options call, standard European
puts, forward contracts and cash
Non standard American
Option


In Americans option exercise can take place at
any time during the life of the option and the
exercise price will be the same
One type of non standard American option is
known as Bermudan option. In this early
exercise is restricted to certain dates during the
life of the option. An example of a Bermudan
option would be an American swap option that
can be exercised only on dates when swap
payments are exchanged
Compound Options



Compound options are options on options
4 main types of compound options, call on call, call
on put, put on put and put on call
Example call on call: on the first exercise date, the
holder of the compound option is entitled to pay the
first strike price and receive a call option. The call
option gives the holder the right to buy the
underlying asset for the second strike price on the
second exercise date. The compound option will be
exercised on the first exercise date only if the value of
the option on that date is greater than the first strike
price
“As you like it “




An as you like it option (sometimes referred
to as a choosers option), has the feature that
after a specified period of time, the holder
can choose whether the option is a call or a
put option.
Suppose that the time at which the charge is
made is t1, the value of the as you like it
option is max (c,p)
Where c is the value of call
Where p is the value of the put option
Barrier option


Barrier option are options where the payoff
depends on whether the underlying assets
price reaches a certain level during a certain
period of time
One of the product is CAPS
Binary options


Binary
options
are
options
with
discontinuous pay offs
A simple example of binary option is a cash
or nothing call. This pays off nothing if the
stock price ends up below the strike price
and pays a fixed amount, if it ends up above
the strike price
“Look back” options



The pay offs from look back options depend
upon the maximum or minimum stock price
reached during the life of the option.
The pay off from a European style look back
call option is the amount by which the final
stock price exceeds the minimum stock stock
price achieved during the life of the option
The pay off from a European style look back
put option is the amount by which the
maximum stock price achieved during the life
of the option exceeds the final stock price.
Asian Options

Asian pay offs are options where the pay off
depends on the average price of the
underlying assets during at least some part
of the life of the option