Transcript Option Pricing Theory and Applications

Option Pricing
Theory &
Financial Options
7/18/2015
What is an option?
 An
option provides the holder with the
right to buy or sell a specified quantity of
an underlying asset at a fixed price (called
a strike price or an exercise price) on or
before the expiration date of the option.
 Since it is a right and not an obligation,
the holder can choose not to exercise the
right and allow the option to expire.
 There are two types of options - call
options (right to buy) and put options (right
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to sell).
Call & Put
•
•
•
•
Buyer of a call option – long call
Seller of a call option – short call
Buyer of a put option – long put
Seller of a put option – Short Put
 Underlying

asset could be
Stocks, bonds, commodity, indices, foreign
currency & real assets.
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Call Options
A
call option gives the buyer of the option
the right to buy the underlying asset at a
fixed price (strike price or , X or K) at any
time on/ before the expiration date of the
option.
 The buyer pays a price for this right – Call
premium
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Call Option

At expiration,

If the value of the underlying asset (S) > Strike
Price(K)
• Buyer exercises the option.
• Call option buyer benefit: S - K

If the value of the underlying asset (S) < Strike Price
(K)
• Buyer does not exercise
Payoff on exercise date is Max [ (ST-K), 0]
 More generally,



the value of a call increases as the value of the
underlying asset increases
the value of a call decreases as the value of the
underlying asset decreases
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Option Types
 American
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& European Types.
Long Call on Powergrid
Profit from buying one PowerGrid European call option:
Call premium = Rs.5, strike price = Rs.100, option life =
2 months
30 Profit (Rs.)
20
10
70
0
-5
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80
90
100
Terminal
stock price (Rs.)
110 120 130
Short Call on PowerGrid
Profit from writing/selling one PowerGrid European call
option: Option Premium = Rs.5, strike price = Rs.100
Profit (Rs.)
5
0
-10
-20
-30
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110 120 130
70
80
90 100
Terminal
stock price (Rs.)
ZERO SUM GAME???
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Put Options

A put option gives the buyer of the option
the right to sell the underlying asset at a
fixed price on/at any time before the expiry
date of the option.
buyer pays a price for this right – put
premium
 The
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Put Options

At expiration,


If the value of the underlying asset (S) < Strike Price(K)
• Put option buyer profit : K-S
If the value of the underlying asset (S) > Strike Price (K)
• Buyer does not exercise

Payoff on exercise date is Max [ (K- ST), 0]

More generally,


the value of a put decreases as the value of the underlying asset
increases
the value of a put increases as the value of the underlying asset
decreases
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Long Put on KSOIL
Profit from buying an KSOIL European put option: Put
premium = Rs.7, strike price = Rs.70
30 Profit (Rs.)
20
10
0
-7
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Terminal
stock price (Rs)
40
50
60
70
80
90 100
Short Put on KSOIL
Profit from writing an KSOIL European put option: option
price = Rs.7, strike price = Rs.70
Profit (Rs.)
7
0
-10
-20
-30
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40
50
Terminal
stock price (Rs.)
60
70
80
90 100
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Position
Long Call/Call Holder
Long Put/Put Holder
Profit/Loss
Unlimited profit
potential & limited loss
to the tune of premium
Short Call/Call Writer
Short Put/Put Writer
Limited profit to the
tune of premium &
unlimited loss
Options Type

An option is in-the-money (ITM) option when it
is profitable for the option holder, if exercised.
 And option is out-of-money (OTM) when the
option holder looses money if exercises the
option.
 An option is at-the-money (ATM) when the
underlying stock price is identical or relatively
close to the option strike price.
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Understanding Call & Put Option
Quotation from Financial Dailies

Nifty options contracts have 3 consecutive
monthly contracts, additionally 3 quarterly
months of the cycle March / June / September /
December and 5 following semi-annual months
of the cycle June / December would be
available, so that at any point in time there would
be options contracts with atleast 3 year tenure
available
 Option Quotation
 NIFTY Option details
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NIFTY-50 Quotations (25th July 2012)








July 12 call : strike price (3600– 6500)
Aug 12 call : strike price ( 3600-6500).
Sep 12 call : strike price ( 4000-6500).
Dec 12 call :strike price (4000-7000).
July 12 Put : strike Price ( 3600-6400)
Aug12 put : Strike Price ( 3600-6500)
Dec 12 put : Strike Price ( 3500-6000)
Variation in Option Premium:
 August call option premium varies from Rs.1505.05 to
0.10
 August put option, premium varies from Rs.0.70 to
Rs.1320
 WHY????
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Option Premium

Two components


Intrinsic Value
Time Value
 Intrinsic
Value: Benefit the option buyer
will get, if the option is exercised now.


Call Option : S0 – E
Put Option: E – S0.
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ITM, OTM & ATM
 Premiums:
ITM > ATM > OTM
 WHY OTM contracts have value???
 Premiums = Intrinsic value + time value


CE-4500-August call premium of Rs.638.80
Spot Nifty is 5118.40
 Intrinsic Value = 5118.40-4500 = 618.4
 Time Value = Call premium – Intrinsic value = 638.80618.4 =20.4
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ITM, OTM & ATM

CE-5000-August call premium of Rs.192.1




Spot Nifty is 5118.40
Intrinsic Value = 5118.40 - 5000 = 118.40
Time Value = Call premium – Intrinsic value = 73.7
CE-5000-September call premium of Rs.254.1

Intrinsic value = ??
Time Value = ??

CE-5900-September, call premium = 3.40
 Intrinsic value = ??
Time Value = ??

PE-5200-August with a put premium of Rs.133.30


Intrinsic value = ??
Time Value = ??
PE-5300-Dec with a put premium of Rs.279.45
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
Intrinsic value = ?? Time Value = ??
ITM, OTM & ATM
 CE-3700-August
is Deep-in-the-money call
 PE-3700-August is Deep-out-of-money put
 CE-6500-
August is Deep-out-of-money call
 PE-6500- August is Deep-in-the-money put
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Option Premium & Time Value
Time premium increases as Time-to-Maturity increases
Option
CE-6000-July
Premium ( in Rs.)
0.05
CE-6000-August
0.40
CE-6000-Sep
1.50
CE-6000-Dec
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Open Interest
 Measures
series.
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the liquidity in the given option
Option Trading & Settlement

The electronic order matching system of NSE
is known as NEAT (National Exchange for
Automated Trading) while that of BSE is known
as DTSS (Derivative Trading and Settlement
System).
 Options can be squared off anytime before the
maturity.
 Exercising an option and squaring of an open
position are different.
 Squaring off means that the trader enters into an
exact opposite contract for the same underlying,
same maturity and same strike price.
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Option Trading & Settlement
Squaring Off

A long call position enters into a short call position.
Initially he gave the call premium and while squaring off
will receive the call premium. Similarly a trader with a
short call will enter into a contract for long call.

Only ITM options are exercised. So when a
trader is exercising his option, he receives the differential
amount between the option strike price and the market
price/settlement price of the underlying asset.
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PUT-CALL RATIO
 Put/call
 It
ratio = put option OI /call option OI
is used to measure the level of public
bullishness or bearishness in the market at
a given time.
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Factors affecting option premium
 Variables


Relating to Underlying Asset
Value of Underlying Asset; as this value
increases, the right to buy at a fixed price
(calls) will become more valuable and the right
to sell at a fixed price (puts) will become less
valuable.
Variance in that value; As the volatility
increases, both calls and puts will become
more valuable because all options have
limited downside and depend upon price
volatility for upside.
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• Volatility is measured by Standard Deviations
Factors affecting option premium
 Variables

Relating to Underlying Asset
Expected dividends on the asset, which are
likely to reduce the price appreciation
component of the asset, reducing the value of
calls and increasing the value of puts.
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Factors affecting option premium
 Variables


Relating to Option
Strike Price of Options; the right to buy (sell)
at a fixed price becomes more (less) valuable
at a lower strike price.
Life of the Option; both calls and puts benefit
from a longer life.
 Level
of Interest Rates; As rate increases,
the right to buy (sell) at a fixed price in the
future becomes more (less) valuable.
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Determinants of Option Value
Factor
Increase in stock price
Increase in strike price
Increase in variance of
underlying asset
Increase in time to expiration
Increase in interest rate
Increase in dividend paid
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Call
Put
premium premium
Determinants of Option Value
Factor
Call
Value
Put
Value
Increase in stock price
↑
↓
↑
↓
↑
↑
Increase in time to expiration
↑
↑
Increase in interest rate
↑
↓
↓
↑
Increase in strike price
Increase in variance of
underlying asset
Increase in dividend paid
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Put Call Parity
C + PV (K) = S + P
 Holds true for European Option.
 If this equation does not hold good then
arbitrage will happen.
 If the underlying asset is expected
generate Dividend ( D)


C + D+ PV (K) = S + P
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Put Call Parity

For American Option Put Call Parity does not
hold good in absolute sense.
S  K  C  P  S  PV ( K )

With dividend,
S  D  K  C  P  S  PV ( K )
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How to measure Volatility What is
Standard Deviation ?
 Deviation
from Mean.
 We take 252 days daily price. Calculate
daily return [(P1-P0)/P0)] or, LN (Pt/Pt-1).
Weekly /monthly price can be taken.
 Praj Industries
 Suppose daily price is taken.
 Standard deviation is 3.99%.
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What is Standard Deviation ?

If 12th July price Rs. 85.05, what would be price
on

13th July 2011

Annual SD = 3.99%
 Deviation for 1 day = 3.99% * Sqrt(1/252) =
0.25%
 Annual STDDEV (Daily Price): 3.99%



Standard Deviation: 1 day 0.25%
Standard Deviation 1 week 0.56%
Standard Deviation 1 Month 1.17%
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Standard Deviation & Price





Price on
13th July 2011: 85.05 ± 1(0.25%*85.05) = 85.05
± 0.21 = 84.84 to 85.26 ( Probability: 68.27%)
13th July 2011: 85.05 ± 2(0.25%*85.05) = 85.05
± 0.43 = 84.62 to 87.05 ( Probability: 95.45%)
13th July 2011: 85.05 ± 3 (0.25%*85.05) =
85.05 ± 0.64 = 84.41 to 85.59 (Probability:
99.73%)
Probability figures comes from the assumption
that stock returns follow normal distribution.
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Standard Normal Distribution
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Bean Machine (Source:Wikipedia)
 The
bean machine is a device invented by
Sir Francis Galton to demonstrate how the
normal distribution appears in nature. This
machine consists of a vertical board with
interleaved rows of pins. Small balls are
dropped from the top and then bounce
randomly left or right as they hit the pins.
The balls are collected into bins at the
bottom and settle down into the pattern
resembling the Gaussian curve
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Option Pricing Model
 Binomial
Option Pricing Model
 Black Scholes Option Pricing Model
 The binomial model is a discrete-time
model for asset price movements, with a
time interval (t) between price
movements.The stock can jump to only
one of two points in each time interval, and
the option value is estimated iteratively.
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Valuing a call option using
Binomial option pricing method
A 3-month call option on the stock has a strike price of
Rs.21. Risk-Free rate of interest = 12% per annum.
Stock Price = Rs.22
Option Price = Rs.1
Stock price = Rs.20
Option Price=?
Stock Price = Rs.18
Option Price = Rs.00
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Binomial Option Pricing Risk-Neutral
Valuation
S0u = 22
ƒu = 1
S0
ƒ
S0d = 18
ƒd = 0
 Since p is the probability that gives a return on the stock
equal to the risk-free rate.
 We can find it from
20e0.12 0.25 = 22p + 18(1 – p )
which gives p = 0.6523
 Alternatively, we can use the formula
0.120.25
e d e
 0.9
p

 0.6523
ud
1.1  0.9
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rT
Valuing the Option Using RiskNeutral Valuation
S0u = 22
ƒu = 1
S0
ƒ
S0d = 18
ƒd = 0
The value of the option is
e–0.120.25 (0.65231 + 0.34770)
= 0.633
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Choosing u and d
One way of matching the volatility is to set
s
u  e
Dt
d 1 u  e
s
Dt
where s is the volatility and Dt is the length
of the time step. This is the approach used
by Cox, Ross, and Rubinstein
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Work -Out
A 6-month call option on the stock has a
strike price of 300. Risk-Free rate of
Interest = 12% per annum. Current market
price is Rs. 310 and u = 1.5 and p =
0.6523. Find out the Call premium ?
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A Two-Step Example
A 6-month call option on the stock has a strike
price of 21.
24.2
22
19.8
20
18

Each time step is 3 months
 X=21, r=12%
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16.2
Valuing a Call Option
D
22
20
1.2823
A
B
2.0257
18
24.2
3.2
E
19.8
0.0
C
0.0
F
16.2
0.0

Value at node B

= e–0.120.25(0.65233.2 + 0.34770) = 2.0257

Value at node A

= e–0.120.25(0.65232.0257 + 0.34770) = 1.2823
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Valuing a Put Option
A 6-month put option on the stock has a
strike price of 300. Risk-Free rate of
Interest = 12% per annum. Current market
price is Rs. 310 and u = 1.5 and p =
0.6523. Find out the put premium ?
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Put Option Valuation
K = 52, S= 50, time period = 2 years in time step =
1yr, r = 5% per annum
Value the Put option. p= 0.6282. Find the put
premium
D
60
72
0
B
50
4.1923
A
48
4
1.4147
E
40
C
9.4636
F
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20
What Happens When an Option is American ?
 At each node, we have to check whether early
exercise is better.
At node B, payoff from early exercise is negative 8 – not
optimal
 At node C, payoff from early exercise is 12. At this node,
the option value is 9.4636
 Hence at node C, the option value will be 12.
72
D
0
60

B
50
5.0894
A
1.4147
40
48
4
E
C
12.0
F
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20
 Everything
remaining as it is, an American
option premium ________ European
Option.
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Option Pricing Model
 The
Black-Scholes model applies when
the underlying asset return distribution is
the normal distribution , and explicitly
assumes that the price process is
continuous and that there are no jumps in
asset prices.
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The Black-Scholes Model
 The
value of a call/put option in the B&S
model can be written as a function of the
following variables:
S = Current value of the underlying asset
K = Strike price of the option
t = Life to expiration of the option
r = Riskless interest rate corresponding to the
life of the option
s2 = Variance in the of the underlying asset
return
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The Black-Scholes Formulas
c  S 0 N ( d1 )  K e
pK e
d1 
d2 
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 rT
 rT
N (d 2 )
N (  d 2 )  S 0 N (  d1 )
ln( S 0 / K )  ( r  s
s
/ 2)T
T
ln( S 0 / K )  ( r  s
s
2
T
2
/ 2)T
 d1  s
T
The Normal Distribution
Normal Distribution Table
 Normsdist( ) function of Ms-Excel also
gives the value.

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Adjusting for Dividends
 If
the dividend yield (y = dividends/ Current
value of the asset) of the underlying asset is
expected to remain unchanged during the life
of the option, the Model is modified

C = S e-yt N(d1) - K e-rt N(d2)
 S 
ln 
+ (r -y+

K
d1 =
s
t
d2 = d1 - s √t
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s2
2
) t
Example

MTNL share is quoting Rs.130 today. The risk-free rate
of interest is 6 per cent, the time to maturity is 3 months,
the exercise price is Rs.140 and the volatility is 20 per
cent per annum. No dividend is expected within 3 months.
S
X
r
s
T
=
=
=
=
=
130
140
0.06 or 6% per annum
0.20 or 20%
0.25 years

d1 = -0.5411
N(d1) = 0.2942

d2 = -0.6411
N(d2) = 0.2607

Call value = 2.29
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Assignment

XYZ Co. share is quoting Rs.130 today. The
risk-free rate of interest is 6 per cent, the time to
maturity is 3 months, the exercise price is
Rs.140 and the volatility is 20 per cent per
annum. SYZ Co. is expected to pay Rs. 3.50 as
dividend 45 days from today. Value the call
Option.
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Option Calculator
Option Calculator which uses Black-Scholes
Option Pricing formula.
 http://www.bseindia.com/derivatives/optioncalc
.asp

Forms a base price.
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Trinomial Tree (Hull 6th edition)
Su
pu
S
pm
S
pd
Sd
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Trinomial Tree (Hull 6th edition)
u  es
pu 
pm
Dt
1 2s 2
d  1/ u

s2 
1

r  2 
 6


2

3
pd  
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3 Dt
Dt
1 2s 2

s2 
1

r  2 
 6


Trinomial Tree (Boyle 1986)
u  es
d  e s
2 Dt

e
d

 ud

Dt
r
2
pu
pm
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2


u e



 ud 


 1  pu  pd
Dt
r
2
pd





2
2 Dt
 1 /u
IMPLIED VOLATILITY
 Option
Premium is available. Find out σ
σ to find out what is going to be the
option premium for other series.
 Use
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112
INDIA VIX
India VIX for 08-Sep-2008
Previo Open High
us
Close
High
Time
32.65
11.25 28.26 15.01 30.77
38.27 56.19
Low
Low
Time
Close
Chang
e
%
Change
-1.88
-5.76%
Volatility Index is a measure of market’s expectation of
volatility over the near term. Volatility is often described
as the “rate and magnitude of changes in prices” and in
finance often referred to as risk.
 Volatility Index is a measure, of the amount by which
an underlying Index is expected to fluctuate, in the near
term, (calculated as annualised volatility, denoted in
percentage e.g. 20%) based on the order book of the
underlying index options.
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113

INDIA VIX
 India
VIX is a volatility index based on the
Nifty 50 Index Option prices. From the
best bid-ask prices of Nifty 50 Options
contracts, a volatility figure (%) is
calculated which indicates the expected
market volatility over the next 30 calendar
days.
 INDIA
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VIX METHODOLOGY
114
Quiz
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115
ESOP & Black Scholes Option
Pricing
 SUZLON
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Annual report (page 50).
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