Description
Describes concepts of options like in the money, at the money, out of money, covered call, naked options, shot call, long call, various option trading strategies, bull spread, bear spread, butterfly option, straddle combination, black scholes option pricing model.
OPTIONS
Group II
Option Basics and Properties
What is Option?
?
An option is a contract, which gives the buyer (holder) the right, but not the obligation, to buy or sell specified quantity of the underlying assets, at a specific (strike) price on or before a specified time (expiration date) The underlying may be physical commodities like wheat / rice / cotton / gold / oil or financial instruments like equity stocks / stock index / bonds
?
Options on basis of Market
?
Exchange traded options (also called "listed options") Exchange traded options have standardized contracts, and are settled through a clearing house with fulfillment guaranteed by the credit of the exchange (As the contracts are standardized, accurate pricing models are often available)
Exchange traded options include: ? Stock options, ? Commodity Options, ? Bond options and Other Interest Rate options ? Index (Equity) Options, and ? Options on Futures Contracts
?
Options on basis of Market
?
Over-the-counter Options (also called "dealer options") These are traded between two private parties, and are not listed on an exchange; the terms of an OTC option are unrestricted and may be individually tailored to meet any business need (In general, at least one of the counterparties to an OTC option is a well-capitalized institution) Option types commonly traded over the counter include: ? Interest Rate options ? Currency options, and ? Options on Swaps or “Swaptions”
?
Two Types of Options
?
Call Option Call Option is gives the holder the right but not the obligation to “Buy” an asset by a certain date for a certain price Put Option A put option gives the holder the right but not the obligation to “Sell” an asset by a certain date for a certain price
?
Option Terminology
Option Price/ Premium Option Price / Premium is the price which the option buyer pays to option seller ? Strike Price The Price that is specified in the options contract ? American Option These options can be exercised at any time up to expiration date ? European Option These options can be exercised only on the expiration date
?
In, At and Out of Money Concepts
CONCEPTS In-the-money CALL OPTION PUT OPTION
Strike price < Spot price Strike price > Spot price of underlying asset of underlying asset Strike price = Spot price Strike price = Spot price of underlying asset of underlying asset Strike price > Spot price Strike price < Spot price of underlying asset of underlying asset
At-the-money
Out-of-the-money
Covered Call and Naked Options
?
Covered Call
A call option position that is covered by an opposite position in the underlying instrument (for example shares, commodities etc), is called a covered call. Writing covered calls involves writing call options when the shares that might have to be delivered (if option holder exercises his right to buy), are already owned E.g A writer writes a call on Reliance and at the same time holds shares of Reliance so that if the call is exercised by the buyer, he can deliver the stock
?
Naked Options
A Naked Option is an option that is not combined with an offsetting position in the underlying stock. Intial Margin on Naked Option is higher of
Use of Options
Hedge Market Restrictions Tax Saving Transaction Cost Leverage
?
? ?
?
?
Options Position – Long Call
Profit from buying one European call option: option price = Rs. 5, Strike Price = Rs.100, Option life = 2 months
30 20 10
Profit (Rs)
70 0 -5
80
90
100 110
Terminal stock price (Rs) 120 130
Options Position – Short Call
Profit from writing one European call option: option price = Rs. 5, Strike Price = Rs.100, Option life = 2 months
Profit (Rs)
5 0
-10 -20 -30
110 120 130 70 80 90 100 Terminal stock price (Rs)
Options Position – Long Put
Profit from buying one European put option: option price = Rs. 7, Strike Price = Rs. 70, Option life = 2 months
30 Profit (Rs) 20 10
0
-7
Terminal stock price (Rs)
40 50 60 70 80 90 100
Options Position – Short Put
Profit from writing one European Put option: Option price = Rs. 7, Strike Price = Rs.70, Option life = 2 months
Profit (Rs)
7 0 -10 -20 -30 40 50 60 70 80 90
Terminal stock price (Rs) 100
Option Trading Strategies
Single Option and A Stock
Profit
Profit
K K
(a)
ST
Profit (b)
ST
Profit
K ST
(c)
a. b. c. d.
K
(d)
ST
Long Position in a Stock combined with short position in a call Short Position in a Stock combined with long position in a call Long Position in a Put combined with long position in a stock Short Position in a Put combined with short position in a stock
Bull Spread using Call Option
Profit
K1
K2
ST
?
Bull Spread with Call Options is created by buying a call option on a stock with certain strike price K1 and selling / writing call option on higher strike price K2
Bull Spread using Put Option
Profit
K1
K2
ST
?
Bull Spreads with Put option can be created by buying a put option with a low strike price K1 and selling a put option with high strike price K2
Bear Spread using Call Option
Profit K1 K2 ST
?
Bear Spreads with Call option can be created by buying a call option with a High strike price K2 and selling a Call option with lower strike price K1
Bear Spread using Put Option
Profit
K1
K2
ST
?
Bear Spreads with Put option can be created by buying a Put option with a High strike price K2 and selling a Put option with lower strike price K1
Butterfly Spread using Call Option
Profit
K1
K2
K3
ST
Butterfly Spread with Call option is created by ? Buying a Call option with relatively low strike price K1 ? Buying a Call option with relatively high strike price K3 ? Sell two Call Option with a strike price K2 halfway between K1 and K3
Butterfly Spread using Put Option
Profit
K1
K2
K3
ST
Butterfly Spread with Call option is created by ? Buying a Put option with relatively low strike price K1 ? Buying a Put option with relatively high strike price K3 ? Sell two Put Option with a strike price K2 halfway between K1 and K3
Straddle Combination
Profit
K
Butterfly Spread is created by ? Buy Call Option at Strike Price K ? Buy Put Option at same Strike Price K
ST
Strip and Strap
Profit Profit
K Strip
ST
K
ST
Strip is created by Strap ? Buying a Call option with strike price K ? Buying a Two Put Option with the same Strike Price K Strap is created by ? Buying a Put option with strike price K ? Buying a Two Call Option with the same Strike Price K
Strangle
Profit
ST
K1
K2
Strangle Spread with Call option is created by ? Buying a Put option with low strike price K1 ? Buying a Call option with high strike price K2
Option Valuation
Valuation of an Option
?
The Value of an Option before expiration depends on 5 factors:
1. 2.
3.
4. 5.
The Price of the Underlying Stock The Exercise price of the Option The time remaining until Expiration The Risk – Free interest rate Possible price movements
Valuation of an Option
?
Limits of Put & Call Options:
American Put Put Price
X
European Put
Xe-r(T-t)
Call Price
Put must lie in this region
X Stock Price
X Stock Price
Black - Scholes Option Pricing Model
Assumptions
?
? ?
?
?
There are no transaction costs (i.e. markets are frictionless) Trading may take place continuously There is no prohibition on short selling The risk free rate is the same for borrowing and lending Assets are perfectly divisible
The Set Up
Securities: Bond: Stock:
dB ? rBdt
dS ? ?Sdt ? ?Sdz
14
Bond
? ? ?
12
Deterministic Exponential Growth Continuous compounding
10 8 6 4 2
Bt ? B0 e rt
0
0
5
10
15
20
25
30
35
40
45
50
The Set Up
Securities: Bond: Stock:
dB ? rBdt
dS ? ?Sdt ? ?Sdz
8
Stock
?
7
? ?
Geometric Brownian Motion Log-Normal Always positive
( ? ? 1 ? 2 ) t ??zt 2
6 5 4 3 2 1 0
St ? S 0 e
0
1
2
3
4
5
6
7
8
9
10
The Set Up
Securities:
Bond:
Stock:
dB ? rBdt
dS ? ?Sdt ? ?Sdz
Consider a derivative security whose price depends on St and t We will call it: c( St , t )
By Ito’s lemma:
dc ? (ct ? ?ScS ? 1 ? 2 S 2 cSS ) dt ? ?ScS dz 2
The Set Up
?
Now we have 3 price processes
dB ? rBdt dS ? ?Sdt ? ?Sdz dc ? (ct ? ?ScS ? 1 ? 2 S 2 cSS ) dt ? ?ScS dz 2
?
Bond: Stock:
Derivative:
Let’s form a portfolio using two of the assets, so that it looks exactly like the third ? Then this portfolio must have the same price as the third ? We can choose any two assets for our portfolio. Let’s choose the stock and derivative, and create a bond
Valuation of an Option
?
Black Scholes Call Option Pricing Model:
?C
= S1N(d1) – xe-r(T-t) N(d2)
? N(.)
= Cumulative Normal Distribution Function ? d1 = ln(S1/x) + (r+0.5 SD2)(T-t) SD ?(T-t) ? d2 = d1-SD ?(T-t)
? Pt
= Ct – St + xe-r(T-t)
Valuation of an Option
European Option
?
Estimation of Risk-Free Rate of Interest:
? By
using the T-Bill as an indicator of Risk Free rate with closest maturity of an option we can calculate the Risk Free Interest Rate ? PTB = 1-0.01 (Bid+Ask) (Days Until Maturity) 2 360 ? To find the corresponding continuously compounded rate er(T-t) = 1/PTB
Valuation of an Option
American Option
?
There are two situations under American Option Valuation:
1.
2.
Non - Dividend paying Stock: Valuation method for European Option and American Options are the same Dividend Paying Stock:
Sensitivity of Option Price
?
Option Prices vary with respect to 5 elements:
? 1.
Price of the Underlying Asset ? 2. Time to Expiration ? 3. Volatility of the Underlying Asset ? 4. Interest Rate
1. Stock Price
?
?
?
Delta is the first derivative of the option with respect to Price of the Stock It measures the sensitivity of the option Price W.R.T. change in Stock Price Delta=?C/ ?S
? (C
is the Option Price) ? (S is the Stock Price)
1. Stock Price
Call Price
Stock Price
1. Stock Price
?
Delta Neutral Position:
?A
portfolio P of a short position of one European Call on a non dividend Stock, combined with a long position of Delta Tends to approach 1.0
Call Price
Stock Price
1. Stock Price
: Measures how Delta changes with changes in the Stock Price
?
Positive and Negative Gamma Portfolios
2. Time to Expiration
Is the negative of the first derivative of the Option Price with respect to the time to expiration
6 4 2 0 -2 -4 -6 -8 -10 -12
? Option Price
Put Theta Call Theta
Call Price
Call Price
Put Price
Stock Price
Days till Expiration
3. Volatility of the Underlying Asset
Is the first derivative of an option price WRT to volatility of the underlying asset
? ?C/
?C
?SD = S ?(T-t) N`(d1)
= Option Price ? SD = Standard Deviation of Price of Underlying ? T = Expiration Date ? t = Current Date ? N(d1) = Normal Dist. Value
4. Interest Rate
: Is the first derivative of an options price with respect to the interest rate
Option Price
Vega
Call or Put Vega
Interest Rate
Stock Price
doc_328195269.ppt
Describes concepts of options like in the money, at the money, out of money, covered call, naked options, shot call, long call, various option trading strategies, bull spread, bear spread, butterfly option, straddle combination, black scholes option pricing model.
OPTIONS
Group II
Option Basics and Properties
What is Option?
?
An option is a contract, which gives the buyer (holder) the right, but not the obligation, to buy or sell specified quantity of the underlying assets, at a specific (strike) price on or before a specified time (expiration date) The underlying may be physical commodities like wheat / rice / cotton / gold / oil or financial instruments like equity stocks / stock index / bonds
?
Options on basis of Market
?
Exchange traded options (also called "listed options") Exchange traded options have standardized contracts, and are settled through a clearing house with fulfillment guaranteed by the credit of the exchange (As the contracts are standardized, accurate pricing models are often available)
Exchange traded options include: ? Stock options, ? Commodity Options, ? Bond options and Other Interest Rate options ? Index (Equity) Options, and ? Options on Futures Contracts
?
Options on basis of Market
?
Over-the-counter Options (also called "dealer options") These are traded between two private parties, and are not listed on an exchange; the terms of an OTC option are unrestricted and may be individually tailored to meet any business need (In general, at least one of the counterparties to an OTC option is a well-capitalized institution) Option types commonly traded over the counter include: ? Interest Rate options ? Currency options, and ? Options on Swaps or “Swaptions”
?
Two Types of Options
?
Call Option Call Option is gives the holder the right but not the obligation to “Buy” an asset by a certain date for a certain price Put Option A put option gives the holder the right but not the obligation to “Sell” an asset by a certain date for a certain price
?
Option Terminology
Option Price/ Premium Option Price / Premium is the price which the option buyer pays to option seller ? Strike Price The Price that is specified in the options contract ? American Option These options can be exercised at any time up to expiration date ? European Option These options can be exercised only on the expiration date
?
In, At and Out of Money Concepts
CONCEPTS In-the-money CALL OPTION PUT OPTION
Strike price < Spot price Strike price > Spot price of underlying asset of underlying asset Strike price = Spot price Strike price = Spot price of underlying asset of underlying asset Strike price > Spot price Strike price < Spot price of underlying asset of underlying asset
At-the-money
Out-of-the-money
Covered Call and Naked Options
?
Covered Call
A call option position that is covered by an opposite position in the underlying instrument (for example shares, commodities etc), is called a covered call. Writing covered calls involves writing call options when the shares that might have to be delivered (if option holder exercises his right to buy), are already owned E.g A writer writes a call on Reliance and at the same time holds shares of Reliance so that if the call is exercised by the buyer, he can deliver the stock
?
Naked Options
A Naked Option is an option that is not combined with an offsetting position in the underlying stock. Intial Margin on Naked Option is higher of
Use of Options
Hedge Market Restrictions Tax Saving Transaction Cost Leverage
?
? ?
?
?
Options Position – Long Call
Profit from buying one European call option: option price = Rs. 5, Strike Price = Rs.100, Option life = 2 months
30 20 10
Profit (Rs)
70 0 -5
80
90
100 110
Terminal stock price (Rs) 120 130
Options Position – Short Call
Profit from writing one European call option: option price = Rs. 5, Strike Price = Rs.100, Option life = 2 months
Profit (Rs)
5 0
-10 -20 -30
110 120 130 70 80 90 100 Terminal stock price (Rs)
Options Position – Long Put
Profit from buying one European put option: option price = Rs. 7, Strike Price = Rs. 70, Option life = 2 months
30 Profit (Rs) 20 10
0
-7
Terminal stock price (Rs)
40 50 60 70 80 90 100
Options Position – Short Put
Profit from writing one European Put option: Option price = Rs. 7, Strike Price = Rs.70, Option life = 2 months
Profit (Rs)
7 0 -10 -20 -30 40 50 60 70 80 90
Terminal stock price (Rs) 100
Option Trading Strategies
Single Option and A Stock
Profit
Profit
K K
(a)
ST
Profit (b)
ST
Profit
K ST
(c)
a. b. c. d.
K
(d)
ST
Long Position in a Stock combined with short position in a call Short Position in a Stock combined with long position in a call Long Position in a Put combined with long position in a stock Short Position in a Put combined with short position in a stock
Bull Spread using Call Option
Profit
K1
K2
ST
?
Bull Spread with Call Options is created by buying a call option on a stock with certain strike price K1 and selling / writing call option on higher strike price K2
Bull Spread using Put Option
Profit
K1
K2
ST
?
Bull Spreads with Put option can be created by buying a put option with a low strike price K1 and selling a put option with high strike price K2
Bear Spread using Call Option
Profit K1 K2 ST
?
Bear Spreads with Call option can be created by buying a call option with a High strike price K2 and selling a Call option with lower strike price K1
Bear Spread using Put Option
Profit
K1
K2
ST
?
Bear Spreads with Put option can be created by buying a Put option with a High strike price K2 and selling a Put option with lower strike price K1
Butterfly Spread using Call Option
Profit
K1
K2
K3
ST
Butterfly Spread with Call option is created by ? Buying a Call option with relatively low strike price K1 ? Buying a Call option with relatively high strike price K3 ? Sell two Call Option with a strike price K2 halfway between K1 and K3
Butterfly Spread using Put Option
Profit
K1
K2
K3
ST
Butterfly Spread with Call option is created by ? Buying a Put option with relatively low strike price K1 ? Buying a Put option with relatively high strike price K3 ? Sell two Put Option with a strike price K2 halfway between K1 and K3
Straddle Combination
Profit
K
Butterfly Spread is created by ? Buy Call Option at Strike Price K ? Buy Put Option at same Strike Price K
ST
Strip and Strap
Profit Profit
K Strip
ST
K
ST
Strip is created by Strap ? Buying a Call option with strike price K ? Buying a Two Put Option with the same Strike Price K Strap is created by ? Buying a Put option with strike price K ? Buying a Two Call Option with the same Strike Price K
Strangle
Profit
ST
K1
K2
Strangle Spread with Call option is created by ? Buying a Put option with low strike price K1 ? Buying a Call option with high strike price K2
Option Valuation
Valuation of an Option
?
The Value of an Option before expiration depends on 5 factors:
1. 2.
3.
4. 5.
The Price of the Underlying Stock The Exercise price of the Option The time remaining until Expiration The Risk – Free interest rate Possible price movements
Valuation of an Option
?
Limits of Put & Call Options:
American Put Put Price
X
European Put
Xe-r(T-t)
Call Price
Put must lie in this region
X Stock Price
X Stock Price
Black - Scholes Option Pricing Model
Assumptions
?
? ?
?
?
There are no transaction costs (i.e. markets are frictionless) Trading may take place continuously There is no prohibition on short selling The risk free rate is the same for borrowing and lending Assets are perfectly divisible
The Set Up
Securities: Bond: Stock:
dB ? rBdt
dS ? ?Sdt ? ?Sdz
14
Bond
? ? ?
12
Deterministic Exponential Growth Continuous compounding
10 8 6 4 2
Bt ? B0 e rt
0
0
5
10
15
20
25
30
35
40
45
50
The Set Up
Securities: Bond: Stock:
dB ? rBdt
dS ? ?Sdt ? ?Sdz
8
Stock
?
7
? ?
Geometric Brownian Motion Log-Normal Always positive
( ? ? 1 ? 2 ) t ??zt 2
6 5 4 3 2 1 0
St ? S 0 e
0
1
2
3
4
5
6
7
8
9
10
The Set Up
Securities:
Bond:
Stock:
dB ? rBdt
dS ? ?Sdt ? ?Sdz
Consider a derivative security whose price depends on St and t We will call it: c( St , t )
By Ito’s lemma:
dc ? (ct ? ?ScS ? 1 ? 2 S 2 cSS ) dt ? ?ScS dz 2
The Set Up
?
Now we have 3 price processes
dB ? rBdt dS ? ?Sdt ? ?Sdz dc ? (ct ? ?ScS ? 1 ? 2 S 2 cSS ) dt ? ?ScS dz 2
?
Bond: Stock:
Derivative:
Let’s form a portfolio using two of the assets, so that it looks exactly like the third ? Then this portfolio must have the same price as the third ? We can choose any two assets for our portfolio. Let’s choose the stock and derivative, and create a bond
Valuation of an Option
?
Black Scholes Call Option Pricing Model:
?C
= S1N(d1) – xe-r(T-t) N(d2)
? N(.)
= Cumulative Normal Distribution Function ? d1 = ln(S1/x) + (r+0.5 SD2)(T-t) SD ?(T-t) ? d2 = d1-SD ?(T-t)
? Pt
= Ct – St + xe-r(T-t)
Valuation of an Option
European Option
?
Estimation of Risk-Free Rate of Interest:
? By
using the T-Bill as an indicator of Risk Free rate with closest maturity of an option we can calculate the Risk Free Interest Rate ? PTB = 1-0.01 (Bid+Ask) (Days Until Maturity) 2 360 ? To find the corresponding continuously compounded rate er(T-t) = 1/PTB
Valuation of an Option
American Option
?
There are two situations under American Option Valuation:
1.
2.
Non - Dividend paying Stock: Valuation method for European Option and American Options are the same Dividend Paying Stock:
Sensitivity of Option Price
?
Option Prices vary with respect to 5 elements:
? 1.
Price of the Underlying Asset ? 2. Time to Expiration ? 3. Volatility of the Underlying Asset ? 4. Interest Rate
1. Stock Price
?
?
?
Delta is the first derivative of the option with respect to Price of the Stock It measures the sensitivity of the option Price W.R.T. change in Stock Price Delta=?C/ ?S
? (C
is the Option Price) ? (S is the Stock Price)
1. Stock Price
Call Price
Stock Price
1. Stock Price
?
Delta Neutral Position:
?A
portfolio P of a short position of one European Call on a non dividend Stock, combined with a long position of Delta Tends to approach 1.0
Call Price
Stock Price
1. Stock Price
: Measures how Delta changes with changes in the Stock Price
?
Positive and Negative Gamma Portfolios
2. Time to Expiration
Is the negative of the first derivative of the Option Price with respect to the time to expiration
6 4 2 0 -2 -4 -6 -8 -10 -12
? Option Price
Put Theta Call Theta
Call Price
Call Price
Put Price
Stock Price
Days till Expiration
3. Volatility of the Underlying Asset
Is the first derivative of an option price WRT to volatility of the underlying asset
? ?C/
?C
?SD = S ?(T-t) N`(d1)
= Option Price ? SD = Standard Deviation of Price of Underlying ? T = Expiration Date ? t = Current Date ? N(d1) = Normal Dist. Value
4. Interest Rate
: Is the first derivative of an options price with respect to the interest rate
Option Price
Vega
Call or Put Vega
Interest Rate
Stock Price
doc_328195269.ppt