Derivative Securities

1
Topic 13

• Derivative Securities

2
Learning Objectives

• LO 13.1 Explain the basic characteristics and
  terminology of options.
• LO 13.2 Discuss the effect of financial factors,
  especially volatility, on the value of options.
• LO 13.3 Explain the rationale behind the
  Black-Scholes option pricing model.
• LO 13.4 Calculate the value of a European call
  option in Excel using the Black-Scholes option
  pricing model.

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2 Excel Features

• None this week.

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3 Option Basics

• An option gives the holder the right, but not the
  obligation, to buy or sell a given quantity of an
  asset on (or before) a given date, at prices
  agreed upon today.
• Exercising the Option
• The act of buying or selling the underlying asset
  
• Strike Price or Exercise Price
• Refers to the fixed price in the option contract
  at which the holder can buy or sell the
  underlying asset.
• Expiry (Expiration Date)
• The maturity date of the option

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Options

• European versus American options
• European options can be exercised only at expiry.
• American options can be exercised at any time up
  to expiry.
• In-the-Money
• Exercising the option would result in a positive
  payoff.
• At-the-Money
• Exercising the option would result in a zero
  payoff (i.e., exercise price equal to spot
  price).
• Out-of-the-Money
• Exercising the option would result in a negative
  payoff.

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3a Call Options

• Call options gives the holder the right, but not
  the obligation, to buy a given quantity of some
  asset on or before some time in the future, at
  prices agreed upon today.
• When exercising a call option, you call in the
  asset.

7
Call Option Pricing at Expiry

• At expiry, an American call option is worth the
  same as a European option with the same
  characteristics.
• If the call is in-the-money, it is worth ST
  E.
• If the call is out-of-the-money, it is worthless
• C MaxST E, 0
• Where
• ST is the value of the stock at expiry (time T)
• E is the exercise price.
• C is the value of the call option at expiry

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Call Option Payoffs
60
Buy a call
40
Option payoffs ()
20
80
120
20
40
60
100
50
Stock price ()
20
Exercise price 50
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Call Option Profits
Buy a call
10
50
10
Exercise price 50 option premium 10
10
3b Put Options

• Put options gives the holder the right, but not
  the obligation, to sell a given quantity of an
  asset on or before some time in the future, at
  prices agreed upon today.
• When exercising a put, you put the asset to
  someone.

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Put Option Pricing at Expiry

• At expiry, an American put option is worth the
  same as a European option with the same
  characteristics.
• If the put is in-the-money, it is worth E ST.
• If the put is out-of-the-money, it is worthless.
• P MaxE ST, 0

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Put Option Payoffs
60
50
40
Option payoffs ()
20
Buy a put
0
80
0
20
40
60
100
50
Stock price ()
20
Exercise price 50
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Put Option Profits
60
40
Option payoffs ()
20
10
Stock price ()
80
20
40
60
100
50
10
Buy a put
20
Exercise price 50 option premium 10
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Option Value

• Intrinsic Value
• Call MaxST E, 0
• Put MaxE ST , 0
• Speculative Value
• The difference between the option premium and the
  intrinsic value of the option.

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3c Selling Options

• The seller (or writer) of an option has an
  obligation.
• The seller receives the option premium in
  exchange.

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Call Option Payoffs
60
40
Option payoffs ()
20
80
120
20
40
60
100
50
Stock price ()
Sell a call
Exercise price 50
20
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Put Option Payoffs
40
20
Option payoffs ()
Sell a put
0
80
0
20
40
60
100
50
Stock price ()
20
Exercise price 50
40
50
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Option Diagrams Revisited
Buy a call
40
Option payoffs ()
Buy a put
Sell a call
Sell a put
10
Stock price ()
50
40
60
100
Buy a call
10
Buy a put
Sell a put
Exercise price 50 option premium 10
Sell a call
40
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2 Option Quotes
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Option Quotes
This option has a strike price of 135
a recent price for the stock is 138.25
July is the expiration month.
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Option Quotes
This makes a call option with this exercise price
in-the-money by 3.25 138¼ 135.
Puts with this exercise price are
out-of-the-money.
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Option Quotes
On this day, 2,365 call options with this
exercise price were traded.
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Option Quotes
The CALL option with a strike price of 135 is
trading for 4.75.
Since the option is on 100 shares of stock,
buying this option would cost 475 plus
commissions.
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Option Quotes
On this day, 2,431 put options with this exercise
price were traded.
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Option Quotes
The PUT option with a strike price of 135 is
trading for .8125.
Since the option is on 100 shares of stock,
buying this option would cost 81.25 plus
commissions.
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5 Combinations of Options

• Puts and calls can serve as the building blocks
  for more complex option contracts.
• If you understand this, you can become a
  financial engineer, tailoring the risk-return
  profile to meet your clients needs.

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Protective Put Strategy (Payoffs)
Protective Put payoffs
Value at expiry
50
Buy the stock
Buy a put with an exercise price of 50
0
Value of stock at expiry
50
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Protective Put Strategy (Profits)
Value at expiry
Buy the stock at 40
40
Protective Put strategy has downside protection
and upside potential
0
-10
40
50
Buy a put with exercise price of 50 for 10
Value of stock at expiry
-40
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Covered Call Strategy
Value at expiry
Buy the stock at 40
Covered Call strategy
0
Value of stock at expiry
40
50
Sell a call with exercise price of 50 for 10
-40
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Long Straddle
Buy a call with exercise price of 50 for 10
40
Option payoffs ()
30
Stock price ()
40
60
30
70
Buy a put with exercise price of 50 for 10
50
A Long Straddle only makes money if the stock
price moves 20 away from 50.
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Short Straddle
This Short Straddle only loses money if the stock
price moves 20 away from 50.
Option payoffs ()
Sell a put with exercise price of 50 for 10
Stock price ()
30
70
40
60
50
Sell a call with an exercise price of 50 for 10
30
40
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6 Put-Call Parity p0 S0 c0 E/(1 r)T
Portfolio payoff
Call
Option payoffs ()
25
Stock price ()
25
Consider the payoffs from holding a portfolio
consisting of a call with a strike price of 25
and a bond with a future value of 25.
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Put-Call Parity
Portfolio payoff
Portfolio value today p0 S0
Option payoffs ()
25
Stock price ()
25
Consider the payoffs from holding a portfolio
consisting of a share of stock and a put with a
25 strike.
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Put-Call Parity
Since these portfolios have identical payoffs,
they must have the same value today hence the
Put-Call Parity c0 E/(1r)T p0 S0
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7 Valuing Options

• The last section concerned itself with the value
  of an option at expiry.

• This section considers the value of an option
  prior to the expiration date.
• A much more interesting question.

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American Call
ST
Profit
Call
Option payoffs ()
25
Time value
Intrinsic value
ST
E
In-the-money
Out-of-the-money
loss
C0 must fall within max (S0 E, 0) 37
Option Value Determinants

• Call Put
• Stock price
• Exercise price
• Interest rate
• Volatility in the stock price
• Expiration date
• The value of a call option C0 must fall within
• max (S0 E, 0)
• The precise position will depend on these factors.

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An Option Pricing Formula

• We will start with a binomial option pricing
  formula to build our intuition.

• Then we will graduate to the normal approximation
  to the binomial for some real-world option
  valuation.

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7a The Black-Scholes Model
Where C0 the value of a European option at
time t 0
r the risk-free interest rate.
N(d) Probability that a standardized, normally
distributed, random variable will be less than or
equal to d.
The Black-Scholes Model allows us to value
options in the real world just as we have done in
the 2-state world.
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The Black-Scholes Model

• Find the value of a six-month call option on
  Microsoft with an exercise price of 150.
• The current value of a share of Microsoft is
  160.
• The interest rate available in the U.S. is r
  5.
• The option maturity is 6 months (half of a year).
• The volatility of the underlying asset is 30 per
  annum.
• Before we start, note that the intrinsic value of
  the option is 10our answer must be at least
  that amount.

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The Black-Scholes Model

• Lets try our hand at using the model. If you
  have a calculator handy, follow along.

First calculate d1 and d2
Then,
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The Black-Scholes Model
N(d1) N(0.52815) 0.7013 N(d2) N(0.31602)
0.62401