Basics of Option Pricing Theory

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Basics of Option Pricing Theory Applications in
Business Decision Making

• Purpose
• Provide background on the basics of Option
  Pricing Theory (OPT)
• Examine some recent applications

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What are options?

• Options are financial contracts whose value is
  contingent upon the value of some underlying
  asset
• Such arrangements are also known as contingent
  claims
• because equilibrium market value of an option
  moves in direct association with the market value
  of its underlying asset.
• OPT measures this linkage

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The basics of options

• Calls and puts defined
• Call privilege of buying the underlying asset at
  a specified price and time
• Put privilege of selling the underlying asset
  at a specified price and time

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The basics of options

• Regional differences
• American options can be exercised anytime before
  expiration date
• European options can be exercised only on the
  expiration date
• Asian options are settled based on average price
  of underlying asset

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The basics of options

• Options may be allowed to expire without
  exercising them
• Options game has a long history
• at least as old as the premium game of 17th
  century Amsterdam
• developed from an even older time game
• which evolved into modern futures markets
• and spawned modern central banks

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Binomial Approach
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As the binomial change process runs faster and
faster, it approaches something known as Brownian
Motion

• Lets have a sneak preview of the Black-Scholes
  model, using a similar example

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Illustration using Black-Scholes
Value of 1st years option 1135.45 Value of
2nd years option 1287.59 NPV 2000
1135.40 1287.59 423.04
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Put-Call Parity

• Consider two portfolios
• Portfolio A contains a call and a bond
• C(S,X,t) B(X,t)
• Portfolio B contains stock plus put
• S P(S,X,t)

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Put-Call Parity

• Consider two portfolios
• Portfolio A contains a call and a bond
• C(S,X,t) B(X,t)
• Portfolio B contains stock plus put
• S P(S,X,t)

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Put-Call Parity

• C(S,X,t) B(X,t) S P(S,X,t)

• News leaks about negative event
• Informed traders sell calls and buy puts

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Put-Call Parity

• News leaks about negative event
• Informed traders sell calls and buy puts
• Arbitrage traders buy the low side and sell the
  high side

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Put-Call Parity
C(S,X,t) B(X,t) S P(S,X,t)

• News leaks about negative event
• Informed traders sell calls and buy puts
• Arbitrage traders buy the low side and sell the
  high side
• Stock price falls the tail wags the dog

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Boundaries on call values C(S,X,t) B(X,t) S
P(S,X,t)

• Upper Bound
• C(S,X,t) lt S

Call
Stock
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Boundaries on call values C(S,X,t) B(X,t) S
P(S,X,t)

• Upper Bound
• C(S,X,t) lt S
• Lower bound
• C(S,X,t) S B(X,t)

Call
Stock
B(X,t)
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Call values C(S,X,t) S - B(X,t) P(S,X,t)
Call
Stock
B(X,t)
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Call values C(S,X,t) S - B(X,t) P(S,X,t)
Call
Stock
B(X,t)
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Call values C(S,X,t) S - B(X,t) P(S,X,t)
Call
Stock
B(X,t)
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Call values C(S,X,t) S - B(X,t) P(S,X,t)
Call
Stock
B(X,t)
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Call values C(S,X,t) S - B(X,t) P(S,X,t)
Call
Stock
B(X,t)
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Call values C(S,X,t) S - B(X,t) P(S,X,t)
Call
Stock
B(X,t)
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Call values C(S,X,t) S - B(X,t) P(S,X,t)
Call
Stock
B(X,t)
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Call values C(S,X,t) S - B(X,t) P(S,X,t)
Call
Stock
B(X,t)
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Call values C(S,X,t) S - B(X,t) P(S,X,t)
Call
Stock
B(X,t)
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Call values C(S,X,t) S - B(X,t) P(S,X,t)
Call
Stock
B(X,t)
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Call values C(S,X,t) S - B(X,t) P(S,X,t)
Call
Stock
B(X,t)
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Keys for using OPT as an analytical tool
C(S,X,t) S - B(X,t) P(S,X,t)
Call
Call
Stock
B(X,t)
Stock
B(X,t)
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Impact of Limited Liability C(V,D,t) V -
B(D,t) P(V,D,t)

• Equity C(V,D,t)
• Debt V - C(V,D,t)

Equity
B(D,t)
V
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Basic Option Strategies

• Long Call
• Long Put
• Short Call
• Short Put
• Long Straddle
• Short Straddle
• Box Spread

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Long Call

0
S
X
XC
- C
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Short Call


C
XC
Long Call
0
S
0
S
X
X
XC
- C
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Long Put


C
XC
Short Call
Long Call
0
S
0
S
X
X
- C
XC

X-P
0
S
X
- P
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Short Put


C
XC
Short Call
Long Call
0
S
0
S
X
X
- C
XC


X-P
P
0
Long Put
S
0
S
X
X
- P
X-P
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Long Straddle


C
XC
Short Call
Long Call
0
S
0
S
X
X
- C
XC


X-P
P
Short Put
0
Long Put
S
0
S
X
X
- P
X-P

X-P-C
0
S
X
-(PC)
XPC
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Short Straddle


C
XC
Short Call
Long Call
0
S
0
S
X
X
- C
XC


X-P
P
Short Put
0
Long Put
S
0
S
X
X
- P
X-P


PC
X-P-C
Long Straddle
XPC
0
0
S
S
X
X
-(PC)
XPC
X-P-C
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Box Spread

• Long call, short put, exercise X
• Same as buying a futures contract at X

0
S
X
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Box Spread

• Long call, short put, exercise X
• Short call, long put, exercise Z

0
S
X
Z
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Box Spread

• You have bought a futures contract at X
• And sold a futures contract at Z

0
S
X
Z
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Box Spread

• Regardless of stock price at expiration
• youll buy for X, sell for Z
• net outcome is Z - X

0
S
X
Z
Z - X
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Box Spread

• How much did you receive at the outset?
• C(S,Z,t) - P(S,Z,t)
• - C(S,X,t) P(S,X,t)

0
S
X
Z
Z - X
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Box Spread

• Because of Put/Call Parity, we know
• C(S,Z,t) - P(S,Z,t) S - B(Z,t)
• - C(S,X,t) P(S,X,t) B(X,t) - S

0
S
X
Z
Z - X
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Box Spread

• So, building the box brings you
• S - B(Z,t) B(X,t) - S
• B(X,t) - B(Z,t)

0
S
X
Z
Z - X
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Assessment of the Box Spread

• At time zero, receive PV of X-Z
• At expiration, pay Z-X
• You have borrowed at the T-bill rate.

0
S
X
Z
Z - X
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Swaps
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Floating-Fixed Swaps
Illustration of a Floating/Fixed Swap
If net is positive, underwriter pays party. If
net is negative, party pays underwriter.
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Floating to Floating Swaps
Illustration of a Floating/Floating Swap
If net is positive, underwriter pays party. If
net is negative, party pays underwriter.
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Parallel Loan
Illustration of a parallel loan
United States
Germany
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Currency Swap
Illustration of a straight currency swap
Step 1 is notional Steps 2 3 are net
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Swaps
Illustration of an Equity Return Swap
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Swaps
Illustration of an Equity Asset Allocation Swap
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Equity Call Swap
Illustration of an Equity Call Swap
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Equity Asset Swap
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Bringing these innovations to the retail level
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PENs
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Equity Call Swap
Illustration of an Equity Call Swap
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Box Spread

• Because of Put/Call Parity, we know
• C(S,Z,t) B(Z,t) S P(S,Z,t)

0
S
X
Z
Z - X
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Box Spread

• C(S,Z,t) B(Z,t) S P(S,Z,t)
• Now, lets subtract the bond from each side
• C(S,Z,t) S P(S,Z,t) - B(Z,t)

0
S
X
Z
Z - X
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Box Spread

• C(S,Z,t) S P(S,Z,t) - B(Z,t)
• Next, lets subtract the put from each side
• C(S,Z,t) - P(S,Z,t) S - B(Z,t)

0
S
X
Z
Z - X
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Box Spread

• C(S,Z,t) - P(S,Z,t) S - B(Z,t)
• Given this, we also know
• - C(S,X,t) P(S,X,t) - S B(X,t)

0
S
X
Z
Z - X
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Box Spread

• So, because of Put/Call Parity, we know
• C(S,Z,t) - P(S,Z,t) S - B(Z,t)

0
S
X
Z
Z - X