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
-1-
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
-2-
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
-3-
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
-4-
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
-5-
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)
-6-
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)
VA
VB
S*<X
0
+X
=X
X-S
+S
=X
S*>X
S-X
+X
=S
0
+S
=S
-7-
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
-8-
Put-Call Parity
S,X,t)
(
P
+
S
,t) =
X
(
B
+
)
C(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
-9-
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”
- 10 -
Boundaries on call values
C(S,X,t) + B(X,t) = S + P(S,X,t)
•  Upper Bound:
Call
C(S,X,t) < S
Stock
- 11 -
Boundaries on call values
C(S,X,t) + B(X,t) = S + P(S,X,t)
•  Upper Bound:
•  Lower bound:
Call
C(S,X,t) < S
C(S,X,t) ≥ S – B(X,t)
B(X,t)
Stock
- 12 -
Call values
Call
C(S,X,t) = S - B(X,t) + P(S,X,t)
Stock
B(X,t)
- 23 -
S
C
P
X
C
P
R
C
P
t
C
P
σ
C
P
Call
Keys for using OPT as an analytical tool
C(S,X,t) = S - B(X,t) + P(S,X,t)
B(X,t)
Stock
- 24 -
Basic Option Strategies
•
•
•
•
•
•
•
Long Call
Long Put
Short Call
Short Put
Long Straddle
Short Straddle
Box Spread
- 25 -
Long Call
$
0
-C
X
S
X+C
- 26 -
Long Call
Short Call
$
C
$
0
-C
0
S
X
X+C
X+C
S
X
- 27 -
$
0
-C
$
0
-P
S
X
X+C
Short Call
Long Call
Long Put
$
C
0
X+C
X
S
X-P
X
S
- 28 -
Long Put
$
0
-C
S
X
X+C
$
Short Call
Long Call
Short Put
X
-P
0
X+C
S
X
$
P
X-P
0
$
C
0
S
X
S
X-P
- 29 -
0
-C
S
X
X+C
Long Put
$
X-P
0
X
-P
$
0
-(P+C)
S
Short Call
$
Short Put
Long Call
Long Straddle
$
C
0
X+C
S
X
$
P
0
X
S
X-P
X-P-C
X
S
X+P+C
- 30 -
0
-C
S
X
X+C
X-P
0
X
-P
Long
Straddle
Long Put
$
$
0
-(P+C)
S
$
C
Short Call
$
X+C
0
S
X
$
P
Short Put
Long Call
Short Straddle
0
X
X-P
$
P+C
X-P-C
X+P+C
0
S
X
S
X+P+C
X
X-P-C
S
- 31 -
Box Spread
•  Long call, short put, exercise = X
•  Same as buying a futures contract at X
$
0
X
S
- 32 -
Box Spread
•  Long call, short put, exercise = X
•  Short call, long put, exercise = Z
$
0
Z
X
S
- 33 -
Box Spread
•  You have bought a futures contract at X
•  And sold a futures contract at Z
$
0
Z
X
S
- 34 -
Box Spread
•  Regardless of stock price at expiration
–  you’ll buy for X, sell for Z
–  net outcome is Z - X
$
0
Z
X
S
Z-X
- 35 -
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
Z
X
S
Z-X
- 36 -
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
Z
X
S
Z-X
- 37 -
Box Spread
•  So, building the box brings you
S - B(Z,t) + B(X,t) - S
= B(X,t) - B(Z,t)
$
0
Z
X
S
Z-X
- 38 -
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
- 39 -
Impact of Limited Liability
•  Equity = C(V,D,t)
•  Debt = V - C(V,D,t)
Equity
C(V,D,t) = V - B(D,t) + P(V,D,t)
B(D,t)
V
- 40 -
Swaps
- 41 -
Floating-Fixed Swaps
Illustration of a Floating/Fixed Swap
Party
Variable
Variable
Underwriter
Fixed
Counterparty
Fixed
If net is positive, underwriter pays party. If net is negative, party pays underwriter.
- 42 -
Floating to Floating Swaps
Illustration of a Floating/Floating Swap
T-Bill
Party
T-Bill
Underwriter
LIBOR
Counterparty
LIBOR
If net is positive, underwriter pays party. If net is negative, party pays underwriter
- 43 -
Parallel Loan
Illustration of a parallel loan
United States
Germany
U.S. Parent
Principal
in $
Debt
service
in $
U.S. subsidiary
of German Firm
Loan
guarantees
German Parent
Principal
in Euro
Debt
service in
Euro
German
subsidiary of
U.S. Firm
- 44 -
Currency Swap
Illustration of a straight currency swap
$1,500,000
$1,500,000
1
1
Borrow in
US, invest
in Europe
€ 1,000,000
€ 1,000,000
German rate x €1,000,000
German rate x €1,000,000
2
2
U.S. rate x $1,500,000
U.S. rate x $1,500,000
Borrow in
Europe,
invest in US
€ 1,000,000
€ 1,000,000
3
3
$1,500,000
$1,500,000
Step 1 is notional
Steps 2 & 3 are net
- 45 -
Swaps
Illustration of an Equity Return Swap
Equity Index
Return*
Investor
Underwriter
Libor ± Spread
*Equity index return includes dividends, paid quarterly or reinvested
- 46 -
Swaps
Illustration of an Equity Asset Allocation Swap
Foreign Equity
Index Return* A
Investor
Underwriter
Foreign Equity
Index Return* B
*Equity index return includes dividends, paid quarterly or reinvested
- 47 -
Equity Call Swap
Illustration of an Equity Call Swap
Equity Index Price
Appreciation*
Investor
Underwriter
Libor ± Spread
* No depreciation—settlement at maturity
- 48 -
Equity Asset Swap
Asset
Income Stream
Equity Index
Return*
Investor
Underwriter
Income Stream
* Equity index return includes dividends, paid quarterly or reinvested
- 49 -
Bringing these innovations to the
retail level
- 50 -
1% Coupon
$5mm +
Appreciation
Fixed
BT
PEFCO
SCPERS
$5 mm
Undisclosed
Flow
Counterpary
PENs
Appreciation
Appreciation
- 51 -
Equity Call Swap
Illustration of an Equity Call Swap
Equity Index Price
Appreciation*
Investor
Underwriter
Libor ± Spread
* No depreciation—settlement at maturity
- 52 -
Nikkei Put Warrants (Bringing
Dep
Public Market
Option Premium
At Beginning
Depriciation
At Maturity
Dep
Flow
Kingdom of Denmark
Flow
Goldman Sachs
Counterparty
innovation to retail)
- 53 -
App
App
Flow
Dep
Price
Public Market
Flow
Kingdom of Denmark
$5mm
+ App
Fixed
BT
1%
PEFCO
SCPERS
$5 mm
Goldman Sachs
Alternative Plan
Dep
Volatility
Dynamic Hedge
- 54 -