- 1 - basics of option pricing theory & applications in business decision making purpose:...
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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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Binomial Approach
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DCF only
Augmented
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As the binomial change process runs faster and faster, it approaches something known as Brownian Motion
Let’s have a sneak preview of the Black-Scholes model, using a
similar example
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Illustration using Black-Scholes
Value of 1st year’s option = $1135.45
Value of 2nd year’s 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)
S*<X S*>X
VA
0+X=X
S-X+X=S
VB
X-S+S=X
0+S=S
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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
C(S,X,t) + B(X,t) = S + P(S,X,t)
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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• Stock price falls — “the tail wags the dog”
C(S,X,t) + B(X,t) = S + P(S,X,t)
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Boundaries on call values C(S,X,t) + B(X,t) = S + P(S,X,t)C(S,X,t) + B(X,t) = S + P(S,X,t)
• Upper Bound:C(S,X,t) < S
Stock
Cal
l
- 18 -
Boundaries on call values C(S,X,t) + B(X,t) = S + P(S,X,t)C(S,X,t) + B(X,t) = S + P(S,X,t)
• Upper Bound:C(S,X,t) < S
• Lower bound: C(S,X,t) ≥ S – B(X,t)
Stock
Cal
lB(X,t)
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Call values C(S,X,t) = S - B(X,t) + P(S,X,t)C(S,X,t) = S - B(X,t) + P(S,X,t)
Stock
Cal
l
B(X,t)
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Call values C(S,X,t) = S - B(X,t) + P(S,X,t)C(S,X,t) = S - B(X,t) + P(S,X,t)
Stock
Cal
l
B(X,t)
- 21 -
Call values C(S,X,t) = S - B(X,t) + P(S,X,t)C(S,X,t) = S - B(X,t) + P(S,X,t)
Stock
Cal
l
B(X,t)
- 22 -
Call values C(S,X,t) = S - B(X,t) + P(S,X,t)C(S,X,t) = S - B(X,t) + P(S,X,t)
Stock
Cal
l
B(X,t)
- 23 -
Call values C(S,X,t) = S - B(X,t) + P(S,X,t)C(S,X,t) = S - B(X,t) + P(S,X,t)
Stock
Cal
l
B(X,t)
- 24 -
Call values C(S,X,t) = S - B(X,t) + P(S,X,t)C(S,X,t) = S - B(X,t) + P(S,X,t)
Stock
Cal
l
B(X,t)
- 25 -
Call values C(S,X,t) = S - B(X,t) + P(S,X,t)C(S,X,t) = S - B(X,t) + P(S,X,t)
Stock
Cal
l
B(X,t)
- 26 -
Call values C(S,X,t) = S - B(X,t) + P(S,X,t)C(S,X,t) = S - B(X,t) + P(S,X,t)
Stock
Cal
l
B(X,t)
- 27 -
Call values C(S,X,t) = S - B(X,t) + P(S,X,t)C(S,X,t) = S - B(X,t) + P(S,X,t)
Stock
Cal
l
B(X,t)
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Call values C(S,X,t) = S - B(X,t) + P(S,X,t)C(S,X,t) = S - B(X,t) + P(S,X,t)
Stock
Cal
l
B(X,t)
- 29 -
Call values C(S,X,t) = S - B(X,t) + P(S,X,t)C(S,X,t) = S - B(X,t) + P(S,X,t)
Stock
Cal
l
B(X,t)
- 30 -
Keys for using OPT as an analytical tool C(S,X,t) = S - B(X,t) + P(S,X,t)C(S,X,t) = S - B(X,t) + P(S,X,t)
Stock
Cal
l
B(X,t) Stock
Cal
l
B(X,t)
S C
X C
t C
C
R C
P
P
P
P
P
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Impact of Limited Liability C(V,D,t) = V - B(D,t) + P(V,D,t)
B(D,t) V
Equ
ity
• Equity = C(V,D,t)• Debt = V - C(V,D,t)
- 42 -
Basic Option Strategies
• Long Call
• Long Put
• Short Call
• Short Put
• Long Straddle
• Short Straddle
• Box Spread
- 43 -
Long Call
S
$
0
- CX
X+C
- 44 -
Short Call
S
$
0
- CX
X+CLon
g C
all
XS
$
0X+CC
- 45 -
Long Put
S
$
0
- CX
X+CX
S
$
0X+CC
Lon
g C
all
Sho
rt C
all
S
$
0X
- P
X-P
- 46 -
Short Put
S
$
0
- CX
X+CX
S
$
0X+CC
Lon
g P
utL
ong
Cal
l
Sho
rt C
all
S
$
0X
- P
X-P
S
$
0
P
XX-P
- 47 -
Long Straddle
S
$
0
- CX
X+CX
S
$
0X+CC
Lon
g P
utL
ong
Cal
l
Sho
rt C
all
S
$
0
P
XX-P
S
$
0X
- P
X-P
Sho
rt P
ut
S
$
0X
-(P+C)
X-P-C
X+P+C
- 48 -
Short Straddle
S
$
0
- CX
X+CX
S
$
0X+CC
Lon
g P
utL
ong
Cal
l
Sho
rt C
all
S
$
0
P
XX-P
S
$
0X
- P
X-P
Sho
rt P
ut
S
$
0X
-(P+C)
X-P-C
X+P+C
Lon
gS
trad
dle $
0X
P+C
X-P-C
X+P+C
S
- 49 -
Box Spread
• Long call, short put, exercise = X• Same as buying a futures contract at X
SX
$
0
- 50 -
Box Spread
• Long call, short put, exercise = X• Short call, long put, exercise = Z
SX
$
0Z
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Box Spread
• You have bought a futures contract at X• And sold a futures contract at Z
SX
$
0Z
- 52 -
Box Spread• Regardless of stock price at expiration
– you’ll buy for X, sell for Z– net outcome is Z - X
SX
$
0Z
Z - X
- 53 -
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)
SX
$
0Z
Z - X
- 54 -
Box SpreadBecause 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
SX
$
0Z
Z - X
- 55 -
Box Spread• So, building the box brings you
S - B(Z,t) + B(X,t) - S = B(X,t) - B(Z,t)
SX
$
0Z
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.
SX
$
0Z
Z - X
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Swaps
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Floating-Fixed Swaps
Fixed
If net is positive, underwriter pays party. If net is negative, party pays underwriter.
Illustration of a Floating/Fixed Swap
Party Underwriter CounterpartyVariable
Fixed
Variable
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Floating to Floating Swaps
LIBOR
If net is positive, underwriter pays party. If net is negative, party pays underwriter.
Illustration of a Floating/Floating Swap
Party Underwriter CounterpartyT-Bill
LIBOR
T-Bill
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Parallel Loan
United States Germany
Loan guarantees
Debt service in $
Illustration of a parallel loan
German Parent
U.S. subsidiary of German
Firm
U.S. Parent
German subsidiary of
U.S. Firm
Principal in $
Debt service in Euro
Principal in Euro
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Currency Swap
German rate x €1,000,000
€ 1,000,000
2 2
U.S. rate x $1,250,000
German rate x €1,000,000
U.S. rate x $1,250,000
1 1
€ 1,000,000
$1,250,000$1,250,000
€ 1,000,000
3 3
$1,250,000
€ 1,000,000
$1,250,000
Illustration of a straight currency swap
Step 1 is notionalSteps 2 & 3 are net
Borrow in US, invest in Europe
Borrow in Europe, invest in US
- 62 -
Swaps
Investor UnderwriterLibor ± Spread
Equity Index Return*
*Equity index return includes dividends, paid quarterly or reinvested
Illustration of an Equity Return Swap
- 63 -
Swaps
Investor Underwriter
Foreign Equity Index Return* A
Illustration of an Equity Asset Allocation Swap
*Equity index return includes dividends, paid quarterly or reinvested
Foreign Equity Index Return* B
- 64 -
Equity Call Swap
Investor Underwriter
Illustration of an Equity Call Swap
Equity Index Price Appreciation*
* No depreciation—settlement at maturity
Libor ± Spread
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Equity Asset Swap
Underwriter
Equity Index Return*
* Equity index return includes dividends, paid quarterly or reinvested
Income Stream
Investor
Income
Stream
Asset
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Bringing these innovations to the retail level
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PENsS
CP
ER
S
BT
Cou
nte
rpar
y
PE
FC
O
$5 mm
$5mm + Appreciation
1% Coupon Fixed Undisclosed Flow
AppreciationAppreciation
- 68 -
Equity Call Swap
Investor Underwriter
Illustration of an Equity Call Swap
Equity Index Price Appreciation*
* No depreciation—settlement at maturity
Libor ± Spread
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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)
SX
$
0Z
Z - X
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Box Spread• C(S,Z,t) + B(Z,t) = S + P(S,Z,t)
Now, let’s subtract the bond from each side:• C(S,Z,t) = S + P(S,Z,t) - B(Z,t)
SX
$
0Z
Z - X
- 72 -
Box Spread• C(S,Z,t) = S + P(S,Z,t) - B(Z,t)
Next, let’s subtract the put from each side:• C(S,Z,t) - P(S,Z,t) = S - B(Z,t)
SX
$
0Z
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)
SX
$
0Z
Z - X
- 74 -
Box Spread• So, because of Put/Call Parity, we know:
C(S,Z,t) - P(S,Z,t) = S - B(Z,t)
SX
$
0Z
Z - X
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