- 1 - Basics of Option Pricing Theory & Applications in Business Decision Making Purpose: Provide background on the basics of Option Pricing Theory (OPT)

- 1 -

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

- 14 -

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)

- 16 -

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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Stock

Cal

l

B(X,t)

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Stock

Cal

l

B(X,t)

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Stock

Cal

l

B(X,t)

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Stock

Cal

l

B(X,t)

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Stock

Cal

l

B(X,t)

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Stock

Cal

l

B(X,t)

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Stock

Cal

l

B(X,t)

- 27 -


Stock

Cal

l

B(X,t)

- 28 -


Stock

Cal

l

B(X,t)

- 29 -


Stock

Cal

l

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)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

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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

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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)

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

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Swaps

Investor UnderwriterLibor ± Spread

Equity Index Return*

*Equity index return includes dividends, paid quarterly or reinvested

Illustration of an Equity Return Swap

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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


Illustration of an Equity Call Swap

Equity Index Price Appreciation*

* No depreciation—settlement at maturity

Libor ± Spread

- 65 -

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


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

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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

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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)

SX

$

0Z

Z - X

Documents

- 1 - Basics of Option Pricing Theory & Applications in Business Decision Making Purpose: Provide background on the basics of Option Pricing Theory (OPT)