- 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

- 2 - 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

- 3 - 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

- 4 - 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

- 5 - 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

- 6 - Binomial Approach

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- 8 - 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

Illustration using Black-Scholes Value of 1st year’s option = $ Value of 2nd year’s option = $ NPV = – = $423.04

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)

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)

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

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)

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)

C(S,X,t) + B(X,t) = S + P(S,X,t) Boundaries on call values C(S,X,t) + B(X,t) = S + P(S,X,t) Upper Bound: C(S,X,t) < S Stock Call

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

C(S,X,t) = S - B(X,t) + P(S,X,t) Call values C(S,X,t) = S - B(X,t) + P(S,X,t) Stock Call B(X,t)

C(S,X,t) = S - B(X,t) + P(S,X,t) Call values C(S,X,t) = S - B(X,t) + P(S,X,t) Stock Call B(X,t)

C(S,X,t) = S - B(X,t) + P(S,X,t) Call values C(S,X,t) = S - B(X,t) + P(S,X,t) Stock Call B(X,t)

C(S,X,t) = S - B(X,t) + P(S,X,t) Call values C(S,X,t) = S - B(X,t) + P(S,X,t) Stock Call B(X,t)

C(S,X,t) = S - B(X,t) + P(S,X,t) Call values C(S,X,t) = S - B(X,t) + P(S,X,t) Stock Call B(X,t)

C(S,X,t) = S - B(X,t) + P(S,X,t) Call values C(S,X,t) = S - B(X,t) + P(S,X,t) Stock Call B(X,t)

C(S,X,t) = S - B(X,t) + P(S,X,t) Call values C(S,X,t) = S - B(X,t) + P(S,X,t) Stock Call B(X,t)

C(S,X,t) = S - B(X,t) + P(S,X,t) Call values C(S,X,t) = S - B(X,t) + P(S,X,t) Stock Call B(X,t)

C(S,X,t) = S - B(X,t) + P(S,X,t) Call values C(S,X,t) = S - B(X,t) + P(S,X,t) Stock Call B(X,t)

C(S,X,t) = S - B(X,t) + P(S,X,t) Call values C(S,X,t) = S - B(X,t) + P(S,X,t) Stock Call B(X,t)

C(S,X,t) = S - B(X,t) + P(S,X,t) Call values C(S,X,t) = S - B(X,t) + P(S,X,t) Stock Call B(X,t)

C(S,X,t) = S - B(X,t) + P(S,X,t) Keys for using OPT as an analytical tool C(S,X,t) = S - B(X,t) + P(S,X,t) Stock Call B(X,t) Stock Call 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 Equity Equity = C(V,D,t) Debt = V - C(V,D,t)

Basic Option Strategies Long Call Long Put Short Call Short Put Long Straddle Short Straddle Box Spread

Long Call S $ 0 - C X X+C

Short Call S $ 0 - C X X+C Long Call X S $ 0 X+C C

Long Put S $ 0 - C X X+C X S $ 0 C Long Call Short Call S $ 0 X - P X-P

Short Put S $ 0 - C X X+C X S $ 0 C Long Put Long Call Short Call S $ 0 X - P X-P S $ 0 P X

Long Straddle S $ 0 - C X X+C X S $ 0 C Long Put Long Call Short Call S $ 0 P X X-P S $ 0 X - P X-P Short Put S $ 0 X -(P+C) X-P-C X+P+C

Short Straddle S $ 0 - C X X+C X S $ 0 C Long Put Long Call Short Call S $ 0 P X X-P S $ 0 X - P X-P Short Put S $ 0 X -(P+C) X-P-C X+P+C Long Straddle $ 0 X P+C X-P-C X+P+C S

Box Spread Long call, short put, exercise = X Same as buying a futures contract at X S X $ 0

Box Spread Long call, short put, exercise = X Short call, long put, exercise = Z S X $ 0 Z

Box Spread You have bought a futures contract at X And sold a futures contract at Z S X $ 0 Z

Box Spread Regardless of stock price at expiration –you’ll buy for X, sell for Z –net outcome is Z - X S X $ 0 Z Z - X

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) S X $ 0 Z Z - X

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 S X $ 0 Z Z - X

Box Spread So, building the box brings you S - B(Z,t) + B(X,t) - S = B(X,t) - B(Z,t) S X $ 0 Z Z - X

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. S X $ 0 Z Z - X

Swaps

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 Counterparty Variable Fixed Variable

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 Counterparty T-Bill LIBOR T-Bill

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

Currency Swap German rate x €1,000,000 € 1,000, U.S. rate x $1,250,000 German rate x €1,000,000 U.S. rate x $1,250, € 1,000,000 $1,250,000 € 1,000, $1,250,000 € 1,000,000 $1,250,000 Illustration of a straight currency swap Step 1 is notional Steps 2 & 3 are net Borrow in US, invest in Europe Borrow in Europe, invest in US

Swaps Investor Underwriter Libor ± Spread Equity Index Return* *Equity index return includes dividends, paid quarterly or reinvested Illustration of an Equity Return Swap

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

Equity Call Swap Investor Underwriter Illustration of an Equity Call Swap Equity Index Price Appreciation* * No depreciation—settlement at maturity Libor ± Spread

Equity Asset Swap Underwriter Equity Index Return* * Equity index return includes dividends, paid quarterly or reinvested Income Stream Investor Income Stream Asset

Bringing these innovations to the retail level

PENs SCPERS BT Counterpary PEFCO $5 mm $5mm + Appreciation 1% Coupon Fixed Undisclosed Flow Appreciation

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) S X $ 0 Z Z - X

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) S X $ 0 Z Z - X

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) S X $ 0 Z Z - X

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) S X $ 0 Z Z - X

Box Spread So, because of Put/Call Parity, we know: C(S,Z,t) - P(S,Z,t) = S - B(Z,t) S X $ 0 Z Z - X