Lecture 22: Other Derivatives

Option Parameters

• Delta: Partial derivative of option price with respect to underlying price: ∂C/∂S

• Gamma: Second partial derivative of option price with respect to underlying: ∂2C/∂S2

• Theta: Partial derivative of option price with respect to time ∂C/∂T. (Equals minus the partial derivative with respect to time remaining until exercise)

• Vega: Partial derivative of option price with respect to volatility ∂C/∂σ

Exercise Price = 20, r=5%, T=1,sigma=.3

-5

0

5

10

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25

0 5 10 15 20 25 30 35 40 45

Stock Price

Ca

ll P

ric

e

Intrinsic Value of Call

Call Price (Black Scholes)

Option Delta

• Option delta is derivative of option price with respect to stock price

• For calls, if stock price is way below exercise price, delta is nearly zero

• For calls, if option is at the money, delta is roughly a half, but price of option may be way below half the price of the stock.

• For calls, if stock price is way above the exercise price, delta is nearly one and one pays approximately stock price minus pdv of exercise price, like buying stock with credit pdv(E)

Call Delta ∂C/∂S

Volatility of Call Return / Volatility of Stock Return, Exercise Price = 20

0

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0 5 10 15 20 25 30 35 40 45

Stock Price

dln

(ca

ll p

ric

e)/

dln

(sto

ck

pri

ce

)

∂log(C)/∂log(S)

Call Gamma ∂2C/∂S2

Call Theta ∂C/∂T

Behavioral Aspects of Options Demand

• Thaler’s mental categories theory• Writing an out-of-the-money call on a stock one

holds, appears to be a win-win situation (Shefrin)• Buying an option is a way of attaining a more

leveraged, risky position• Lottery principle in psychology, people

inordinately attracted to small probabilities of winning big

• Margin requirements are circumvented by options

Swaps and Risk Management

• Swaps are simple exchanges between parties of one risk for another

• Started in 1981, now in trillions of dollars worldwide

Precursor to Swaps:Parallel Loan Agreements

• Breakdown of Bretton Woods fixed exchange rates in 1973 led to new ways to manage exchange rate risks

• US firm with UK subsidiary lends dollars to a UK firm with US subsidiary. They lend pounds to US firm.

• Hedges exchange rate risk• Long-term, to terms of parties, hence better than

futures market hedging

Problems with Parallel Loan Agreements

• Default Risk: loans are independent instruments, so default by one party does not release other from obligated payments

• Balance sheet impact: parallel loans will inflate the balance sheet, which leads to possible problems with financial covenants, with public perception of safety of their stock

Efforts to Stabilize Earnings

• Currency swings caused major changes in income statements of firms.

• Zeckhauser and Patel show that firms rarely lose earnings, truncated distribution of earnings change

• GE showed steadily growing earnings for last 20 years

Swaps as Parallel Loan Agreements “Stapled Together”

• First privately arranged swaps occurred in mid 1970s

• First public introduction of a currency swap between IBM and World Bank 1981.

Interest Rate Swaps

• Swap fixed for floating

• A bank with a lot of long-term investments and short-term deposits may swap the short-term deposits for long-term

Swaps on Telerate ScreenTreasury-LIBOR Swap

2 Yr. T+70 T+74

3 Yr. T+74 T+77

4 Yr. T+74 T+78

5 Yr. T+74 T+79

7 Yr. T+73 T+79

10 Yr. T+73 T+78

Spiders

MITTs: Market Index Target Term Securities

• Traded on AMEX, often issued by Merrill Lynch• “What if I told you there’s an investment that will

give you no downside but an unlimited upside?”• Example: Five-year MITTs issued in 1992 for $10

pay promise to pay the $10(1+x*1.15) back in five years, where x is percentage increase in the S&P if positive, otherwise zero.

• Downsides: You get no dividends, your floor of $10 is pretty low given that interest rates were around 5% a year in 1992

Futures Options

Why Options on Futures?

• Futures market more liquid, up-to-date, than spot market

• Easier to hedge an options position in futures market than in spot market

Futures on REITs

• August 1998 Chicago Mercantile Exchange announced plan for futures on the S&P Real Estate Trust Composite Index

• Screen-traded rather than open-outcry