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
25
20
Call Price
15
Intrinsic Value of Ca
10
Call Price (Black Sch
5
0
0
5
10
15
20
25
-5
Stock Price
30
35
40
45
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
∂log(C)/∂log(S)
Volatility of Call Return / Volatility of Stock Return, Exercise Price = 20
25
dln(call price)/dln(stock price)
20
15
10
5
0
0
5
10
15
20
25
Stock Price
30
35
40
45
Call Gamma
2
2
∂ C/∂S
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 shortterm deposits for long-term
Swaps on Telerate Screen
Treasury-LIBOR Swap
2 Yr. T+70
3 Yr. T+74
4 Yr. T+74
5 Yr. T+74
7 Yr. T+73
10 Yr. T+73
T+74
T+77
T+78
T+79
T+79
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