Introduction to

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Introduction to OPTIONS S. J. Chang,
Ph.D. College of Business Illinois State
University
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Options Volatility - Diversity -
Uncertainty/Flexibility - Choice - Option
Financial Options, Managerial Options, Real
Options, ... Options are everywhere... Life
is full of options! Yes, options are valuable
and they are your right. But, options do not
come free. Options are contingent and
short-lived. Exercising options is also
costly. So, you have to make decision!
Financial, Marketable Options Calls and Puts A
contingent, derivative security (financial
contract) which gives its holder a right to buy
(call) or sell (put) the underlying asset at a
specified price (strike, exercise price) on or
before the expiration date.
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Why Options? Like futures and other financial
derivatives, options exist mainly for hedging
and speculation purposes To eliminate the
uncertainly about future prices of underlying
assets To realize the earnings
potential based on judgments and expectations
about the price move directions. Buying,
Selling, and Writing Options The act of
originating and selling an option is called
writing. Who writes options? Anyone.
(IBM stock calls and puts are not issued by
IBM) Exercise Buying or selling the
underlying asset according to the
terms of the option. American vs. European
Expiration/maturity/delivery date ... Also,
option price is often referred to as option
premium.
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Underlying Assets for Options Common Stock
More than 1,500 stocks in the U.S. Indexes
SP100 (OEX), SP500 (SPX), DJIA, Oil Index,
NYSECI, Computer Tech Index, Institutional
Index, Gold/Silver Index, Utilities Index, OTC
Index, ... (over 50 indices) Foreign Currencies
PS, DM, JY, FF, Euro, ... Commodities wheat,
barley, coffee, cocoa, sugar, potatoes, gold,
silver, aluminum, lead, nickel, crude oil,
... Futures T-Bond futures, E futures, DM
futures, corn futures, Options Exchanges
Most options are traded on organized exchanges,
although increasingly many option contracts are
traded in OTC markets. CBOE (since 1973) and
PHLX, AMEX, PSE, NYSE in the U.S. EOE, LIFFE,
LME, MATIF, EUREX, SIMEX, TSE, BMF, .. Options
Clearing Corporation --- SEC/CFTC
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More Terminologoies In-the-money,
Out-of-the-money, At/Around/Near-the-money
Covered vs Naked (Call writer does not own the
stock or put writer has no short position in
the stock.) Open Interest Number of
outstanding option contracts Dividend
Protection Usually, exercise price and number of
shares are adjusted for stock splits and stock
dividends. However, while many OTC options are
protected against cash dividends, CBOE options
are not. Exotic Options Asian options,
Bermuda options, barrier options, lookback
options, knock-in/out options, ...
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Intrinsic, Theoretical Value of Options
Immediately Recognizable Value (or option value
at expiration) - value ignoring the value of
the remaining time (floor value). X40,
S50 C50-4010 P0
S30 C0 P40-3010 To the holder C
Max (S-X, 0) P Max (X-S, 0)
Hockey-Stick Lines Limited loss potential and
Unlimited gain potential! I.R.V. ______
_______________ S Question Why do options
always sell at prices greater than 0? As long as
there is time left before expiration, there is
always some chance that the underlying stock
price will move such that the option value will
be greater than today's I.R.V. Remember, no
matter how bad the future stock price will be,
option holder has limited loss. Corollary
The longer the time left, the greater the option
value!
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Put-Call Parity A well-regulated,
no-arbitrage relationship between a European
call and the corresponding put written on a
no-dividend stock. Consider the following two
portfolio strategies 1. A StockPut Portfolio
(buy a stock and a put) Value at expiration
S Max(X-S, 0) Max (X, S) 2. A CallTBill
Portfolio (buy a corresponding call and a X
TBill) Value at expiration Max (S-X, 0) X
Max (S, X) (Graphically, ...?) Since the two
positions give the same value (outcome)
regardless of the stock price at the option's
expiration, their initial costs should be
identical (to preclude arbitrages)! Cost of
Position 1 S P Cost of Position 2 C
X/(1r)t gt C X.e-rt Therefore, S P C
X/(1r)t gt S P C Xe-rt
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Put-Call Parity Beginning of Financial
Engineering! S P C Xe-rt A collection of
related securities (financial products) can be
synthesized (engineered) into a particular type
of security. C S - X.e-rt P Call
Stock Put Borrowing P -S X.e-rt C
Put ShortStock Call Lending S C
X.e-rt - P Stock Call ShortPut
Borrowing X.e-rt S P - C TBill Stock
Put ShortCall Dividend-Adjusted Put-Call
Parity (Dividends paid at T) S P C
(DX)/(1r)t gt S P C (DX)e-rt An
American call on a no-dividend stock should never
be exercised before the expiration (By
exercising early, you give up the time
value) CACE However, American puts may
always be exercised PAgtPE
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No-Arbitrage Valuation of a Call Option (A
Binomial Case) Consider a common stock (S40)
and its at-the-money call with X40 and t1
year. Further suppose r5 and ST in one
year will be either 60 or 20. That means the
call value at the expiration will be 20 if
ST60 and 0 if ST20. Now, for some reason,
you buy 1/2 share of the stock and write one
of these calls. Then, at expiration If
ST60, stock position value30 call position
value -20 net 10 If ST20, stock position
value10 call position value 0 net 10
No-arbitrage condition implies that this hedged
portfolio should yield no more or less than
the market (riskfree) rate Since the Cost is
(20 - C) and the Return is 10, 20 - C
10/(1.05) C 20 - 10/1.05 10.48
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No-Arbitrage (Binomial) Valuation (Example
Contd) Magic Number 1/2 (share) gt Hedge
Ratio Number of shares to be bought and held
per call written in order to produce a
hedged portfolio outcome. Delta (neutral) ?
or ? Essentially, the hedge ratio (?) represents
the sensitivity of call value in response to
stock price changes. ? gt ?C/?S Range of call
value/Range of stock value
(20-0)/(60-20)20/401/2 gt ?C/?S gt
Nd1 (In Black-Scholes OPM) For calls, 0lt?lt1
??1 for in-the-money calls, ?0.5 for
at- the-money calls, and ??0 for
out-of-the-money calls. 1/? Number of calls
to write per share of stock to create a
hedge (?1/2 -gt 1/?2 buy 1 share and write 2
calls)
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Black-Scholes Option Pricing Model A
mathematical model for pricing European options
(1973) Assumptions 1. Lognormally distributed
stock prices 2. Continuous security trading 3.
Frictionless, equilibrium market (no transactions
costs or taxes, perfect divisibility of
securities, no arbitrages) 4. No dividends 5.
Availability of a short-term, constant riskfree
rate When stock prices follow a lognormal
distribution, short-time stock price changes
(returns) follow a normally distributed
stochastic process called a Wiener Process or
Geometric Brownian Motion dS/S ?dt ?dw
dw z?dt (zstandard normal, dt?tshort time
period) gt C S.N(d1) - X.e-rt.N(d2) P
-S.N(-d1) X.e-rt.N(-d2)
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Black-Scholes OPM C S.N(d1) - X.e-rt.N(d2)
P -S.N(-d1) X.e-rt.N(-d2) d1
ln(S/X)(r?2/2)t / ??t d2
ln(S/X)(r-?2/2)t / ??t d1 - ??t N(x)
cumulative probability for a standard normal
variable 0ltN(x)lt1, for a finite value for x
N(0)0.5, N(?)1 Since d2ltd1, N(d2)ltN(d1)
-? ________0_______? d1 ln(S/X)gt0,
when the call is in and the put is out
(SgtX) ln(S/X)0, when the option is at the money
(SX) ln(S/X)lt0, when the call is out and the
put is in (SltX) Typically, -1ltln(S/X)lt1
From the simple put-call parity to the B-S
OPM C S - X.e-rt P ----gt C S.N(d1) -
X.e-rt.N(d2)
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Black-Scholes OPM (Contd) CS.N(d1)-X.e-rt.N(d2
), d1ln(S/X)(r?2/2)t/??t and d2d1-??t
Dividend Adjustment Stock price declines on
the ex-dividend day by the dividend. gt For
option valuation, use the stock price reduced by
the present value of all the dividends
during the option's life. gt With a
(continuous) dividend yield "q," replace S by
Se-qt Then, C S.e-qt.N(d1) - X.e-rt.N(d2)
d1ln(S/X)(r-q?2/2)t/??t and
d2d1-??t Determinants/Ingredients of Option
Valuation C ? (S, X, t, ?, r, D) P ? (S, X,
t, ?, r, D) -
- - -
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Observability/Availability of S, X, t, ?, r, D
gt ? ? ? an ex ante constant volatility
measure of stock returns Ex post, historical
estimate of ? time-dependent, inaccurate
"WISD" (Weighted Implied Standard Deviation)
- Volatility inferred from the actual option
value. - Average of ISD's for outstanding
calls with lower weights on deep in's and
out's Estimated through iterative
process. gt Accurate estimation of volatility
provides a strong trading/speculation motive
... feeds on volatility! Empirical
Applications of Black-Scholes Model Generally
very accurate for at-the-money options. Not very
good for deep in's and out's. Not very good when
t -gt 0. Not very good when ? is too large or too
small.
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Black-Scholes OPM Sensitivity Measures of Call
Value Delta ?(?) ?C/?S?Hedge
RatioN(d1) 0lt?lt1 N(d2)probability that
the call is exercised Gamma ?(?)
?2C/?S2??/?Sgt0 (curvature - convexity)
Delta's sensitivity to changes in stock price
For high ? calls, frequent adjustments are needed
to maintain a ?-neutral position. The
higher ?, the higher the call value to the
holder. Theta ?(?) ?C/?tgt0, ?C/?(-t)lt0
Value Erosion / Time Decay Daily ? gives the
next day call price, other things equal. ? is
the highest for at-the-money options. Vega,
Lambda (?, ?), or Kappa (?, ?) ?C/??gt0 Call
value sensitivity to the underlying
volatility. Highest for at-the-money
options. Declines with the remaining time
until expiration. Rho ?(?) ?C/?rgt0
weak (lt0 for futures options) Delta Neutral,
Gamma Neutral, Vega Neutral Hedges ?-neutral
hedge from small changes in S ?-neutral hedge
from large changes in S ?-neutral hedge from
volatility changes of S
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Portfolio Hedge Using Stock Index Options (An
Example) Hedge your 1 mil stock portfolio
against the possible market decline.
Portfolio Beta2.0, Rf2, DY on the Portfolio
and the Index4, Spot SP100 is 250 now but
declines to 230 in 3 months. Using 3-month OEX
(SP100) puts, you want to hedge the portfolio
for a min. value of 0.9 mil. How many puts to
buy? 1 mil. can be covered by
1,000,000/(250x100)40 puts. However, since the
pfolio is twice as volatile as the market
(?2), you need to buy 40x280 puts. What
strike price? The strike price should be the
index value at which your portfolio value falls
to the minimum desired value (.9mil.). Use a
dividend-adjusted CAPM for the 3-mon period
PortfolioReturn 2 (IndexReturn - 2) x Beta
e.g., If the Index increases by 10 points (4)
from 250 to 260 over the 3-mon
period, the portfolio return (net of dividends)
is 2(4-2)x26. So, the portfolio
value rises from 1 mil. to 1.06 mil.
Minimum desired value is 0.90 mil. A 10 value
decline. -10 2 (IndexReturn-2)x2 gt
Index Return-4 Value will decline to
0.90 mil. when the Index falls by 4 or 10
points to 240. gt X! What happens when the
Index declines to 230 (-8)? P'folioReturn2(-8
-2)x2-18 (-180,000 0.82mil.) Gain on
puts (240-230)x100x8080,000 -gt 0.90 mil.!
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Currency Options (Since 1982, PHLX) A company
due to receive sterling at a future point can
hedge its risk by buying put options on
sterling which mature at that time. This
guarantees that the value of the sterling
will not be less than the strike price, while
allowing the company to benefit from any
appreciation in sterling value. Likewise, a
firm due to pay sterling in the future can buy
call options. "PHLX 31,250 British Pounds 166
JUN Call quoted at 0.58" Contract Size
31,250, Exercise Price 166
31,250x1.6651,875.00 Expiration Month June
(Saturday preceding the 3rd Wed) Call Price
0.58 31,250x0.0058181.25 (Spot 30-day rate
1.6130) A foreign currency is similar to a
stock paying a known dividend. Riskfree
interest rate in foreign currency (f) ? Dividend
yield (q) With Sspot value of 1 unit of the
foreign currency in US, C S.e-ft.N(d1) -
X.e-rt.N(d2) d1ln(S/X)(r-f?2/2)t/??t and
d2d1-??t (rdomestic interest rate, fforeign
interest rate) Recall from the Interest Rate
Parity that the forward exchange rate F (with
the same maturity as the option) is determined
by Cost of Carry domestic interest cost (r)
- foreign interest gain (f) (as you borrow a
domestic fund to invest in a foreign currency).
With FS.e(r-f)t, the currency option valuation
can be rewritten C e-rt.F.N(d1) - X.N(d2)
d1ln(F/X)(?2/2)t/??t and d2d1-??t
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Options on Futures The holder of a futures
call has the right to acquire a long position in
the underlying futures plus a cash amount
equal to the current futures price minus the
exercise price. --- When a futures put is
exercised, the holder acquires a short
position in the underlying futures plus a cash
amount equal to the exercise price minus the
current futures price. Futures options traded
in the U.S. are normally "American" and referred
to by the futures delivery month, not by the
option expiration (futures options generally
expire on or a few days before the futures
delivery date). Commodities, Stock
Indices, Currencies, Interest Rates, ... Why
futures options instead of (spot) options? -
More convenient to make/take delivery of a
futures contract than the asset. - Easier to
obtain price information on futures (for some
assets, futures liquidity is far
greater than spot liquidity). - Options With
futures, you lock yourself in the contract price
(obligation), but with futures options, you
have the right (flexibility) to exercise. - You
can achieve better (closer to perfect) hedging
outcome with futures options than with
spot options. (Black's) Valuation of Futures
Options Treat the underlying futures as a
stock paying a dividend yield of qr
(Futures' risk-neutral return ? zero). Then,
with FFutures Price, C e-rt.F.N(d1) -
X.N(d2) d1ln(F/X)(?2/2)t/??t and
d2d1-??t
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Option Trading Strategies Building Blocks for
Strategies For graphical profiles Calls and
Puts (long and short) hockey-stick lines
Exercise prices deflecting points Unhedged
spot positions straight lines Riskfree
(T-Bill) positions horizontal lines (LongTB
Lending, Short TB Borrowing) Covered Call
Writing I.R.V. (Immediately Recognizable
Value Value at Expiration) Stock
C Call Written Covered Call
X S (Immediate Cash Flow -
Return Potential)
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Straddles A Long (Short) Straddle Buy (sell)
one call and one corresponding put.
I.R.V. (CP5, X50) Call
Put A Long Straddle
X50 40 60 Stock
Price -5 A Short Straddle
-10 Long Straddle S - Wide
move Short Straddle Narrow S range
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Strangles A Long (Short) Strangle Buy (sell)
one out-of-the-money call and one corresponding
out-of-the-money put.
I.R.V. (CP3, S 50, Xc55, Xp45)
Call
Put Long Strangle 39
45 S50 55 61 Stock Price
-3
-6 Long Strangle S - Wide
move Short Strangle Narrow S range
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Butterfly Spreads A Long (Short) Strangle Buy
(sell) one in-the-money call and sell (buy) one
corresponding in-the-money put.
I.R.V. (CP7, S 50, Xc45, Xp55) 7
Stock Price 45 55
-7
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Bullish Money Spread (Augmented) Bullish Money
Spread