February 28, 2008
Option Pricing
Review
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© 2004 South-Western Publishing
Lecture Objectives

Wrap up Greek application
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
Review Option Greeks and Delta application
Review Option valuation
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Delta in Directional and Speed Market
Binomial
BSOPM
Directional Market

Whether we are bullish or bearish indicates
a directional market

Delta measures exposure in a directional
market
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–
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Bullish investors want a positive position delta
Bearish speculators want a negative position
delta
Speed Market

The speed market refers to how quickly we
expect the anticipated market move to
occur
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4
Not a concern to the stock investor but to the
option speculator
Speed Market (cont’d)
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
In fast markets you want positive gammas

In slow markets you want negative gammas
Gamma (cont’d)
 2C  c
c  2 
S
S
 2 P  p
p  2 
S
S
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Gamma (cont’d)
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
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As calls become further in-the-money, they
act increasingly like the stock itself
For out-of-the-money options, option prices
are much less sensitive to changes in the
underlying stock
An option’s delta changes as the stock
price changes
Gamma (cont’d)

Gamma is a measure of how often option
portfolios need to be adjusted as stock
prices change and time passes
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Options with gammas near zero have deltas
that are not particularly sensitive to changes
in the stock price
For a given striking price and expiration,
the call gamma equals the put gamma
Functions and their derivatives

Call price as a function of the following
variables:
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–
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Stock price
Time
Volatility
Interest rate
Striking price constant
Stock price and Delta

Measure of Option Sensitivity to small
changes in the stock
For a call option:

For a put option:

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C
c 
S
P
p 
S
Measure of Option Sensitivity

Delta indicates the number of shares of
stock required to mimic the returns of the
option
–
E.g., a call delta of 0.80 means it will act like 0.80
shares of stock
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If the stock price rises by $1.00, the call option will
advance by about 80 cents
Call and Put deltas
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
For a European option, the absolute values
of the put and call deltas will sum to one

Delta is exactly equal to N(d1)
S, K and delta
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The delta of an at-the-money option
declines linearly over time and approaches
0.50 at expiration
The delta of an out-of-the-money option
approaches zero as time passes
The delta of an in-the-money option
approaches 1.0 as time passes
Hedge Ratio

Delta is the hedge ratio
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Assume a short option position has a delta of
0.25. If someone owns 100 shares of the stock,
writing four calls results in a theoretically
perfect hedge
Likelihood of Becoming In-theMoney

Delta is a crude measure of the likelihood
that a particular option will be in the money
at option expiration
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E.g., a delta of 0.45 indicates approximately a
45 percent chance that the stock price will be
above the option striking price at expiration
Position Derivatives


The position delta is the sum of the deltas
for a particular security
Delta neutrality means the combined deltas
of the options involved in a strategy net out
to zero
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Important to institutional traders who establish
large positions using straddles, strangles, and
ratio spreads
Position Risk
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
Position risk is an important, but often
overlooked, aspect of the riskiness of
portfolio management with options

Option derivatives are not particularly
useful for major movements in the price
of the underlying asset
Importance of risk monitoring
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18
Barings Bank was among the oldest merchant
banking companies in England, having been founded
in 1762 as the 'John and Francis Baring Company' by
Sir Francis Baring. In 1806 his son Alexander Baring
joined the firm and they renamed it Baring Brothers &
Co., merging it with the London offices of Hope & Co.,
where Alexander worked with Henry Hope. It collapsed
in 1995 after one trader, Nick Leeson, lost $1.4 billion
in speculation primarily on futures contracts.
Binomial Pricing
Long stock + Write N calls
Long stock + Buy N Puts
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Price the options if Rf=10%
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
Solve for N (number of calls)

The cost of the portfolio today and value of the
portfolio in one year (40/1.1)
Pricing puts
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
Solve for N (number of puts) (2)

The cost of the portfolio today and value of the
portfolio in one year (50/1.1) P=.23
Risk Neutrality
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For multi-period binomial pricing:
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Build the stock price binomial tree
Calculate the intrinsic value of the option at the end
period (C=S-K; P=K-S)
Find the risk neutral probabilities of the two
branches (U and D use % S1u/S0 and S1d/S0)
The Black-Scholes option pricing
model
C  SN ( d1 )  Ke  RT N ( d 2 )
where
2
 S  
ln 
R


2
K 
d1 
 T
and
d 2  d1  
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T


T

put/call parity

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Writing a call exposes you to risk,
therefore you cover your position with a
long stock. What is the outcome of this
strategy?
……………Synthetic short put…………
Cash Flows and time value
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The Put/Call Parity Relationship

We now know how the call prices, put
prices, the stock price, and the riskless
interest rate are related:
K
C  P  S0 
t
(1  r )
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