fundamentals of futures and options markets, 8th ed, ch 10, copyright © john c. hull 2013...
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Fundamentals of Futures and Options Markets, 8th Ed, Ch 10, Copyright © John C. Hull 2013
Properties of Stock Options
Chapter 10
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Fundamentals of Futures and Options Markets, 8th Ed, Ch 13, Copyright © John C. Hull 2013
The Concepts Underlying Black-Scholes
The option price and the stock price depend on the same underlying source of uncertainty
We can form a portfolio consisting of the stock and the option which eliminates this source of uncertainty
This involves maintaining a delta neutral portfolio
The portfolio is instantaneously riskless and must instantaneously earn the risk-free rate
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Fundamentals of Futures and Options Markets, 8th Ed, Ch 13, Copyright © John C. Hull 2013
The Black-Scholes Formulas(See page 309)
TdT
TrKSd
T
TrKSd
dNeKdNSc rT
10
2
01
210
)2/2()/ln(
)2/2()/ln( where
)( )(
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The N(x) Function
N(x) is the probability that a normally distributed variable with a mean of zero and a standard deviation of 1 is less than x
See tables at the end of the bookFundamentals of Futures and Options Markets, 8th Ed, Ch 13, Copyright © John C. Hull 2013 4
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The Probabilities The delta of a European call on a
non-dividend paying stock is N (d 1)
The variable N (d 2) is the probability of exercise
0.72*10-0.6(10) =1.2 Same as (12-10)*0.6
Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 5
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The Costs in Delta Hedgingcontinued
Delta hedging a written option involves a “buy high, sell low” trading rule
Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 6
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First Scenario for the Example: Table 18.2 page 384
Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 7
Week Stock price
Delta Shares purchased
Cost (‘$000)
CumulativeCost ($000)
Interest
0 49.00 0.522 52,200 2,557.8 2,557.8 2.5
1 48.12 0.458 (6,400) (308.0) 2,252.3 2.2
2 47.37 0.400 (5,800) (274.7) 1,979.8 1.9
....... ....... ....... ....... ....... ....... .......
19 55.87 1.000 1,000 55.9 5,258.2 5.1
20 57.25 1.000 0 0 5263.3
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Properties of Black-Scholes Formula
As S0 becomes very large c tends to S0 – Ke-rT and p tends to zero
As S0 becomes very small c tends to zero and p tends to Ke-rT – S0
What happens as s becomes very large? What happens as T becomes very large?
Fundamentals of Futures and Options Markets, 8th Ed, Ch 13, Copyright © John C. Hull 2013 8
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Fundamentals of Futures and Options Markets, 8th Ed, Ch 13, Copyright © John C. Hull 2013
Risk-Neutral Valuation
The variable m (expected appreciation in the stock value) does not appear in the Black-Scholes equation
The equation is independent of all variables affected by risk preference
This is consistent with the risk-neutral valuation principle
When we use Monte Carlo to value an option we typically assume risk neutrality
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Fundamentals of Futures and Options Markets, 8th Ed, Ch 13, Copyright © John C. Hull 2013
Applying Risk-Neutral Valuation
1. Assume that the expected return from an asset is the risk-free rate
2. Calculate the expected payoff from the derivative
3. Discount at the risk-free rate
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Fundamentals of Futures and Options Markets, 8th Ed, Ch 10, Copyright © John C. Hull 2013
American vs European Options
An American option is worth at least as much as the corresponding European option
C cP p
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Fundamentals of Futures and Options Markets, 8th Ed, Ch 10, Copyright © John C. Hull 2013
Lower Bound for European Call Option Prices; No Dividends (Equation 10.4, page 238)
c max(S0 – Ke –rT, 0)
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Fundamentals of Futures and Options Markets, 8th Ed, Ch 10, Copyright © John C. Hull 2013
Lower Bound for European Put Prices; No Dividends (Equation 10.5, page 240)
p max(Ke –rT – S0, 0)
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Fundamentals of Futures and Options Markets, 8th Ed, Ch 10, Copyright © John C. Hull 2013
Put-Call Parity; No Dividends
Consider the following 2 portfolios: Portfolio A: European call on a stock + zero-
coupon bond that pays K at time T Portfolio C: European put on the stock + the stock
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Values of Portfolios
Fundamentals of Futures and Options Markets, 8th Ed, Ch 10, Copyright © John C. Hull 2013
ST > K ST < K
Portfolio A Call option ST − K 0
Zero-coupon bond K K
Total ST K
Portfolio C Put Option 0 K− ST
Share ST ST
Total ST K
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The Put-Call Parity Result (Equation 10.6, page 241)
Both are worth max(ST , K ) at the maturity of the options
They must therefore be worth the same today. This means that
c + Ke -rT = p + S0
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Fundamentals of Futures and Options Markets, 8th Ed, Ch 16, Copyright © John C. Hull 2013
Put-Call Parity Results: Summary
:Futures
:Stock PayingdNondividen
0
0
rTrT
rT
eFpKec
SpKec
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Fundamentals of Futures and Options Markets, 8th Ed, Ch 10, Copyright © John C. Hull 2013
Early Exercise
Usually there is some chance that an American option will be exercised early
An exception is an American call on a non-dividend paying stock, which should never be exercised early
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An Extreme Situation
For an American call option:
S0 = 100; T = 0.25; K = 60; D = 0Should you exercise immediately?
What should you do if You want to hold the stock for the next 3
months? You do not feel that the stock is worth holding
for the next 3 months?
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Fundamentals of Futures and Options Markets, 8th Ed, Ch 10, Copyright © John C. Hull 2013
Reasons For Not Exercising a Call Early (No Dividends)
No income is sacrificed You delay paying the strike price Holding the call provides
insurance against stock price falling below strike price
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Bounds for European or American Call Options (No Dividends)
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Bounds for European and American Put Options (No Dividends)
Fundamentals of Futures and Options Markets, 8th Ed, Ch 10, Copyright © John C. Hull 2013 22
S0 S0