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

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

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

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

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

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

Fundamentals of Futures and Options Markets, 8th Ed, Ch 10, Copyright © John C. Hull 2013 16

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?

Fundamentals of Futures and Options Markets, 8th Ed, Ch 10, Copyright © John C. Hull 2013 19

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)

Fundamentals of Futures and Options Markets, 8th Ed, Ch 10, Copyright © John C. Hull 2013 21

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