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

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

Properties of Stock Options

Chapter 10

1


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

2


The Black-Scholes Formulas(See page 309)

TdT

TrKSd

T

TrKSd

dNeKdNSc rT

10

2

01

210

)2/2()/ln(

)2/2()/ln( where

)( )(

3

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


First Scenario for the Example: Table 18.2 page 384


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


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

9


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

10


American vs European Options

An American option is worth at least as much as the corresponding European option

C cP p

11


Lower Bound for European Call Option Prices; No Dividends (Equation 10.4, page 238)

c max(S0 – Ke –rT, 0)

12


Lower Bound for European Put Prices; No Dividends (Equation 10.5, page 240)

p max(Ke –rT – S0, 0)

13


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

14

Values of Portfolios


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

15

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



Put-Call Parity Results: Summary

:Futures

:Stock PayingdNondividen

0

0

rTrT

rT

eFpKec

SpKec

17


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

18

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?



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

20

Bounds for European or American Call Options (No Dividends)


Bounds for European and American Put Options (No Dividends)


S0 S0

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

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