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

2. 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
Fundamentals of Futures and Options Markets, 8th Ed, Ch 13, Copyright © John C. Hull 2013

3. The Black-Scholes Formulas (See page 309)
Fundamentals of Futures and Options Markets, 8th Ed, Ch 13, Copyright © John C. Hull 2013

4. 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 book
Fundamentals of Futures and Options Markets, 8th Ed, Ch 13, Copyright © John C. Hull 2013

5. 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

6. The Costs in Delta Hedging continued
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

7. First Scenario for the Example: Table 18.2 page 384
Week
Stock price
Delta
Shares purchased
Cost (‘$000)
Cumulative
Cost ($000)
Interest
49.00
0.522
52,200
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
5263.3
Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012

8. 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

9. 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
Fundamentals of Futures and Options Markets, 8th Ed, Ch 13, Copyright © John C. Hull 2013

10. Applying Risk-Neutral Valuation
Assume that the expected return from an asset is the risk-free rate
Calculate the expected payoff from the derivative
Discount at the risk-free rate
Fundamentals of Futures and Options Markets, 8th Ed, Ch 13, Copyright © John C. Hull 2013

11. American vs European Options
An American option is worth at least as much as the corresponding European option
C  c
P  p
Fundamentals of Futures and Options Markets, 8th Ed, Ch 10, Copyright © John C. Hull 2013

12. Lower Bound for European Call Option Prices; No Dividends (Equation 10
Lower Bound for European Call Option Prices; No Dividends (Equation 10.4, page 238)
c  max(S0 – Ke –rT, 0)
Fundamentals of Futures and Options Markets, 8th Ed, Ch 10, Copyright © John C. Hull 2013

13. Lower Bound for European Put Prices; No Dividends (Equation 10
Lower Bound for European Put Prices; No Dividends (Equation 10.5, page 240)
p  max(Ke –rT – S0, 0)
Fundamentals of Futures and Options Markets, 8th Ed, Ch 10, Copyright © John C. Hull 2013

14. 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
Fundamentals of Futures and Options Markets, 8th Ed, Ch 10, Copyright © John C. Hull 2013

15. Values of Portfolios ST > K ST < K Portfolio A Call option
Zero-coupon bond K
Total ST
Portfolio C
Put Option
K− ST
Share
Fundamentals of Futures and Options Markets, 8th Ed, Ch 10, Copyright © John C. Hull 2013

16. 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

17. Put-Call Parity Results: Summary
Fundamentals of Futures and Options Markets, 8th Ed, Ch 16, Copyright © John C. Hull 2013

18. 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
Fundamentals of Futures and Options Markets, 8th Ed, Ch 10, Copyright © John C. Hull 2013

19. An Extreme Situation For an American call option:
S0 = 100; T = 0.25; K = 60; D = 0
Should 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

20. 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
Fundamentals of Futures and Options Markets, 8th Ed, Ch 10, Copyright © John C. Hull 2013

21. Bounds for European or American Call Options (No Dividends)
Fundamentals of Futures and Options Markets, 8th Ed, Ch 10, Copyright © John C. Hull 2013

22. Bounds for European and American Put Options (No Dividends)
Fundamentals of Futures and Options Markets, 8th Ed, Ch 10, Copyright © John C. Hull 2013