Options, continued

1
Options, continued

• March 10, 2004

2
Put-Call Parity

• Consider the payoff on a portfolio consisting of
  a call plus the (present value of the) exercise
  price
• At the exercise date of the call, the payoff
  equals the exercise price of the call or the
  price of the stock, whichever is greater.
• Were assuming European options here (no early
  exercise)

3
(No Transcript)
4
(No Transcript)
5

• Now consider a portfolio consisting of a holding
  of the stock, plus a put (with the same exercise
  price and exercise date as the call).

6
(No Transcript)
7
(No Transcript)
8

• The payoffs (orange line) are the same!
• Since the payoffs on the two portfolios are the
  same, so must be their current values.
• (Otherwise there would exist an arbitrage be
  sure you understand how you would set up the
  arbitrage to exploit a discrepancy).

9

• Put-call parity, applied to current values
• Call premium PV of the exercise price
• Put premium current stock price.
• Implication if you can value a call, you can
  value a put via put-call parity.

10
Option Valuation

• value (price) intrinsic value time value
• intrinsic value value if exercised now
• time value whatever is left over ...
• Time value cant be negative, since otherwise you
  would throw away an out of the money option, and
  exercise an in the money option.

11
(No Transcript)
12
Bounds on option value

• obvious the value of a call is (1) always
  positive (2) always less than value of stock (3)
  always greater than or equal to its intrinsic
  value
• Less obvious call value gt stock price - present
  value of exercise price.

13
A lower bound on call value
14
Derivation via Put-Call Parity

• You can derive this bound via put-call parity.
• Claim C gt S - PVE,
• or C - S PVE gt 0
• Put-call parity C - S PVE P,
• so the claim reduces to P gt 0.
• The current price of a put is always positive.

15
Direct derivation

• If C gt S - PVE werent satisfied, there would be
  an arbitrage
• Suppose C lt S - PVE.
• (1) buy the call, and (2) invest the present
  value of the exercise price, and (3) short the
  stock.
• This generates a positive amount of cash now.

16

• If the call matures in-the-money, exercise it and
  cover the short position. Zero cash at the
  exercise date.
• If the call matures out-of-the-money, throw it
  away. Use cash to close out the short position.
  Positive cash at the exercise date.
• So its an arbitrage profitable no matter what
  happens to the stock price!

17
Major determinants of option value

• exercise price -- the higher the exercise price,
  the lower (higher) the value of a call (put)
• exercise date -- the farther in the future is the
  exercise date, the greater is the value of either
  a call or a put
• stock price volatility -- the greater the stock
  price volatility, the greater the value of either
  a call or a put

18
Early exercise of options

• Is it ever optimal to exercise a call (in the
  case of an American option) early?
• No (on non-dividend-paying stock).
• Conclusion an American call is effectively the
  same as a European call.
• In particular, they have the same price.

19

• However, this isnt necessarily true if the stock
  pays a dividend (since the holder of an
  unexercised call doesnt get the dividend).
• Also, its not necessarily true of puts.
• Well discuss this later. Now lets think about
  calls on non-dividend-paying stock

20

• Heres why its never optimal to exercise a call
  on a (non-dividend-paying) stock Suppose the
  call is in the money (otherwise you certainly
  dont want to exercise it).
• Assume the price of the stock now is S0, and that
  the price of the stock at the exercise date is ST
  (not known now, of course).

21

• If you exercise now, the current value of your
  portfolio is S0 - X.
• If you hold the stock, the subsequent change is
  ST - S0.
• Add these together. The sum equals ST - X
• By not exercising the call, you get
• max (ST - X, 0), which is at least as good.

22

• So if you exercise the call now and the stock
  subsequently drops below the exercise price,
  youll wish you hadnt exercised.
• This is the basis for the time value of a call
  the call is worth more unexercised because that
  way youre protected against the price dropping
  below the exercise price.

23
Suppose you exercise and sell the stock

• You might think the preceding exercise isnt
  realistic, since you could lock in a profit by
  exercising the call now and selling the stock,
  whereas if you hold the call the stock price
  might drop, in which case the call might expire
  out of the money

24

• This makes 2 changes at once exercising the
  call now and selling the stock now,
• vs. exercising the call (or not exercising it, if
  its out of the money) at the exercise date and
  selling the stock then.

25
Instead

• You want to compare apples to apples.
• Instead, compare
• (a) early exercise and sale of stock now with
• (b) exercise at the exercise date plus a short
  sale now.
• Delaying exercise plus an immediate short sale
  dominates exercise now plus selling the stock now

26
(No Transcript)
27
However

• If the stock pays dividends before the exercise
  date, you might want to exercise early so as to
  get the dividend.
• Most simply, think of a liquidating dividend.
• With puts, if you exercise early you get paid, so
  the time value of money suggests early exercise.

28
Black-Scholes Model

• example
• S0 100
• X 95
• r .1
• T .25
• s .5

29
Black-Scholes formula
30

• N(x) is the value of the cumulative normal
  distribution evaluated at x.
• This is available in Excel under the function
  Normsdist(x)
• (Hit fx button, choose statistical functions)
• Black-Scholes value of call 13.70