Nov 11 – DERIVATIVES
• A DERIVATIVE INSTRUMENT IS ONE IN WHICH THE
PERFORMANCE IS DETERMINED BY THE
PERFORMANCE OF ANOTHER INSTRUMENT
- STOCK OPTIONS
- FUTURES CONTRACTS
- FORWARD CONTRACTS
- SWAPS
OPTIONS
Option Contract - Contract which gives the owner the right to buy
or sell an asset at some determined price within a specified
period of time.
Option Contract - Buyer is the holder and the seller is the writer.
OPTIONS
Call Option - gives the holder the right to buy a given number of shares of a particular
stock at a specified price on or before a given date.
Put Option - Gives the holder the right to sell a given number of shares of stock at a
specified price on or before a given date.
Exercise or Strike Price - The price at which the option holder may buy or sell 100
shares of stock under the option contract.
Expiration Date - Last day on which the option can be exercised.
1
VALUE OF CALL OPTION AT EXPIRATION
V = Max [0, S - X]
V = Price of Call
S = Price of Stock
X = Exercise or Strike Price
V = f(S, X, T, rrf, 2)
T = Time
2 = Variance of Stock Returns
rf = Risk Free Rate
CALL OPTION ON IBM
IBM
Stock
Price STRIKE EXP VOL LAST
178.50 160 Dec 376 25.10
178.50 165 Dec 1763 18
178.50 170 Jan 466 11
178.50 180 Jan 1746 7.50
178.50 190 Jan 885 6.80
Exercise Value at Option Excess Above
Price Price Expiration Price Expiration Value
178.50 160 Dec 18 .50 25.10 6.60
178.50 170 Jan 8.50 11 2.50
2
Ted Westfall was considering the purchase of 100 shares of Stopgap Corporation
common stock selling at $32.40 per share on the last day in October. As an alternative,
Len Griffen, Ted’s neighbor, suggested that Ted consider a Stopgap option instead.
Together they examined the following information that was obtained from their broker.
Exercise
Price Calls Puts
30 6 2
35 3.50 4.75
What are Ted’s profits and rates of return if he makes the following purchases and
subsequently closes his position at expiration given the stock prices as indicated below?
a. A call with an exercise price of 30. The stock ends up at 41.90.
b. A call with an exercise price of 35. The stock ends up at 33.
c. A put with an exercise price of 30. The stock ends up at 37.
d. A put with an exercise price of 35. The stock ends up at 29.
3
OPTION PRICING
The Black Scholes Option Pricing Model rests on the concept of a riskless hedge
Riskless Hedge
By buying the appropriate amount of stock and selling calls, one can insure the same pay
off at expiration of the stock-call portfolio regardless of where the price of the stock
ends up.
At expiration - the range of stock prices
Range Stock Price
Low $30
High $50
Range $20
OPTION PRICING
Assume the exercise price is $35. X = $35
At Expiration
Range Stock Price Value of Call
Low $30 $ 0
High $50 $15
Range $20 $15
The range of payoffs need to be identical to have a perfect hedge.
By buying .75 shares of stock and selling one call, the range of payoffs is equal.
4
OPTION PRICING
Range Stock Price Call Value
Low $22.50 $ 0
High $37.50 $15
Range $15 $15
Portfolio - buy .75 shares of stock and sell one call.
Stock Value of Sell One Final
Range Price .75 Shares Call of Portfolio
Low $30 $22.50 - $0 = $22.50
High $50 $37.50 - $15 = $22.50
$22.50 is the value of the portfolio at expiration if the price of the stock falls anywhere
between $30 & $50.
VALUING STOCK OPTIONS
If there is one year to expiration and the price of the stock equals $40 per share, we could
calculate the value of the call option now.
1. We know the value of the portfolio of long .75 shares of stock and short one call on the
share is worth $22.50 at expiration. Since this is certain then we can discount the
value of the portfolio at the risk free rate. rrf = 8%
2. Value of the portfolio one year from expiration
Value of Portfolio = $22.50 * 1/1.08 = $20.83
3. Call Value = .75(40) - 20.83 = $9.17
5
BLACK SCHOLES OPTION PRICING MODEL
V = S [N(d1)] - X e –r rf *T [N(d2)]
ln(S/X) + [rrf + (2/2)]* T
d1 = -------------------------------------
(T)1/2
d2 = d1 - (T)1/2
V = Current Value of Call
S = Current Price of underlying Stock.
N(di) = Probability that a random draw from a standard normal distribution will be less
than di
X = Exercise Price
rf = Risk Free Rate
T = Time
ln (S/X) = Natural Logarithm of S/X
2 = Variance of the rate of return of the stock
BLACK SCHOLES ASSUMPTIONS
• The stock underlying the call option provides no dividends during the life of the
option.
• no transactions costs
• Risk free rate is known and constant
e = Exponential Function 2.7183
• The call option can only be exercised on its expiration date.
• Assumes that there are no dividends are paid.
• Security trading takes place in continuous time
6
7
Areas for a Standard Normal Distribution
An entry in the table is the area under the curve, between z = O and a positive value of z. Areas for negative values of z are obtained
by symmetry.
Area = Probability
0 z
Second Decimal Place of z
8