OPTIONS

1
OPTIONS

• Call Option
• Put Option
• Option premium
• Exercise (striking) price
• Expiration date
• In, out-of, at-the-money options
• American vs European Options

2
Option Valuation

• Valuation of a call option at Expiration
  maxP-X, 0

Vc
P
X
Valuation of a put option at expiration maxX -
P, 0
Vp
P
X
3
Option Valuation (Contd)
Binominal Call Pricing (one period) 70 40 P0
50 45 -10 70 - 50 20 V0 ? 0 70 -
45 25 5 Hedge Ratio 20 - 0
20 4 HR number of calls sold for each stock
bought Buy 1 shr of stock, sell 1.25 calls If
P145, portfolio value 45 If P170,
portfolio value 70 - 20(1.25)45 Return
45/(50-1.25Vc)-1 0.10 Vc 7.27
4
Option Valuation (Contd
Binominal Call Pricing (two periods) P298.00
V248.00 P170.00 V124.55 P263.00
V213.00 P050.00 V011.60 P263.00
V213.00 P145.00 V14.73 P240.50 V
20
5
Option Valuation (Contd
At T1, If P1 70.00 HR (98.00 -
63.00)/(48.00 - 13.00) 1 Buy 1 stock, sell 1
call If P2 98.00 Port. Value 98 - 48 50
P2 63.00 Port. Value 63 - 13 50 1Return
50/(70 - V1) 1.1 V1 24.55 At T1, If P1
40.50 HR (63.00 - 40.50)/(13.00) 1.73 Buy 1
stock, sell 1.73 call If P2 63.00 Port. Value
63 - 1.73x13 40.50 P2 40.50 Port.
Value 40.50 - 0 40.50 1Return 40.50/(40.50
- 1.73V1) 1.1 V1 4.73
6
Option Valuation (Contd
At T0 HR (70.00 - 45.00) / (24.55 - 4.73)
1.26 Buy 1 stock, sell 1.26 call If P1 70.00
Port. value 70 - 1.26x24.55 39.07 P1
45.00 Port. Value 45 - 1.26x4.73
39.07 Return 39.07 / (50 - 1.26V0) 1.1 V0
11.60
7
Black and Scholes OPM
d1 and d2 are deviations from the expected value
of a unit normal distribution. N(d) is the
probability of getting a value below d.
8
Black and Scholes Eg.
P0 50.00 X 50.00 Rf 10 ?0.60 d1
ln(50/50) 0.10 (1/2)0.602 1 / 0.60
0.28 / 0.60 0.4667 d2 0.4667 - 0.60
-0.1333 N(0.4667) 0.6796 N(-0.1333)
0.4470 Vc 50 (0.6796) - 50 e-0.10 (0.4470)
13.76
9
Put-Call Parity
Buy a share at P, sell a call, buy a put at
the same exercise price (X) as call. Value of
Portfolio if PltX PgtX Stock P P call
0 X-P put X-P 0 Portfolio X
X Therefore the value of the portfolio today
must be equal to the PV of X P Vp -VC X/(1
Rf) or Vp Vc X/(1 Rf) - P
10
Option Investment Strategies
Writing covered calls - buy stock, write cals
Synthetic long Buy call, sell put
11
Option Investment Strategies
Straddle simultaneously buying puts and
calls with the same X and t on the same
underlying asset
Long Straddle
Short Straddle
12
Options Delta, Gamma, and Theta
Delta Rate of change in position value in
response to a change in the value of the
underlying asset. Gamma Rate of change in
delta in response to change in the value of the
underlying asset. Theta Change in position
value as time to expiration gets closer (other
things being the same)
delta zero gamma
13
Portfolio Insurance
Investing in a portfolio of stocks and a
put option on the portfolio simultaneously.
The problem is when you cannot find a put option
on your portfolio.
14
Portfolio Insurance Contd
Alternatively one can combine stock portfolio
with the risk free asset to have the same
portfolio insurance, using OPM
N(d1) slope of the call option value. It gives
the fall in position value for a decline of 1 in
stock value. For portfolio insurance, invest 1
-N(d1) in t-bills, and N(d1) in the risky
portfolio. Potential problem