# FECLecture4 - Financial Engineering Club at Illinois

```FEC
FINANCIAL ENGINEERING CLUB
MORE ON OPTIONS
AGENDA
 Put-Call Parity
 Combination of options
REVIEW
 Option - a contract sold by one party (option writer) to another party (option
holder). The contract offers the buyer the right, but not the obligation, to buy
(call) or sell (put) a security or other financial asset at an agreed-upon price (the
strike price) during a certain period of time or on a specific date (exercise date).
Call options give the option to buy at certain price, so the buyer would want the
stock to go up.
 Ex: Groupon
Put options give the option to sell at a certain price, so the buyer would want the
stock to go down.
 Ex: Auto Insurance Policy
WHY USE OPTIONS?
Versatility
 Make profit when market goes up or down
Hedging
 Limit any losses in your investments
DIFFERENT TYPES OF PURCHASES
Portfolio
Cash
Cash
Net Cost as
Outflows at Outflows at of Time 0
Time 0
Time T
1. Outright
Purchase
0
0
0,
2. Long
--Forward
with
Forward
Price F(0,T)
3. Synthetic Call(K,T) –
Forward
Put(K,T)
PV(0, )
 On the left is a table where the net cost
at time ‘0.’ The cash flows occur only at
time 0 and time T.
 Note that all of those three portfolios
end up giving you a share of stock at
time T
 If you just rearrange the variables, you
will get the same formula every time.
K
Call(K,T) –
Put(K,T)
+PV(K)
PORTFOLIOS 1 AND 2
0 = PV(0, )
The price of the stock So is equal to the Present value of the forward price.
If this is the case, then
0, = FV(0 )
The forward price is equal to the current price of the stock.
*All of the stocks have to be non dividend paying stocks
Portfolio
1
2
Cash outflow at
time 0
Cash outflow at
time t
Net cost as of time
0
0
0
0,
PV(0, )
PORTFOLIOS 2 AND 3
PV(0, ) = Call(K,T) – Put(K,T) + PV(K)
Subtract PV(K) on both sides.
Call(K,T) – Put(K,T) = PV(0, )– PV(K)
We can simply combine the present values together
Call(K,T) – Put(K,T) = PV(0, – K)
Portfolio
Cash outflow at
time 0
Cash outflow at
time t
0,
2
3
Call(K,T) – Put(K,T)
K
Net cost as of time
0
PV(0, )
Call(K,T) – Put(K,T)
+PV(K)
ANALYSIS OF PORTFOLIO 2 AND 3
Original Equation: PV(0, )= Call(K,T) – Put(K,T) + PV(K)
You have to buy a share of stock at time T with either portfolio upfront.
Under portfolio 2, you pay nothing upfront but under portfolio 3, you have to pay
Portfolio
Outflow at time 0
2
---
3
Call(K,T) – Put(K,T)
Outflow at time T
0,
K
PORTFOLIOS 1 AND 3
1. 0 = Call(K,T) – Put(K,T) + PV(K)
Rearrange it to have:
2. Call(K,T) – Put(K,T) = 0 –PV(K)
Substitute S0 with PV(F(0,T)) from Portfolio 1 and 2
3. Call(K,T) – Put(K,T) = PV(0, -K)
THE PUT-CALL PARITY
Call(K,T) – Put(K,T) = 0 –PV(K)
The net cost of buying an asset on a future date should be the same.
If it wasn’t the case, you can make a lot of money buying an asset at a lower cost
and selling it at a higher cost.
If there were two forward contracts based on the same asset and having the same
expiration date available, one with forward price of \$100 and the other with \$104,
you would earn \$4. You would make profit on a no-risk basis. This opportunity is
called arbitrage.
Therefore, we must assume there is a no-arbitrage pricing.
 A static price relationship between the prices of European put and call
options of the same class.
 These option and stock positions must all have the same return or else an
arbitrage opportunity would be available to traders.
 Any option pricing model that produces put and call prices that don't
satisfy put-call parity should be rejected as unsound because arbitrage
opportunities exist.
ALL THE SAME
 Last Lecture: − = − , , , 1 +  , , , 1 − ()
 This Lecture: Call(K,T) – Put(K,T) = 0 –PV(K)
In summary, the equation provides a simple test for
various option pricing models. If you cannot produce
the put-call parity equation, then the option model
presented is flawed.
PUT-CALL PARITY EXAMPLE
 Given the following information:

Forward price = \$163.13

150-strike European call premium = \$23.86

150-strike European put premium = \$11.79
 The risk-free annual effective rate of interest is X. Determine X.
 Call(K,T) – Put(K,T) = PV(F0,T - K)
 23.86 – 11.79 =
 i = 8.78%
1
* (163.13 – 150)
1+
COMBINING OPTIONS
 Payoff graphs for four basic positions
Profit
Price
= ( − )+ = max{ − , 0},
= ( − )+ = max{ − , 0}
profit
 Favor both sides of an issue at once
 Combination of an at-the-money put and an atthe-money call
STRANGLE
 Similar as straddle, but at lower financing cost
 Combination of an out-of-the-money put and
an out-of-the-money call
 Combination of a written straddle (a short call + a short put)
and an out-of-the-money long put + an out-of-the-money
long call (i.e. a strangle)
 Make a profit if the price doesn’t change very much
 Provide insurance for big price changes
105
105
90-105-110
 The weights of the long put and the long call are determined by the location of the peak
 Example:
 A 105-strike written call
 Buy 0.25 units of a 90-strike call and 0.75 units of a 110-strike call for each unit of the 105strike call that you write
 Buy a call and sell it at a higher price, or but a put and sell it at a higher price
 You think the price will increase
100
100-strike short call
 We think the price will decline
 A mirror image of bull spread
100
110
(1) A long 100-strike call and a short 110-strike call
(2) A short 100-strike put and a long 110-strike put
The strategy is to receive a guaranteed payoff,
regardless of changes in the market price
A box spread with a guaranteed payoff of \$10.00
 An unequal number of options at different strike prices are bought and sold
 The strategy is that the price won’t change very much, but the investors wants
insurance in case the price declines
COLLAR
 Combination of a long put and short call at a higher price
 The investor wants a constant payoff for a range of spot prices, and an increasing
payoff as the spot price decreases
COLLARD STOCK
 Combination of owning the stock and buying a collar with the stock
COMBINATION OF OPTIONS
(1) A 100-110 bull spread using call options
• Current spot price of \$100
Net Premium = 15.79 – 11.33 = 4.46
COMBINATION OF OPTIONS
120
• Current spot price of \$100
Net Premium = 15.79 – 7.96 + 18.55 – 7.95 = 18.43
COMBINATION OF OPTIONS
(3) An 80-120 strangle
• Current spot price of \$100
Net Premium = 2.07 + 7.95 = 10.02
COMBINATION OF OPTIONS
(4) A straddle using at-the-money options
• Current spot price of \$100
Net Premium = 15.79 + 7.96 = 23.75
COMBINATION OF OPTIONS
(5) A collar with a width of \$10 using 90-strike and 100-strike options
• Current spot price of \$100
Net Premium = 4.41 -15.97 = -11.38
COMBINATION OF OPTIONS
(6) A ratio spread using 90-strike and 110-strike options, with a payoff
of 20 at a spot price at expiration = 110, and a payoff of 0 at a spot
price at expiration = 120
• Current spot price of \$100
Buy one 90-strike call, and write three 110-strike call
Net Premium = 21.46 – 3 * 11.33 = -12.53
COMBINATION OF OPTIONS
with insurance using options that are out-of-the money by \$10
• Current spot price of \$100
Net Premium = -15.79 – 7.96 + 4.41 + 11.33 = -8.01
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