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
𝑆0
𝑆0
𝐹0,𝑇
2. Long
--Forward
with
Forward
Price F(0,T)
3. Synthetic Call(K,T) –
Forward
Put(K,T)
PV(𝐹0,𝑇 )
 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
𝑆0 = PV(𝐹0,𝑇 )
The price of the stock So is equal to the Present value of the forward price.
If this is the case, then
𝐹0,𝑇 = FV(𝑆0 )
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
𝑆0
𝑆0
𝐹0,𝑇
PV(𝐹0,𝑇 )
PORTFOLIOS 2 AND 3
PV(𝐹0,𝑇 ) = Call(K,T) – Put(K,T) + PV(K)
Subtract PV(K) on both sides.
Call(K,T) – Put(K,T) = PV(𝐹0,𝑇 )– PV(K)
We can simply combine the present values together
Call(K,T) – Put(K,T) = PV(𝐹0,𝑇 – K)
Portfolio
Cash outflow at
time 0
Cash outflow at
time t
𝐹0,𝑇
2
3
Call(K,T) – Put(K,T)
K
Net cost as of time
0
PV(𝐹0,𝑇 )
Call(K,T) – Put(K,T)
+PV(K)
ANALYSIS OF PORTFOLIO 2 AND 3
Original Equation: PV(𝐹0,𝑇 )= 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
the excess of the call premium over the put premium upfront.
Portfolio
Outflow at time 0
2
---
3
Call(K,T) – Put(K,T)
Outflow at time T
𝐹0,𝑇
K
PORTFOLIOS 1 AND 3
1. 𝑆0 = Call(K,T) – Put(K,T) + PV(K)
Rearrange it to have:
2. Call(K,T) – Put(K,T) = 𝑆0 –PV(K)
Substitute S0 with PV(F(0,T)) from Portfolio 1 and 2
3. Call(K,T) – Put(K,T) = PV(𝐹0,𝑇 -K)
THE PUT-CALL PARITY
Call(K,T) – Put(K,T) = 𝑆0 –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.
WHAT’S SO IMPORTANT ABOUT IT?
 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: −𝑆𝑡 = −𝑐 𝑆, 𝐾, 𝑡, 𝑇1 + 𝑝 𝑆, 𝐾, 𝑡, 𝑇1 − 𝑃𝑉(𝐾)
 This Lecture: Call(K,T) – Put(K,T) = 𝑆0 –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+𝑖
COMBINING OPTIONS
 Payoff graphs for four basic positions
Profit
Price
𝐶𝑎𝑙𝑙𝐼𝑛𝑡𝑟𝑖𝑛𝑠𝑖𝑐 = (𝑆 − 𝐾)+ = max{𝑆 − 𝐾, 0},
𝑃𝑢𝑡𝐼𝑛𝑡𝑟𝑖𝑛𝑠𝑖𝑐 = (𝐾 − 𝑆)+ = max{𝐾 − 𝑆, 0}
STRADDLE
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
BUTTERFLY SPREAD
 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
ASYMMETRIC BUTTERFLY SPREAD
105
105
90-105-110
asymmetric butterfly spread
 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
BULL SPREAD
 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
BEAR SPREAD
100
100-strike short call
 We think the price will decline
 A mirror image of bull spread
100
110
100-110 bear spread
BOX SPREAD
(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
RATIO SPREAD
 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
(2) A 100-120 box spread
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
(7) A butterfly spread with a straddle using at-the-money options and
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