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

1. Outright Purchase 2. Long Forward with Forward Price F(0,T)

Cash Outflows at Time 0 Cash Outflows at Time T Net Cost as of Time 0

-- 𝑆 0 3. Synthetic Forward Call(K,T) – Put(K,T) K 𝐹 0,𝑇 PV( 𝐹 𝑆 0 0,𝑇 ) Call(K,T) – Put(K,T) +PV(K)  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.

PORTFOLIOS 1 AND 2

𝑆 0 = PV( 𝐹 0,𝑇 ) The price of the stock So is equal to the Present value of the forward price.

If this is the case, then 𝐹 0,𝑇 = FV( 𝑆 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

𝑆 0

Cash outflow at time t

𝐹 0,𝑇

Net cost as of time 0

𝑆 0 PV( 𝐹 0,𝑇 )

PORTFOLIOS 2 AND 3

PV( 𝐹 0,𝑇 ) = Call(K,T) – Put(K,T) + PV(K) Subtract PV(K) on both sides.

Call(K,T) – Put(K,T) = PV( 𝐹 0,𝑇 )– PV(K) We can simply combine the present values together Call(K,T) – Put(K,T) = PV( 𝐹 0,𝑇 – K)

Portfolio

2 3

Cash outflow at time 0

Call(K,T) – Put(K,T)

Cash outflow at time t

𝐹 0,𝑇 K

Net cost as of time 0

PV( 𝐹 0,𝑇 ) Call(K,T) – Put(K,T) +PV(K)

ANALYSIS OF PORTFOLIO 2 AND 3

Original Equation: PV( 𝐹 0,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

2 3

Outflow at time 0

-- Call(K,T) – Put(K,T)

Outflow at time T

𝐹 0,𝑇 K

PORTFOLIOS 1 AND 3

1. 𝑆 0 = Call(K,T) – Put(K,T) + PV(K) Rearrange it to have: 2. Call(K,T) – Put(K,T) = 𝑆 0 –PV(K) Substitute S0 with PV(F(0,T)) from Portfolio 1 and 2 3. Call(K,T) – Put(K,T) = PV( 𝐹 0,𝑇 -K)

THE PUT-CALL PARITY



Call(K,T) – Put(K,T) =

𝑆 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.

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: −𝑆 𝑡 = −𝑐 𝑆, 𝐾, 𝑡, 𝑇 1 + 𝑝 𝑆, 𝐾, 𝑡, 𝑇 1  This Lecture: Call(K,T) – Put(K,T) = 𝑆 0 –PV(K) − 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(F 0,T - K)  1 23.86 – 11.79 = 1+𝑖 * (163.13 – 150)  i = 8.78%

Profit

COMBINING OPTIONS

 Payoff graphs for four basic positions Price 𝐶𝑎𝑙𝑙 𝐼𝑛𝑡𝑟𝑖𝑛𝑠𝑖𝑐 = (𝑆 − 𝐾)+ = max{𝑆 − 𝐾, 0}

,

𝑃𝑢𝑡 𝐼𝑛𝑡𝑟𝑖𝑛𝑠𝑖𝑐 = (𝐾 − 𝑆)+ = max{𝐾 − 𝑆, 0}

STRADDLE

 Favor both sides of an issue at once  Combination of an at-the-money put and an at the-money call profit

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 105 strike 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 A box spread with a guaranteed payoff of $10.00

The strategy is to receive a guaranteed payoff, regardless of changes in the market price

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 • Current spot price of $100 120 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