fec financial engineering club. more on options agenda put-call parity combination of options

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 Outflows at Time 0

Cash Outflows at Time T

Net Cost as of Time 0

1. Outright Purchase

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

--- PV()

3. Synthetic Forward

Call(K,T) – Put(K,T)

K 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

= PV()

The price of the stock So is equal to the Present value of the forward price.

If this is the case, then

= FV()

The forward price is equal to the current price of the stock.

*All of the stocks have to be non dividend paying stocksPortfolio Cash outflow at

time 0Cash outflow at time t

Net cost as of time 0

1

2 PV()

PORTFOLIOS 2 AND 3

PV() = Call(K,T) – Put(K,T) + PV(K)

Subtract PV(K) on both sides.

Call(K,T) – Put(K,T) = PV()– PV(K)

We can simply combine the present values together

Call(K,T) – Put(K,T) = PV( – K)

Portfolio Cash outflow at time 0

Cash outflow at time t

Net cost as of time 0

2

3 Call(K,T) – Put(K,T)

K Call(K,T) – Put(K,T) +PV(K)

ANALYSIS OF PORTFOLIO 2 AND 3

Original Equation: PV()= 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 Outflow at time T

2 ---

3 Call(K,T) – Put(K,T) K

PORTFOLIOS 1 AND 3

1. = Call(K,T) – Put(K,T) + PV(K)

Rearrange it to have:

2. Call(K,T) – Put(K,T) = –PV(K)

Substitute S0 with PV(F(0,T)) from Portfolio 1 and 2

3. Call(K,T) – Put(K,T) = PV(-K)

THE PUT-CALL PARITY

Call(K,T) – Put(K,T) = –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:

This Lecture: Call(K,T) – Put(K,T) = –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 = * (163.13 – 150)

i = 8.78%

COMBINING OPTIONS

Payoff graphs for four basic positions

Price

Profit

,

STRADDLE

Favor both sides of an issue at once

Combination of an at-the-money put and an at-the-money call

profit

Written

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

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

105

10590-105-110

asymmetric butterfly spread

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

We think the price will decline

A mirror image of bull spread

100-strike short call

100

100

110

100-110 bear spread

BOX SPREAD

A box spread with a guaranteed payoff of $10.00

(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

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

Net Premium = 15.79 – 11.33 = 4.46

• Current spot price of $100


(2) A 100-120 box spread

Net Premium = 15.79 – 7.96 + 18.55 – 7.95 = 18.43

120



(3) An 80-120 strangle

Net Premium = 2.07 + 7.95 = 10.02• Current spot price of $100


(4) A straddle using at-the-money options

Net Premium = 15.79 + 7.96 = 23.75• Current spot price of $100


(5) A collar with a width of $10 using 90-strike and 100-strike options

Net Premium = 4.41 -15.97 = -11.38• Current spot price of $100



(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

Buy one 90-strike call, and write three 110-strike callNet Premium = 21.46 – 3 * 11.33 = -12.53


(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

Net Premium = -15.79 – 7.96 + 4.41 + 11.33 = -8.01


fec financial engineering club. more on options agenda put-call parity combination of options

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