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 PortfolioCash 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) Synthetic Forward Call(K,T) – Put(K,T) KCall(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 PortfolioCash outflow at time 0 Cash outflow at time t Net cost as of time 0 1 2

PORTFOLIOS 2 AND 3 PortfolioCash outflow at time 0 Cash outflow at time t Net cost as of time 0 2 3Call(K,T) – Put(K,T)KCall(K,T) – Put(K,T) +PV(K)

ANALYSIS OF PORTFOLIO 2 AND 3 PortfolioOutflow at time 0Outflow at time T Call(K,T) – Put(K,T)K

PORTFOLIOS 1 AND 3

THE PUT-CALL PARITY

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 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 = $  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.

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 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 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 bull spread using call options Net Premium = – = 4.46 Current spot price of $100

COMBINATION OF OPTIONS (2) A box spread Net Premium = – – 7.95 = Current spot price of $100

COMBINATION OF OPTIONS (3) An strangle Net Premium = = Current spot price of $100

COMBINATION OF OPTIONS (4) A straddle using at-the-money options Net Premium = = Current spot price of $100

COMBINATION OF OPTIONS (5) A collar with a width of $10 using 90-strike and 100-strike options Net Premium = = Current spot price of $100

COMBINATION OF OPTIONS 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 call Net Premium = – 3 * =

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 Net Premium = – = Current spot price of $100