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# Class Note for FINOPMGT 413 at UMass(1)

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Date Created: 02/06/15

Week 5 Options Basic Concepts Definitions 12 Although many different types of options some quite exotic have been introduced into the market we shall only deal with the simplest plainvanilla options like calls and puts Call A call gives you the option to buy an underlying asset at a fixed price called the strike or exercise price before or on a certain date called the maturity or the expiration date of the option Put A put gives you the right to sell the underlying asset Definitions 22 A European option is one that can be exercised only on the maturity date An American option is one that can be exercised any time before the exercise date The American option price is equal to or greater than the European option price Why As before we will denote S as the spot exchange rate and F as the fonNard Example Example A call on the BP at a strike of 1 60 expiring say on 12152005 This call gives you the option to buy 1 BP for 160 on 12152005 You will exercise the call if the BP is stronger than the strike BP gt 16 on that date Let us denote ST as the exchange rate on 121505 X as the strike and T as the exercise date Then you will exercise the option if ST gtX For example if the exchange rate at maturity is 170BP you will exercise the call and your payoff will be ST X170160010 Payoff of the Call and Put When Exercised Denote CT and PT we can write the payoff of the call and put on the exercise date as follows Payoff on call at T CT 0 if STltX and CT ST X if STgtX Thus CT MaxSTXO Payoff on put at T PT 0 if XltST and PT X ST if XgtST Thus PT MaxX STO Payoff Diagrams The payoff diagram represents the payoff of the option on the expiration date T as a function of the price of the underlying security To trade options it is essential to understand payoff diagrams Examples How would you draw the payoff of the following options or portfolio s of options 1 LongShort call of strike X 2 LongShort Put with strike X 3 Long C and Long P both with same strike X 4 Long C and short P both with same strike X Long Call with Strike160 Payoff STXO2 Short Call with Strike160 Payoff STX 04 X16 sT20 XST 02 Long Put with Strike160 Payoff sT14 X16 Short Put with Strike160 Payoff XST 06 sT10 X16 Other Examples The basic payoff s of a longshort call and a longshort put can be combined into much more complicated payoff structures Examples Please draw the payoffs 1 Straddle A straddle is an option position of long 1 C long 1 P where the call and put have the same strikes This position allows the trader to take a view on volatility 2 Bull Spread Long 1 call of strike 160 short 1 call of strike 180 This allows the trader to take a bullish position but with a capped upside The capped upside makes the position cheaper to implement Some Terminology Intrinsic value value of the option if it is immediately exercised For a call the intrinsic value is StX For a put the intrinsic value is X St An atthemoney option is one with intrinsic value equal to zero An inthemoney option is one whose intrinsic value is positive An outthemoney option is one whose intrinsic value is negative Issues What is the relation between the call put forwards and the spot We can derive this by assuming that the prices of the options the spot and the forward should not allow for arbitrage Put Call Parity 15 Let us figure out today s t0 price of a portfolio of1 long call and 1 short put C P with the same strike X and maturity T What is the payoff of this portfolio of CP Payoff of CP f ST gt X then the call is exercised and the put is not so the combined payoff is STX 81quot f STltX then the put is exercised and the call isn t so the combined payoff is XST 81quot Put Call Parity 25 Thus for any spot rate at tT the payoff on the portfolio of C P is STX Suppose the underlying asset is 1 British Pound BP This means that buying a call and selling a put is the same as receiving one BP at maturity of value ST for a price of X Can we replicate this payoff using the underlying securities That is can we replicate this payoff by trading the foreign exchange and borrowinglending at the domestic and foreign interest rates PutCall Parity 35 Here is one strategy buy the present value of lBP at t0 and hold it until tT and borrow the present value ofX dollars If the option has a maturity of T days then the value of this portfolio is PVS PVX S1 rT360 X1rT360 What would be the total payoff on this portfolio of PVS PVX The payoff on this portfolio is exactly STX ie the same as that on CP Note that when you take the present value of the foreign currency you need to discount at the foreign interest rate PutCall Parity 45 Thus to prevent arbitrage it must be that the following is true for currency options CP S1 rT360 XI1rT360 We have figured out a relationship between C P and S This is called the PutCall Parity Put Call Parity 55 We can also write the relation for currency options in terms of the forward price FS1rT3601rT360 Substituting for S in the put call parity we get CP F1rT360 X1rT360 F X1rT360 Thus we can either express the put call parity in terms of the spot rate or the forward rate Summary of PutCall Parity These are the important points to note from PutCall Parity 1 There is a precise relation between the prices of the call and the put of the same strike given by CPS1rn360 X1rn360 If the observed option prices do not follow this relation then there exists an arbitrage t easier in practice to construct an arbitrage using the futures C PFX1rn360 Thus given the price of the call one can deduce the price of the put or vice versa 3 If SX then the prices of the atthemoney call and put will be equal to each other only if rr The call will be more expensive than the put if rgtr and PgtC if rgtr

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