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Mechanics of Options Markets Liuren Wu Zicklin School of Business, Baruch College Fall, 2007 (Hull chapter: 8) Liuren Wu Options Markets Mechanics Option Pricing, Fall, 2007 1 / 44 Outline 1 Definition 2 Payoffs 3 Mechanics 4 Other option-type products 5 Data source 6 Option behaviors Liuren Wu Options Markets Mechanics Option Pricing, Fall, 2007 2 / 44 Definitions and terminologies An option gives the option holder the right/option, but no obligation, to buy or sell a security to the option writer/seller I for a pre-specified price (the strike price, K) I at (or up to) a given time in the future (the expiry date ) An option has positive value. Comparison: a forward contract has zero value at inception. Option types I A put option gives the holder the right to sell a security. The payoff is + (ST − K) when exercised at maturity. I A call option gives the holder the right to buy a security. The payoff is + (K − ST ) when exercised at maturity. I American options can be exercised at any time priory to expiry. I European options can only be exercised at the expiry. Liuren Wu Options Markets Mechanics Option Pricing, Fall, 2007 3 / 44 More terminologies Moneyness: the strike relative to the spot/forward level I An option is said to be in-the-money if the option has positive value if exercised right now: F St > K for call options and St < K for put options. F Sometimes it is also defined in terms of the forward price at the same maturity (in the money forward): Ft > K for call and Ft < K for put F The option has positive intrinsic value when in the money. The + + intrinsic value is (St − K) for call, (K − St ) for put. F We can also define intrinsic value in terms of forward price. I An option is said to be out-of-the-money when it has zero intrinsic value. F St < K for call options and St > K for put options. F Out-of-the-money forward: Ft < K for call and Ft > K for put. I An option is said to be at-the-money spot (or forward) when the strike is equal to the spot (or forward). Liuren Wu Options Markets Mechanics Option Pricing, Fall, 2007 4 / 44 More terminologies The value of an option is determined by I the current spot (or forward) price (St or Ft ), I the strike price K, I the time to maturity τ = T − t, I the option type (Call or put, American or European), and I the dynamics of the underlying security (e.g., how volatile the security price is). Out-of-the-money options do not have intrinsic value, but they have time value. Time value is determined by time to maturity of the option and the dynamics of the underlying security. Generically, we can decompose the value of each option into two components: option value = intrinsic value + time value. Liuren Wu Options Markets Mechanics Option Pricing, Fall, 2007 5 / 44 Payoffs versus P&Ls + For European options, the terminal payoff can be written as (ST − K) for + calls and (K − ST ) for puts at expiry date T . Since options have positive value, one needs to pay an upfront price (option price) to possess an option. The P&L from the option investment is the difference between the terminal payoff and the initial price you pay to obtain the option. Do not confuse the two. The textbook likes to talk about P&Ls, but I like to talk about payoffs — Different perspectives: I P&Ls: If I buy/sell an option today, how much money I can make under different scenarios? What’s my return? I Payoffs: If I desire a certain payoff structure in the future, what types of options/positions I need to generate it? Liuren Wu Options Markets Mechanics Option Pricing, Fall, 2007 6 / 44 An example: Call option on a stock index Consider a European call option on a stock index. The current index level (spot St ) is 100. The option has a strike (K) of $90 and a time to maturity (T − t) of 1 year. The option has a current value (ct ) of $14. Is this option in-the-money or out-of-the-money (wrt to spot)? What’s intrinsic value for this option? What’s its time value? If you hold this option, what’s your terminal payoff? I What’s your payoff and P&L if the index level reaches 100, 90, or 80 at the expiry date T ? If you write this option and have sold it to the exchange, what does your terminal payoff look like? I What’s your payoff and P&L if the index level reaches 100, 90, or 80 at the expiry date T ? Liuren Wu Options Markets Mechanics Option Pricing, Fall, 2007 7 / 44 Payoffs and P&Ls from long/short a call option (St = 100, K = 90, ct = 14) 30 30 20 20 10 10 0 0 −10 −10 P&L from long a call Payoff from long a call −20 −20 −30 −30 60 70 80 90 100 110 120 60 70 80 90 100 110 120 Spot at expiry, S Spot at expiry, S T T 30 30 20 20 10 10 0 0 −10 −10 P&L from short a call Payoff from short a call −20 −20 −30 −30 60 70 80 90 100 110 120 60 70 80 90 100 110 120 Spot at expiry, S Spot at expiry, S T T + Long a call pays off, (ST − K) , bets on index price going up. Shorting a call bets on index price going down. Liuren Wu Options Markets Mechanics Option Pricing, Fall, 2007 8 / 44 Another example: Put option on an exchange rate Consider a European put option on the dollar price of pound (GBPUSD). The current spot exchange rate (St ) is $1.6285 per pound. The option has a strike (K) of $1.61 and a time to maturity (T − t) of 1 year. The 1-year forward price (Ft,T ) is $1.61. The dollar continuously compounding interest rate at 1-year maturity (rd )is 5%. The option (pt ) is priced at $0.0489. From the above information, can you infer the continuously compounding interest rate at 1-year maturity on pound (rf )? Is this option in-the-money or out-of-the-money wrt to spot? What’s the moneyness in terms of forward? In terms of forward, what’s intrinsic value for this option? What’s its time value? If you hold this option, what’s your terminal payoff, if the dollar price of pound reaches 1.41, 1.61. or 1.81 at the expiry date T ? Liuren Wu Options Markets Mechanics Option Pricing, Fall, 2007 9 / 44 Another example: Put option on an exchange rate (r −r )(T −t) Review the forward pricing formula: Ft,T = St e d f . 1 rf = rd − T −t ln(Ft,T /St ) = .05 − ln(1.61/1.6285)/1 = 6.14%. I Recall covered interest rate parity: Annualized forward return 1 ( T −t ln(Ft,T /St )) on exchange rates equals interest rate differential (rd − rf ) between the two currencies. + Long a put option pays off, (K − ST ) , and bets on the underlying currency (pound) depreciates. Shorting a call option bets on pound appreciates. How does it differ from betting using forwards? Liuren Wu Options Markets Mechanics Option Pricing, Fall, 2007 10 / 44 Payoffs and P&Ls from long/short a put option (St = 1.6285, Ft,T = 1.61, K = 1.61, pt = 0.0489) 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 −0.1 −0.1 −0.2 −0.2 P&L from long a put Payoff from long a put −0.3 −0.3 −0.4 −0.4 −0.5 −0.5 1 1.2 1.4 1.6 1.8 2 1 1.2 1.4 1.6 1.8 2 Spot at expiry, S Spot at expiry, S T T 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 −0.1 −0.1 −0.2 −0.2 P&L from short a put Payoff from short a put −0.3 −0.3 −0.4 −0.4 −0.5 −0.5 1 1.2 1.4 1.6 1.8 2 1 1.2 1.4 1.6 1.8 2 Spot at expiry, S Spot at expiry, S T T Liuren Wu Options Markets Mechanics Option Pricing, Fall, 2007 11 / 44 What derivative positions generate the following payoff? 20 20 120 15 15 115 10 10 110 5 5 105 0 0 100 Payoff Payoff Payoff −5 −5 95 −10 −10 90 −15 −15 85 −20 −20 80 80 85 90 95 100 105 110 115 120 80 85 90 95 100 105 110 115 120 80 85 90 95 100 105 110 115 120 Spot at expiry, S Spot at expiry, S Spot at expiry, S T T T 20 20 −80 15 15 −85 10 10 −90 5 5 −95 0 0 −100 Payoff Payoff Payoff −5 −5 −105 −10 −10 −110 −15 −15 −115 −20 −20 −120 80 85 90 95 100 105 110 115 120 80 85 90 95 100 105 110 115 120 80 85 90 95 100 105 110 115 120 Spot at expiry, S Spot at expiry, S Spot at expiry, S T T T Liuren Wu Options Markets Mechanics Option Pricing, Fall, 2007 12 / 44 Assets underlying exchanged-traded options Stocks Stock indices Index return variance (new) Exchange rate Futures Liuren Wu Options Markets Mechanics Option Pricing, Fall, 2007 13 / 44 Specification of exchange-traded options Expiration date (T ) Strike price (K) European or American Call or Put (option class) OTC options (such as OTC options on currencies) are quoted differently.
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