DERIVATIVES: OPTIONS Reference: John C. Hull, Options, Futures and Other Derivatives, Prentice Hall

Types of Options A call is an option to buy A put is an option to sell A European option can be exercised only at the end of its life An American option can be exercised at any time 2

Option Positions Long call Long put Short call Short put 3

Long Call Profit from buying one European call option: option price = $5, strike price = $100. 30 20 10 -5 70 80 90 100 110 120 130 Profit ($) Terminal stock price ($) 4

Short Call Profit from writing one European call option: option price = $5, strike price = $100 -30 -20 -10 5 70 80 90 100 110 120 130 Profit ($) Terminal stock price ($) 5

Long Put Profit from buying a European put option: option price = $7, strike price = $70 30 20 10 -7 70 60 50 40 80 90 100 Profit ($) Terminal stock price ($) 6

Short Put Profit from writing a European put option: option price = $7, strike price = $70 -30 -20 -10 7 70 60 50 40 80 90 100 Profit ($) Terminal stock price ($) 7

K = Strike price, ST = Price of asset at maturity Payoffs from Options (Terminal Value) European Options, Cost of option is NOT included K = Strike price, ST = Price of asset at maturity Payoff ST K SHORT CALL LONG CALL SHORT PUT LONG PUT 8

MAX (ST - K, 0) MIN (K- ST , 0) MAX (K- ST , 0) MIN (ST - K, 0) Payoff ST K SHORT CALL LONG CALL SHORT PUT LONG PUT 9

Assets Underlying Exchange-Traded Options Stocks Foreign Currency Stock Indices Futures 10

Specification of Exchange-Traded Options Expiration date Strike price European or American Call or Put (option class) 11

Terminology Moneyness : At-the-money option In-the-money option Out-of-the-money option 12

Notation c : European call option price C : American Call option price p : European put option price S0 : Stock price today K : Strike price T : Life of option : Volatility of stock price C : American Call option price P : American Put option price ST :Stock price at option maturity D : Present value of dividends during option’s life r : Risk-free rate for maturity T with cont. comp. 13

– – + + – – + + + + + + + + – – + + – – + + Variable S0 K T  r D c p Effects, in the price of an option, when one variable increases, and all the others remain unaltered European American Variable S0 K T  r D c p C P – – + + – – + + + + ? + + + + – – + + – – + + Option decreases in value if D increases Option increases in value if S0 increases 14

Some Useful Relationships… A call option can never be worth more than the stock c ≤ S0 C ≤ S0

A put option can never be worth more than K p ≤ K P ≤ K

American vs European Options An American option is worth at least as much as the corresponding European option C  c P  p 17

For a European put we know that at maturity the put option cannot be worth more than K It follows then, that today, the price cannot be more than the present value of K, thus p ≤ K e-rT

Lower Bound for The Value of a European Call Option When There Are No Dividends c  S0 –Ke -rT Proof on next page [using an arbitrage argument] 19

Consider two portfolios: [A] a European call option + cash = Ke –rT and [B] one share of the stock In case of portfolio A, the cash can be invested at the risk-free rate, thus, at time = T, we will have K dollars We have two possible outcomes at time =T [1] If ST > K, I will exercise the option (spend K) and now I have a stock whose value is ST OR [2] If ST < K there is no point in exercising the option (option=worthless); so I end up with K dollars Thus, at time T, the value of portfolio A is MAX (K, ST)

c + Ke –rT > S0 (QED) Consider two portfolios: [A] a European call option + cash = Ke –rT and [B] one share of the stock The value of portfolio A is MAX (K, ST) (at time T) AND the value of portfolio (B), at time = T, is obviously, ST Clearly, (A) is more valuable than (B) at maturity ((in other words, B can be equal to A but never more)), therefore, in the absence of arbitrage opportunities the same situation has to be valid today (time = 0); namely, (A) is more valuable than (B), thus c + Ke –rT > S0 (QED)

Lower Bound for The Value of a European Put Option When There Are No Dividends p  Ke –rT - S0 Proof on next page [using an arbitrage argument] 22

Consider two portfolios: [C] a European put option + one share and [D] cash = Ke –rT In case of portfolio D, the cash can be invested at the risk-free rate, thus, at time = T, we will have K dollars We have two possible outcomes at time =T for position [C] [1] If ST < K, I will exercise the option (sell the stock at K) and now I have a portfolio whose value is K OR [2] If ST > K there is no point in exercising the option (option=worthless); so I end up with a value = ST Thus, at time T, the value of portfolio C is MAX (K, ST)

By a reasoning similar to the one employed before we conclude that Portfolio C is worth more than portfolio D (whose value, at T, is K no matter what); and thus, the same should hold at time = 0 (today) Thus, p + S0 > Ke –rT (QED)

Value of A and C Portfolios Time = T ST > K ST < K Portfolio A Call option ST − K cash K Total ST Portfolio C Put Option K− ST Share All options are European (CANNOT be exercised before T), thus, since the value of the two portfolios( A and B) is the same at time = T, they must have the same value today (time = 0) This leads to the so-called…. 25

The Put-Call Parity Theorem Both portfolios are worth max(ST , K ) at maturity They must therefore be worth the same today. This means that (today) c + Ke -rT = p + S0 European Options; No dividends 26

S0 - K < C - P < S0 - Ke -rT The Put-Call Parity Theorem (OR Result) Is Only Valid For EUROPEAN Options; For AMERICAN Options We Can Show That For The Case In Which There Are No Dividends S0 - K < C - P < S0 - Ke -rT 27

European options; D > 0 If Dividends are included (D is the PV of the Dividend during the life of the option) then we have… European options; D > 0 c + D + Ke -rT = p + S0 American options; D > 0 S0 - D - K < C - P < S0 - Ke -rT