OPTIONS MARKETS

Options Similar to futures; however, they give the buyer (holder) the right but not the obligation to buy/sell the underlying asset at a specified future date and at a specified exercise (strike) price) The buyer pays the price for this right (option premium), which is determined by the supply and demand in the market The seller (writer ) sells the right and receives the premium but has the obligation to buy/sell the underlying in the future if the option is exercised by the buyer

Options A call option is an option (the right but not the obligation) to buy a certain asset by a certain date for a certain price (the strike price) A put option is an option (the right but not the obligation) to sell a certain asset by a certain date for a certain price (the strike price) An American option can be exercised at any time during its life A European option can be exercised only at maturity Underlying: Stocks, Foreign Currency, Stock Indices, Futures Example LIFFE: Equity-based futures and options (FTSE 100 Index, FTSE 250 Index, FTSE Eurotop, MCSI Euro Index, MSCI Pan-Euro Index, MSCI European); Individual equities; Commodity futures and options (Robusta Coffee, White Sugar, Feed Wheat, Milling Wheat, Rapeseed and Corn).

4 basic option positions Long call Short call Long put Short put

Positions at maturity Long call option: The buyer (holder) has the right but not the obligation to BUY the underlying asset at a specified future date and at a specified exercise (strike) price) Short call option: The seller has the obligation to deliver the underlying asset at a specified future date and at a specified exercise (strike) price) Long put option: The buyer (holder) has the right but not the obligation to SELL the underlying asset at a specified future date and at a specified exercise (strike) price) Short put option: The seller (holder) has the obligation to BUY the underlying asset at a specified future date and at a specified exercise (strike) price)

Actions The buyer of an option may: → Exercise the option → Liquidate the option → Let the option to expire unexercised The seller of an option may: → Wait for the option to expire → Liquidate the option

Closing a position All open positions may be closed out by doing an opposite transaction with an option of the same series (same characteristics, underlying, strike, maturity) Assume you have bought a call option on the FTSE-ASE20 that matures in December, with a strike price of 400 index points. You paid for the option 20 index points. To close the position you must sell a call option on the FTSE-ASE20 that matures in December, with a strike price of 400 points (If you sell the option for more that 20 index points you will make a profit).

Covered and naked calls A covered call is a cal option where the seller already owns the underlying and can deliver it if the buyer exercises the option. A naked call is a cal option where the seller does not already own the underlying and has to buy it from the market in order to deliver if the buyer exercises the option. Selling naked calls is a dangerous investment practice since the losses can be very high.

Option specifications Expiration date Strike price European or American Call or Put (option class) E.g. a European January call on stock X with a strike of $100.

Trading options Over The Counter (OTC): The major participants are banks the make the market (market makers) big multinationals, etc. The contracts are not standardized and “tailor-made” for clients Organized exchanges: Most organized exchanges use market makers to facilitate options trading; A market maker quotes both bid and ask prices when requested; The market maker does not know whether the individual requesting the quotes wants to buy or sell

Option price = intrinsic value + time value Intrinsic Value: The quantity by which the current price of the underlying is higher from the strike price In other words, the value of the option if it was exercised today E.g. January call at 240: P = 254 pence, strike = 240 pence. Thus the intrinsic value is 14 pence What is the rest (25-14 = 11 pence) ? The rest is the time-value of the option

Intrinsic Value «in-the-money» options: when they have positive intrinsic value «at-the-money» options: price = strike price «out-of-the-money» options: when they have ‘negative’ intrinsic value If St > E call option is «in the money» put option is «out of the money» If St < E call option is «out of the money» put option is «in the money» Εάν St = E call & put are «at the money»

Time Value Time value declines as we approach maturity, until it decays (time decay) The decline rate is not linear and it increases as we approach matutrity

Margins Margins are required when options are sold For example when a naked call option is written the margin is the greater of: A total of 100% of the proceeds of the sale plus 20% of the underlying share price less the amount (if any) by which the option is out of the money A total of 100% of the proceeds of the sale plus 10% of the underlying share price Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007 8.16

Warrants Warrants are options that are issued (or written) by a corporation or a financial institution The number of warrants outstanding is determined by the size of the original issue & changes only when they are exercised or when they expire Warrants are traded in the same way as stocks The issuer settles up with the holder when a warrant is exercised When call warrants are issued by a corporation on its own stock, exercise will lead to new treasury stock being issued Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007 8.17

Executive Stock Options Option issued by a company to executives When the option is exercised the company issues more stock Usually at-the-money when issued They become vested after a period of time (usually 1 to 4 years) They cannot be sold They often last for as long as 10 or 15 years Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007 8.18

Convertible Bonds Convertible bonds are regular bonds that can be exchanged for equity at certain times in the future according to a predetermined exchange ratio Very often a convertible is callable The call provision is a way in which the issuer can force conversion at a time earlier than the holder might otherwise choose Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007 8.19

Example: Long in a call option Long position on a 3-month European call option on stock ABC with a strike price of $120, option price of $4, current stock price of $118 Standardization: Each contract is for 100 stocks. We have the right to buy in 3 months 100 shares of ABC at $120 For this right we pay today $4 per stock (i.e. $400) Assume that in 3 months the stock price is up by 15% approximately at about $135. Exercise?

Example: Long in a call option Exercise and buy 100 shares at $120 Sell in the market the shares for $135 Gain: $15 per share Cost $4 per share Net profit: $11 per share Net profit: $1,000 (returns of aprox. 375%)

Example: Long in a call option Assume that in 3 months the stock price is down by 15%, at about $100. Exercise? If exercised we will pay $120 for a share that is now worth $100 and loose $2,000. We do not exercise: Loss of $400 100% of our initial capital.

Example: Long in a call option Assume that in 3 months the stock price is up by 3%, at about $121.5. Exercise? If exercised we will pay $120 for a share that is now worth $121.5 and gain $150. Cost: $400 Loss: $400 - $150 = $250 60% of our initial capital.

Example: Long in a put option Assume that you buy a 2-month European put option on currency Χ at a strike of $0.64/Χ. Each contract is standardized at 62,500Χ. The premium is $0.02 for each X. Position: In two months we have the right but NOT the obligation To Sell currency X At an exchange rate of $0.64/Χ We pay for this right now $1.250 ($0.02 x 62.500 Χ)

Example: Long in a put option Assume that in 2 months the rate is 0.58/Χ Profit or Loss? We buy spot at $0.58 for $36,250 ($0.58 x 62,500) Exercise the option and deliver for $40.000 ($0.64 x 62,500) Profit $ 3,750 ($40,000 - $36,250) Net Profit $2.500 ($ 3,750 - $1,250)

Example: Long in a put option Assume that in 2 months the rate is 0.68/Χ Profit or Loss? DO NOT Exercise the option and sell (deliver) for $0.64 a currency that is now worth $0.68 Loss: $1,250 100% of initial capital

Returns from options (ignoring the premium) St = price of underlying at maturity, E = strike price Return of long call at maturity: = St - E if St > E = 0 if St  E Returns of a short call at maturity: = - (St - E) if St > E Return of long put at maturity: = 0 if St > E = E - St if St < E Returns of a short put at maturity: = 0 if St > E = - (E - St) if St < E

Example of index options Athens Stock Exchange: Calls & Puts FTSE/ASE – 20, FTSE/ASE – 40 Price in units, multiplier 5 Euro (2.5 now) Table: Trading activity on January 2007 option contracts → underlying FTSE/ASE-20 → 8 December 2006, time: 14.25) → Current FTSE/ASE-20 price: 2385 points

Option Valuation Binomial Trees Consider a stock with a current price of $10 Assume that it is known with certainty that in 3 months it will be worth either $11 or $9 How much will a 3-month European option should be worth if the risk free interest rate is 8% and the strike price is $10.5?

In 3 months If p = $11 then the value of the option will be $0.5 (St - E = 11 – 10.5) If p = $9 then the value of the option will be $0 (Not exercised) What is the value of the call (f) today?

How to get it? Assume no arbitrage and create a risk free portfolio with the stock and the option in such a way that there is no uncertainty in 3 months E.g. Long Δ shares, Short 1 call option If at maturity stock price goes from $10 to $11 → stock value will be $11x Δ → option value will be $0.5 → Portfolio Value: $(11Δ – 0.5) If at maturity stock price goes from $10 to $9 → stock value will be $9 x Δ → option value will be $0. → Portfolio Value: $(9Δ - 0)

How much is Δ; The portfolio will be riskless only if we choose Δ in such a manner that the final value is equal for both prospects: 11Δ – 0.5 = 9Δ  Δ = 0.25 In other words we must buy 0.25 for every stock we sell for the portfolio to be riskless If f is the value of the option today then the value of the portfolio today will be: 10Δ – f = 10(0.25) – f

The value of the portfolio If at maturity stock price goes from $10 to $11 → Portfolio Value: $(11x0.25 – 0.5) = $2.25 If at maturity stock price goes from $10 to $9 → Portfolio Value: $(9 x 0.25 – 0) = $2.25 A riskless portfolio must return the risk free rate and its Present Value will be: PV = FV e-rt = 2.25e-(0.08)(3/12) = 2.205 Since the portfolio value today is 10(0.25) – f Since the Present value of the portfolio is 2.205 Then 10(0,25) – f = 2,205 Solving for f : f = 0,295

Generalization Consider a stock with a current price of S Assume that it is known with certainty that in T months the price: → Will increase from S to Su and the call to fu → Will decrease from S to Sd, and the call to fd How much will a T-month European option should be worth if the risk free interest rate is r% and the strike price is E?

How to get it? Assume no arbitrage and create a risk free portfolio with the stock and the option in such a way that there is no uncertainty in T months E.g. Long Δ shares, Short 1 call option If at maturity stock price goes from S to Su → stock value will be Su x Δ → option value will be fu → Portfolio Value: SuΔ – fu If at maturity stock price goes from S to Sd → stock value will be Sd x Δ → option value will be fd → Portfolio Value: SdΔ – fd

How much is Δ; The portfolio will be riskless only if we choose Δ in such a manner that the final value is equal for both prospects: Su Δ – fu = Sd Δ - fd Δ = ( fu - fd ) / ( Su - Sd )

The value of the portfolio A riskless portfolio must return the risk free rate and its Present Value will be: PV = FV e-rt = (SuΔ – fu) e-rT Since the portfolio value today is: SΔ – f The Present Value and the value today must be equal: SΔ – f = (SuΔ – fu)e-rT  f = SΔ – (SuΔ – fu)e-rT

The value of the option Substitute Δ Δ = ( fu - fd ) / ( Su - Sd ) And solve for the option price: f = e-rT [ p fu + ( 1-p ) fd ] Where: p = (erT – d ) / (u – d)

p = (erT – d ) / (u – d) Δ = ( fu - fd ) / ( Su - Sd ) Αντικατάσταση Δ Και λύση ως προς την τιμή του δικαιώματος: f = e-rT [ p fu + ( 1-p ) fd ] Όπου: p = (erT – d ) / (u – d)

In the numerical example: Stock from 10 to 11 from S to Su u = 1.1 (10 x 1.1 = 11) Stock from 10 to 9 from S to Sd d = 0.9 (10 x 0.9 = 9)

r = 8%, T = 0.25, fu = 0.5, fd = 0 p = (erT – d ) / (u – d) f = e-rT [ p fu + ( 1-p ) fd ] f = e-(0.08)(0.25)[(0.601) 0.5+(1-0.601) 0] f = 0.295

Generalization for two steps: Each step will last Δt and the stock price will:

Repetitions show that: fu = e-rΔT [ p fuu + (1-p) fud ] fd = e-rΔT [ p fud + (1-p) fdd ] f = e-rΔT [ p fu + (1-p) fd ] Replace the first two in the third: f = e-2rΔT [ p2fuu + 2p(1-p)fud + (1-p)2 fdd ]

In practice When we built a binomial tree we choose u and d that matches the true volatility of the underlying (σ = standard deviation) u = e σ√ΔΤ d = e -σ√ΔΤ The real probability of an increase is μ = expected return) q = (eμΔΤ – d ) / (u – d) Cox, Ross, Rubinstein (1979, Journal of Financial Economics, 7) In practice an analyst will divide the life time of an option to steps of duration Δt (e.g. one month = 30 steps) and in every step there will be two possibilities (up, down) The analyst will end up with 31 possible final stock prices and 230 (over a billion) possible price paths

Exercise Stock price = $100 In each of the following 2 6-month periods will go up or down by 10% R=8% What is the value of a 12-month European call with E= $100?

Solution fu = e-rΔT [ p fuu + (1-p) fud ] fd = e-rΔT [ p fud + (1-p) fdd ] f = e-rΔT [ p fu + (1-p) fd ] f = e-2rΔT [ p2fuu + 2p(1-p)fud + (1-p)2 fdd ] p = (erΔT – d ) / (u – d)

Solution u = ? 100 x u = 110 → u = 110 / 100 = 1.1 d = ? 100 x d = 90 → d = 90 / 100 = 0.9 p = (e0.08(6/12) – 0.9 ) / (1.,1 – 0.,9) = 0.70 f = e-2(0.08)(6/12) [0.72 (21) + 2(0.7)(0.3)0 + 0.32 0] f = $9.61

Alternatively fu = e-rΔT [ p fuu + (1-p) fud ] fd = 0 f = e-(0.08)(6/12) [ 0,7 (14.2) + 0.3 (0) ] f = $9.61

Αποτίμηση - Black & Scholes συνεχή διαπραγμάτευση και ότι οι μεταβολές των τιμών (αποδόσεις) των μετοχών ακολουθούν μία κανονική κατανομή. Το υπόδειγμα μπορεί να αναπτυχθεί και για άλλου τύπου δικαιώματα.

Αποτίμηση - Black & Scholes Εάν ορίσουμε ως St την τρέχουσα τιμή του υποκείμενου τίτλου (π.χ. μετοχής) στην λήξη, και E την τιμή εξάσκησης, η αξία ενός δικαιώματος call στην λήξη του συμβολαίου είναι (St -E) εάν St >E, και (0) εάν St E. Εφ' όσον στην λήξη δεν υπάρχει χρονική αξία οι (St -E) και (0) είναι η εσωτερική αξία του δικαιώματος. Με άλλα λόγια, η εσωτερική αξία ενός δικαιώματος call στην λήξη του συμβολαίου είναι: max {0, (St -E)}

Αποτίμηση - Black & Scholes Άρα, η παρούσα αξία (PV) του δικαιώματος call (η αξία του σήμερα) θα πρέπει να είναι: max {0, PV(St -E)} Η πραγματική παρούσα αξία είναι αβέβαιη και άρα η σχέση απεικονίζει την παρούσα αξία της προσδοκώμενης εσωτερικής αξίας του δικαιώματος στην λήξη. Με άλλα λόγια η αξία ενός δικαιώματος εξαρτάται σε σημαντικό βαθμό από την πιθανότητα εκτέλεσης του δικαιώματος.

Αποτίμηση - Black & Scholes Η αξία ενός δικαιώματος call σήμερα (c) θα ισούται με: c = PV {St N(d1) - E N(d2)} Ο όρος N(d1) μπορεί να ειδωθεί σαν την πιθανότητα εκτέλεσης του δικαιώματος, και ο όρος St N(d1) ως η προσδοκώμενη παρούσα αξία της μελλοντικής τρέχουσας τιμής, δοθέντος St >E.

Αποτίμηση - Black & Scholes Γενικά, ο όρος N(d) είναι η αθροιστική συνάρτηση πιθανότητας της κανονικής κατανομής και μπορεί να υπολογισθεί από τους σχετικούς στατιστικούς πίνακες, εφ' όσον υπολογισθεί το d. Με άλλα λόγια N(d1) και N(d2) είναι οι πιθανότητες αποκλίσεων μεγέθους d1 και d2 σε μία τυποποιημένη κανονική κατανομή.

Αποτίμηση - Black & Scholes Εάν ορίσουμε τον χρόνο που απομένει μέχρι την λήξη του δικαιώματος ως (t) και το αγοραίο επιτόκιο ως (r), η εξίσωση μπορεί να ξαναγραφεί (με συνεχή ανατοκισμό) ως εξής: c = St e-rt N(d1) - Ee-rt N(d2) Όμως ο όρος (St e-rt) θα πρέπει να είναι ίσος με την σημερινή τρέχουσα τιμή της μετοχής, S0. c = St N(d1) - Ee-rt N(d2)

Αποτίμηση - Black & Scholes Εάν η μεταβλητότητα (τυπική απόκλιση) των αποδόσεων της υποκείμενης μετοχής ορισθεί ως (), μπορεί να δειχθεί ότι:

Αποτίμηση - Black & Scholes Με παρόμοια μεθοδολογία μπορεί να δειχθεί ότι η αξία ενός δικαιώματος put (p) θα είναι: p = Ee-rt N(-d2) - S0 N(-d1) Εάν ξέρουμε την αξία ενός call πάνω σε μία μετοχή, σύμφωνα με το θεώρημα του put-call parity, μπορεί να δειχθεί ότι η αξία ενός δικαιώματος put (p) μπορεί να υπολογισθεί και ως εξής: p = Ee-rt - S0 + c