OPTIONS MARKETS. Options Similar to futures; however, they give the buyer (holder) the right but not...

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

Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007 8.17

Margins• Margins are required when options are sold• For example when a naked call option is written the

margin is the greater of:1 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

2 A total of 100% of the proceeds of the sale plus 10% of the underlying share price


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


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


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

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?


• 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%)


• 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.


• 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 = 0 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

• 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)

• Αντικατάσταση Δ

Δ = ( fu - fd ) / ( Su - Sd )

• 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)p = (e(0.08) (0.25) – 0.9 ) / (1.1 – 0.9) p = 0.601

• 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


• 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


= e-(0.08)(6/12) [0.7(21) + 0.3(0)] = $14.2

• fd = 0

• f = e-(0.08)(6/12) [ 0,7 (14.2) + 0.3 (0) ]

f = $9.61

OPTIONS MARKETS. Options Similar to futures; however, they give the buyer (holder) the right but not...

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