Session IX:

Stock Options: Properties, Mechanics and Valuation

Lecturer: Dr. Jose Olmo

Module: Economics of Financial Markets

MSc. Financial Economics

Department of Economics, City University, London

Stock options can be primarily divided in call options and put options. These are

defined as follows.

Definition.- A call option gives the holder the right to buy the underlying asset by a

certain date (maturity) at a certain price (strike price).

The holder of a call option has a long position on the asset if its price lies beyond the

strike price at a certain time. Investors can take two positions on a call option. They can

buy the option and obtain the right to hold the underlying stock for some price determined

in advance (strike price), or they can short the call option and obtain the value of the call

and have a short position on the underlying stock if its price raises beyond the strike price.

In the first case the gains of the holder are unbounded (as long as the asset price can reach

unbounded positive values) while losses are bounded by the initial price of the call option.

In the case of the short position the opposite situation occurs. There are unbounded losses

for the investor with the short position if the stock price is greater than the strike, and the

gains are bounded by the call price.

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In the other side of the market it is put options.

Definition.- A put option gives the holder the right to sell an asset by a certain date

(maturity) for a certain price (strike price).

Investors can take two positions in this security as well. An investor with a long position

in a put option has the right to sell the stock at the strike price if the price of the stock goes

below the strike. In this case losses and gains are bounded for the holder of the put option.

For the short position in the put gains and losses are also bounded and are symmetric to

the long position.

Both derivatives can be further classified as European options and American Options.

European options are distinguished because the option cannot be exercised before maturity

while American options allow the investor to close out the position before the expiration

date. In turn the prices and payoffs of each of them are different. For the case of a long

position on a European call option the payoffs at maturity are given by

max(ST −K, 0)

with T denoting maturity, ST the terminal price of the stock and K the strike price. The

payoffs of a long position on a European put option are

max(K − ST , 0).

The corresponding prices for the short positions are

−max(ST −K, 0) = min(K − ST , 0),

and

−max(K − ST , 0) = min(ST −K, 0).

Regarding the value of the underlying stock, call options can be classified as

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• In the money: St > K,

• At the money: St ∼ K,

• Out of the money: St < K.

The same criterion yields the following classification for put options:

• In the money: St < K,

• At the money: St ∼ K,

• Out of the money: St > K.

More sophisticated options are

• digital options:

VT =

1 if ST > K

0 otherwise

• Pay Later Options: There is no cost at buying, but

VT =

−c0 + ST −K if ST > K

0 otherwise

• Asian Options: These options depend on the path followed by the underlying stock.

Within this class it can be found the following.

– Lookback Call: No fixed strike, but min0≤t≤T St.

– Lookback Put: No Strike, but max0≤t≤T St.

– Knock-out Option: Right to exercise option expires if underlying falls below or

jumps above a threshold.

– Kick-in Option: The right to exercise the option exists only if the underlying

falls below or jumps above a threshold value.

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– Index Options: The underlying is an index or basket of assets.

– Corridor Option: There is a gain while the stock is inside a band determined by

a lower and an upper threshold.

There are other types of options depending on the type of underlying security. If this is

a basket of assets the option is called basket or index option. The underlying can be also

a futures contract, foreign currency or bonds (convertible options).

Mechanics of Stock Options

The price of stock options are affected by a number of factors given by market conditions.

• The current stock price, S0.

• The strike price, K.

• The time to expiration, T.

• The volatility of the stock price, σS.

• The risk-free interest rate, r.

• The dividends expected during the life of the option.

Stock options are largely traded in major stock exchanges. In the United States the

exchanges trading stock options are the Chicago Board Options Exchange, the Philadelphia

Stock Exchange, the Pacific Exchange, and the International Securities Exchange.

These securities are distinguished by the time to expiration of the option and by the

strike price. Hence when an investor shows interest in buying an option on a stock she is

offered a range of different maturities and strike prices for the stock. Trade on stock options

is organized in cycles. There are three cycles denoted January, February and March cycles.

The January cycle consists of the months January, April, July, and October; February

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consists of February, May, August, and November. The March cycle consists of March,

June, September, and December. The exact maturity date for a stock option is the first

Saturday immediately after the third Friday of the expiration month. If an investor wishes

to buy stock options before the expiration date for that month has been reached the stock

exchange offers options trading with expiration date this month, the following month and

the next two months in the cycle. If she approaches the market after the expiration date for

that month the range of options offered by the exchange is for maturities in the next month,

the next-but-one month and the next two months of the expiration cycle. For example if

an investor is interested in buying options on GM stock at the beginning of October she

will have the possibility of buying options with maturity on October, November, January

and April. If the expiration date for October has passed she will be offered trading on

November, December, January and April.

The other variable describing trading in stock options in financial markets is the strike

price. The exchange normally chooses the strike prices spaced $2.50, $5, or $10 apart. If

the stock is priced in the range between $5 and $25 the spacing for the set of strike prices

determining the corresponding options is $2.5. For stocks in the range $25 and $200 the

spacing is $5 and for stock prices above $200 the spacing is $10.

The main role in options transactions in stock exchanges is played by the agents op-

erating in stock exchanges: brokers and market makers. A market maker for a certain

option is an individual belonging to the exchange who quotes its offer and bid price on the

option. The bid is the price at which the market maker is willing to buy the option and

the offer is the price at which the market maker is prepared to sell. These agents guarantee

the existence of liquidity given they are obliged by the exchange to buy or sell a number

of options when required. Market makers obtain profits from the bid-offer spread. This

spread is also regulated by the exchange to avoid large fluctuations in the options price

that hinder price discovery of the stock option.

The mechanics of trading on stock options is similar to trading on other financial instru-

ments in the exchange as futures contracts. There are two types of orders: market orders

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that are executed immediately and limit orders that are executed at specified threshold

prices. Investors place their buy or sell orders with a broker that in turn contacts a floor

broker of the exchange that communicates the order to the specialist (market maker). Bro-

kers charge a commission to the investor that varies from a fixed amount to some floating

amount depending on the volume of the operation.

For stock options with large maturity horizons (more than nine months) brokers allow

investors to buy on margin. This means that investors can deposit only 50% of the purchase

in stock options on a margin account held with the broker and borrow the rest from the

broker. This agent charges a commission to the investor. The broker in turn opens a

margin account with a floor broker of the exchange that opens a margin account with

Options Clearing Corporation. The counterpart of the Clearinghouses for futures markets.

In the last two decades trading options on over-the-counter markets has earned pop-

ularity. The flexibility and lack of regulation these markets offer are an attractive decoy

for investors wishing to find financial opportunities suiting their necessities. This involves

choosing exercise dates, strike prices, and contract sizes that are different from those traded

by the exchange. These over-the-counter markets also offer flexibility in the design of the

stock option giving rise to the existence of exotic options.

Properties of stock options

Strike price

If a call option is exercised at maturity its payoff is the amount the terminal stock price

exceeds the strike price. Therefore stock options determined on high strike prices are less

valuable that options where the strike price is lower, given the rest of the variables remain

fixed. For the put option the payoffs are given by the amount the strike exceeds the stock

price, hence more valuable put options are those with higher strike prices, keeping constant

the rest of the variables.

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Time to maturity

We distinguish in this case between European and American options. For an American

option both the call and the put option are more valuable as the time to expiration increases.

This type of options can be exercised at any time up to maturity. Therefore holders of

options with the furthest expiration date have advantage over owners of options with closer

maturities. The holder of the longer American option has always the opportunity of closing

the position in the option at the same time than the holders of shorter maturities. In fact

it has more possibilities given by the longer expiration of its option.

For European options there is not a definitive rule. Typically longer expiration dates

yield more valuable options. However this is not always true if stocks offer dividend yields.

An stock is less valuable after paying dividends because these are paid out from the equity

of the stock. For call options the payment of dividends should lower the value of a call

that expiries after dividends are paid. By the same argument the value of put options with

maturities after dividends are paid should increase with maturity.

Uncertainty

The terminal price of the stock is unknown. The higher the uncertainty about the future

price of the underlying stock the higher the price of the option. In the case of holding a

stock a high level of uncertainty implies a risk premium required by the stock holder. In

turn the expected value of the stock must be higher than it would be from a risk-free

investment. This risk premium offsets the presence of uncertainty.

In the case of call options the holder benefits from increases in the value of the stock

but it does not suffer from large losses because the downside risk of the call is limited. For

the put option the downside risk comes from rises in the price of the stock. This is limited

by definition of the put. Both options increase value when the uncertainty about the future

value of the underlying augments.

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Risk-free interest rate

The rate of interest driving the economy affects stock options in different ways. On one

hand high levels of interest rates convey higher risk premiums from risky investments. On

the other hand the discounted value of future payoffs is lower. Therefore an increase in

interest rates is usually accompanied by falls in stock prices. This implies a decrease in the

value of the stock and an increase in the value of the put. The other situation, decreases

in the interest rates, yield a boost in the call price and decreases in put prices.

Upper and lower bounds for option prices

By definition the price of a call option ct can never be higher than the value of the

underlying St. Otherwise arbitrage strategies arise by buying the stock and selling the call

option. Then

ct ≤ St.

By a similar argument the price of a put option pt cannot be higher than the strike price

K. Otherwise the strategy of buying the stock and selling the put yields a riskless profit.

Then

pt ≤ K.

Furthermore, for European puts the value of the option at maturity cannot be higher

than the strike. Otherwise the holder of the put faces a secure loss. Then pT ≤ K, and

discounting to the past yields

pt expR0(T−t) ≤ K.

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Lower bounds for European options

Let us denote St the price at time t of a non-dividend paying stock. The lower bound

for a European call option is

St −K exp−R0(T−t),

with R0 the risk-free interest rate.

Otherwise arbitrage opportunities arise from shorting the stock and buying the call

option. An investor following this strategy receives St − ct dollars that invests in the risk-

free asset to obtain at maturity (St − ct) expR0(T−t). If ST ≥ K the investor exercises the

option to return the asset and makes a net profit of (St − ct) expR0(T−t)−K that is greater

than zero if St − K exp−R0(T−t) > ct. If ST < K the call option is not exercised and the

investor’s net profit is (St − ct) expR0(T−t)−ST that is greater than zero.

Then the value of a European call option is bounded in the following range.

St −K exp−R0(T−t) ≤ ct ≤ St.

The lower bound can be re-arranged to yield

max(St −K exp−R0(T−t), 0) ≤ ct ≤ St.

Analogously, a lower bound for a European put option on a non-dividend-paying stock is

K exp−R0(T−t)−St,

with R0 the risk-free interest rate.

Otherwise arbitrage strategies surge by borrowing pt plus St to buy a stock and a put

option. At the expiration date the arbitrageur must pay the lender (pt + St) expR0(T−t).

If ST ≥ K the put is not exercised and the arbitrageur makes a net profit of ST − (pt +

St) expR0(T−t). On the other hand if ST < K the net profit is K − (pt + St) expR0(T−t).

A European put option satisfies the following.

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max(K exp−R0(T−t)−St, 0) ≤ pt ≤ K exp−R0(T−t).

The value of both European options at any time before maturity depends on the value of

the underlying asset and on the time to maturity. We saw before that the more the time to

expiration date more expensive the option. Both options have an intrinsic value regardless

the time to maturity. For the European call this intrinsic value is max(ST − K, 0). The

value of the call can be expressed as

ct = Ict + Tct,

with Ict the intrinsic value of the call and Tct the corresponding time value. Time value

vanishes as the expiration date of the call option approaches.

For the European put option the intrinsic value is max(K − ST , 0). Then

pt = Ipt + Tpt.

The time to maturity influences the value of each option in a different way.

Put-Call parity relationship

The prices of a call option and a put option depend on the underlying asset. Both

options are in opposite sides of the market. In the former the owner holds the right for

holding the asset at certain price while in the put option the right consists on selling the

underlying asset at the strike price.

Given ct is a lower bound of St −K exp−R0(T−t), the no-arbitrage price of the call can

be expressed as

ct = St −K exp−R0(T−t) +x

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with x ≥ 0.

The value of x is the price of a put option with the same expiration date and strike

than the corresponding call option.

Consider two portfolios A and B based on the same underlying stock that provide the

same payoffs at maturity.

• Portfolio A: a call option ct and the cash necessary to pay the strike price for the

stock at the expiration date: K exp−R0(T−t).

• Portfolio B : a stock St and a put option pt.

Both portfolios have terminal payoffs of max(ST , K). For ST ≥ K the first portfolio

has payoffs of ST paid with the proceeds of K exp−R0(T−t). The second portfolio provides

the same payoff given the put option is not exercised. For ST < K the payoffs of portfolio

A are K given the call option is not exercised. On the other hand Portfolio B’s payoffs are

K given by the proceeds of exercising the put option.

By no-arbitrage arguments if both portfolios provide the same payoffs the value at any

time before maturity should be the same. This yields

ct + K exp−R0(T−t) = St + pt.

This is denominated the put-call parity relationship. From this formula is straightforward

to derive the time value of both options. For the call option,

Tct = K −K exp−R0(T−t) +pt.

For the put option,

Tpt = K exp−R0(T−t)−K + ct.

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Suppose the price of the call option satisfies ct > St + pt − K exp−R0(T−t). At time t an

arbitrageur can device a portfolio with zero net initial outlay by selling short the call option

and borrowing (St +pt−ct) to buy the stock and the put. At the expiration date if ST ≥ K

the arbitrageur must deliver the stock by the exercise of the call option, and receives K in

return. It also pays back the loan at interest R0. Then the terminal balance is

−(St + pt − ct) expR0(T−t) +K.

This is strictly greater than zero if ct > St + pt −K exp−R0(T−t).

For ST < K the call is not exercised and the arbitrageur sells the stock at price K from

the exercise of the put. Then the gain at maturity is as before

−(St + pt − ct) expR0(T−t) +K.

American Options

These options are characterized by the possibility of exercising the option before ma-

turity. The early exercise leads to different strategies and consequences depending on the

nature of the option. For an American call option there is no advantage in exercising the

right before maturity if the goal is to hold the asset until maturity. In this case it is always

better to hold the call option and benefit from the coverage offered by the call for adverse

fluctuations of the underlying below the strike price. Note the present value of the strike

price is higher as the option is exercised earlier. If the holder’s intention is to exercise the

option to sell the asset he is better off by selling the option than by selling the stock before

maturity. Again the investor benefits from possible future downturns of the stock’s value.

Then

Ct ≤ ct,

with Ct the value of the American call option, and ct the corresponding European call.

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On the other hand a long position on an American option provides more freedom than

a European option and includes the latter as an special case. The value of the American

option must be equal or greater. Then

Ct ≥ ct.

The value of an American call option is the same than the value of a European call option.

In turn the bounds for the European call option are also satisfied by the American call

option.

max(St −K exp−R0(T−t), 0) ≤ ct ≤ St.

For a put option the situation is different. In this case exercising the option means obtaining

cash that can be further invested at the risk-free rate. Therefore it can be more profitable

to exercise the put option before maturity than waiting until the expiration date. This

implies that

Pt ≥ pt

with Pt the price of the American put option. The lower bound for the European put

option, K exp−R0T −St is strengthen by

K − St ≤ Pt

given the put option can be immediately exercised. To see this suppose the inequality holds

in the reverse direction. A long position in the put and in the stock has always less value

than the strike price. If St ≤ K the American put is exercised to obtain K. This contradicts

the proposed inequality. If St > K the put is not exercised but pt ≥ 0 by definition and

the proposed inequality is not satisfied either.

These results on American options yield the following upper and lower bounds.

St −K ≤ Ct − Pt ≤ St −K exp−R0(T−t).

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The second inequality is immediate from the put-call parity relationship given Ct = ct and

Pt ≥ pt. The first inequality is shown by considering two portfolios. Portfolio A consisting

on a call option and cash for a value of K, and Portfolio B consisting on an stock and an

American put option. If St − K > Ct − Pt the no-arbitrage arguments asserts that the

payoffs of Portfolio B are greater than the payoffs of A. If St ≥ K at the expiration date

the put option is not exercised and the payoffs should satisfy ST > K expR0T +ST given

the put option is not exercised. If St < K at expiration the put is the option exercised and

the inequality implies K > K expR0T that is false if R0 ≥ 0.

Effects of Dividends

Stock options issued on stocks paying dividends have lower value than options corre-

sponding to identical stocks without paying dividends. The payment of dividends subtracts

value to the stock and in turn diminishes value to attached options assuming fixed strike

prices.

For the case of European options the value of the dividends must be taken into account

in order to find the no-arbitrage put-call parity relationship. Using the same no-arbitrage

arguments as for the non-dividend case is easy to find the following lower bounds for a call

and a put option based on dividend-paying stocks. These are

ct ≥ St −Dt −K exp−R0(T−t),

and

pt ≥ K exp−R0(T−t) +Dt − St

with Dt the value at time t of the dividends.

The put-call parity in this case takes this form.

ct + Dt + K exp−R0(T−t) = pt + St.

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The proof is as follows. Consider at time t a long position on a call option ct, and a short

position on a stock St and a put option pt. The proceeds, S0 + p0 − c0, is invested at the

risk-free interest rate R0 until the expiration date T. If ST > K the call option is exercised

having to pay K for the stock that is used to pay the short position on the asset. The value

of the dividends at maturity T must be also paid to the investor with the long position

on the stock. In these circumstances the put option is not exercised. Therefore under

no-arbitrage opportunities the payoffs of this strategy with zero initial outlay are

(St + pt − ct) expR0(T−t)−K −DT = 0.

Then the put-call parity relationship at t is

ct + K exp−R0(T−t) +Dt = pt + St

with Dt the value at time t of the dividends. This relation holds for every period until

expiration of the options.

If ST < K the call option is not exercised, and only the put option plays a role at

expiration. The strike price K is paid for the stock that is used to close out the short

position on the stock. The dividends at maturity T must be paid to the investor holding

the long position. Under no-arbitrage opportunities the payoffs of this strategy must be

zero, and the same relation holds as before.

The corresponding lower and upper bounds for American options and stocks paying

dividends are

St −K −DT ≤ Ct − Pt ≤ St −K exp−R0(T−t).

The proof for these inequalities is the same as for the case of non-dividends. The relation

between the value of European options and an American options also hold in this case

(Ct = ct, Pt ≥ pt).

REFERENCES 16

References

[1] Bailey, R.E., (2005). The Economics of Financial Markets. Ed. Cambridge University

Press, New York (Chapters 18,19).

[2] Hull, J.C., (2006). Options, Futures and Other Derivatives. Ed. Prentice Hall (6th

ed.), New Jersey (Chapters 8,9).