Derivatives: Options - Financial Markets and Intermediariesdocenti.luiss.it/protected-uploads/310/2012/01/20120123184042-Fin… · Option Basic Relations Strategies Involving Options

OverviewOption Basic Relations

Strategies Involving OptionsThe Binomial Model and Risk Neutral Valuation

Warrants and Convertibles

Derivatives: OptionsFinancial Markets and Intermediaries

Paolo Vitale

LUISS University

pvitale@luiss

Paolo Vitale Derivatives: Options




Outline

Overview

Option Basic Relations

Strategies Involving Options

The Binomial Model and Risk-Neutral Valuation






The Characteristics of OptionsOptions Pay-offsMoneyness, Intrinsic Value and Time ValueDeterminants of Option Values

Options

Options on stocks were first traded on an organized exchangein 1973.

Since then, there has been a dramatic growth in options mar-kets.

Options are now traded

? on many different exchanges around the world and

? over the counter by banks and financial institutions.

The underlying assets include stocks, stock indexes, foreigncurrencies, debt instruments, commodities, and futures.






Options (cont.ed)

There are two main classes of options:

Definition (Call Option)

A call option gives the holder the right to buy the underlying assetby (at a) certain date for a certain price.

Definition (Put Option)

A put option gives the holder the right to sell the underlying assetby (at a) certain date for a certain price.

Terminology: The price in the contract is known as the exerciseprice or strike price.

Terminology: The date in the contract is referred to as the expira-tion date, exercise date or maturity.






Options (cont.ed)

There are two main types of options:

Definition (American Option)

An American option can be exercised at any time up to the expi-ration date.

Definition (European Option)

A European option can only be exercised on the expiration date.

Remark: The terms American and European do not refer to thelocation of the option or the exchange.

Remark: While most traded options are American options, it issimpler to analyze European ones, and usually the proper-ties of the former are derived from those of the latter.






Options versus Futures Contracts

Options and futures differ in two main respects:

The holder of an option possesses the right to exercise,

? i.e. she can buy or sell the underlying asset at (by) a givendate for a certain price,

but differently from futures she is not obliged to do it.

Whereas it costs nothing to enter into a forward or futurescontract, an investor must pay to purchase an option contract.






Summary

An option presents the following contractual specification:

Option type: Call or put, American or European;

Underlying: The asset to be delivered;

Contract size: The quantity to be delivered;

Maturity: The time at which delivery is to take place;

Strike price: The price to be paid at delivery.

Terminology: An option series consists of all the options of a givenclass with the same maturity and strike price.






Options’ Pay-offs: An Example

James buys 100 European call options on IBM stock with a strikeprice (X ) of $140.

The option is European and can be exercised only at maturity.

At maturity (time T ) we have that:

? If the stock price (ST ) is below the strike price or equal to thestrike price (ST ≤ X ), James will not exercise;

? If the stock price (ST ) is above the strike price (ST > X ), hewill exercise the 100 options.

Suppose the current option price (ct) is $5.

? If ST = 155 James will gain a total profit of $1000, as with aninitial investment of $500 he will be able to buy for a cost of$14,000 100 shares which are worth $15,500.

? If instead ST = 130 James will lose his investment of $500.






Options’s Pay-offs: An Example (cont.ed)

6

ST

-

Total profits

1400

10

155

-5

........................

Figure 4: Profits from a long position in a call option

1

Above a representation of the total profits of a long positionin the call option as function of the final stock price, ST .

Similar representations can be drawn for short positions andfor put options.






Pay-offs and Option Positions

In every option contract there are two sides:

? The investor who takes the long position buying the option.

? The investor who takes the short position selling (writing) it.

The writer of the option receives cash upfront, but has po-tential liabilities later.

His or her profits are the reverse of those of the purchaser:

? The owner of an option faces no downside risk;

? the writer potentially faces unlimited losses.

Remark: While the purchaser of a call (put) option hopes that thestock price will increase (fall), the writer of a call (put)option hopes that the stock price will fall (increase).






Moneyness

An option is said to be in the money if it would lead to apositive cash flow to the holder if it were exercised now.

An option is said to be out of the money if it would lead to anegative cash flow to the holder if it were exercised now.

An option is said to be at the money if it would lead to zerocash flow to the holder if it were exercised now.

Example: Suppose at the outset the price of IBM stock is $138.Then, the call options purchased by James would be outof the money.






Intrinsic Value and Time Value

The intrinsic value of an option is the maximum of zero andthe value it would have if it were exercised now.

For a call option it is the maximum of zero and the differencebetween the current price of the underlying asset, St , and thestrike price, X ,

max[0,St − X ].

For a put option it is the maximum of zero and the differencebetween the strike price, X , and the current price of the un-derlying asset, St , max[0,X − St ].

The time value of an option is the difference between the ac-tual option price and its intrinsic value.

Such difference exists as the holder my find it optimal to waitrather than exercise immediately.






Intrinsic Value and Time Value of a Call Option6

-StX

intrinsic

value

Out of the In themoney money

Option

value

Value of

call option

...

Time

valuej

sY

Figure 4: Intrinsic and Time Value of a Call Option

1

Above a representation of the intrinsic value and actual valueof a call option as function of the current stock price, St .

The distance between the kinked line and the continuous linecorresponds to the time value of the option, which is positiveeven if the option is out of the money!






Factors Affecting Option Prices

There are several factors which affect option prices:

The current stock (underlying asset) price, St ;

The strike price, X ;

The time to expiration, T − t;

The volatility of the stock price, σ;

the risk-free (continuously compounding) interest rate, r ;

the dividends expected during the life of the option, D.






Option Prices and Fundamental Factors

In the following Table we present a summary of the effect of chan-ges in parameter values on option prices:

European European American AmericanVariable Call, c Put, p Call, C Put, P

Stock Price, St + - + -Strike Price, X - + - +Time to Maturity, T − t ? ? + +Volatility, σ + + + +Risk-free rate, r + - + -Dividends, D - + - +






Option Prices, Maturity, Stock and Strike Price

Stock and strike price: If exercised at time τ the pay-off of acall (put) option is the amount by which the stock price ex-ceeds the strike price, Sτ − X (strike price exceeds the stockprice, X − Sτ ).

As St rises (falls) and X falls (rises) the expected pay-off of acall (put) option increases and with this its current value.

Time to maturity: When time to expiration augments Ameri-can options become more valuable as the opportunities to ex-ercise them increase.

The same cannot be said of European options as the opportu-nities to exercise are independent of time to maturity.






Option Prices and Stock Price Volatility

Volatility: As the volatility of the stock price rises the chancethat the stock price will be either very large or very small rises.

The owner of call (put) option benefits from an increase (fall)in the stock price but faces no downside risk if the stock pricedecreases (rises).

Example: Consider a European call option which is currently at themoney. Let us consider two scenarios:

1 Suppose there is a 50% chance the stock price rises (falls) by5% before maturity (low volatility scenario);

2 Suppose there is a 50% chance the stock price rises (falls) by10% before maturity (large volatility scenario).

Question: Under which scenario is the option worth more?Paolo Vitale Derivatives: Options





Call Option Pay-off with Low and High Volatility

6

-STX

-5%-10% +5% +10%

Call option

pay-off

...........

.....................

1

1

2

2

Figure 4: Pay-off of a Call Option under Different Volatilities

1

In the low stock price state the return on the option is zero(no downside risk) in both scenarios (1 and 2).

In the high stock price state the return on the option is largerin the large volatility scenario (2).

As the expected return on the option is larger in the large vo-latility scenario (2) the option is pricer with greater volatility.






Option Prices, Risk-Free Interest Rate and Dividends

Risk-free interest rate: As the risk-free interest rate augmentstwo effects coexist:

1 the present value of future pay-off decreases;

2 others things equal, the stock price grows at a larger rate.

It can be shown that these two effects have a positive (nega-tive) impact on the price of call (put) options.

Dividends: When dividends are paid-out the stock price falls.Thus, their rise affects option prices as a fall in the stock price.





Lower and Upper Bounds on Option PricesEarly Exercise of American OptionsPut-Call Parity

Some More Notation

We introduce some notation to describe the characteristics ofoptions and their relation to the corresponding prices.

Let t denote the current time, T the option maturity, S thecurrent price of the underlying asset, and X the strike price.

Then, at time t we indicate with:

? c(t,S ;T ,X ) the price of a European call option with maturityT and strike price X ;

? p(t,S ;T ,X ) the price of the equivalent European put option;

? C (t,S ;T ,X ) the price of the equivalent American call option;

? P(t,S ;T ,X ) the price of the equivalent American put option.






Lower Bounds on Option Prices

Options always have a non-negative value,

c(t,S ;T ,X ) ≥ 0 p(t,S ;T ,X ) ≥ 0,

C (t,S ;T ,X ) ≥ 0 P(t,S ;T ,X ) ≥ 0.

This is because their final pay-offs are non-negative.

American options are at least as valuable as their Europeancounterparts,

C (t,S ;T ,X ) ≥ c(t,S ;T ,X ),

P(t,S ;T ,X ) ≥ p(t,S ;T ,X ).

This is because exercising an American option at maturitypay-offs at least as high as its European counterpart.






Upper Bounds on Option Prices

A call option is worth no more than the underlying asset,

c(t,S ;T ,X ), C (t,S ;T ,X ) ≤ S .

A put option is worth no more than the present value of thestrike price,

p(t,S ;T ,X ), P(t,S ;T ,X ) ≤ X e−r(T−t),

where r is the continuously compounding interest rate.

To prove these upper and lower bounds we apply arbitragearguments which are valid irrespective of any assumption onthe dynamics of the value of the underlying asset.






Upper Bounds on Option Prices (Proof)

We prove the first inequality considering the following portfolio.

Position Investment Pay-off

ST ≤ X ST > X

Long 1 unit of the asset S ST ST

Short 1 call option −c 0 −(ST − X )

Total S − c ST ≥ 0 X > 0

Then, the statement (c ≤ S) must be true to avoid arbitrage.

With a similar argument you can prove the second inequality.






More Lower Bounds on Option Prices

A call option is worth no less than the present value of itsexercise pay-off,

c(t,S ;T ,X ), C (t,S ;T ,X ) ≥ S − X e−r(T−t).

A put option is worth no less than the present value of itsexercise pay-off,

p(t,S ;T ,X ), P(t,S ;T ,X ) ≥ X e−r(T−t) − S .






More Lower Bounds on Option Prices (Proof)

We prove the first inequality considering the following portfolio.


ST ≤ X ST > X

Short asset −S −ST −ST

Long call option c 0 ST − XLend p.v. of X Xe−r(T−t) X X

Total c − S + Xe−r(T−t) X − ST ≥ 0 0

Then, the statement (c ≥ S − Xe−r(T−t)) must be true to avoidarbitrage.

With a similar argument you can prove the second inequality.






Upper Bounds for A Call Option Price

6

S

-

c

X e−r(T−t) X

Figure 4: Bounds for a Call Option Price

1

A graphical representation of the upper and lower bounds on a calloption price, c , as function of the underlying asset price, S,

max{0, S − X e−r(T−t)} ≤ c(t,S ;T ,X ) ≤ S .

In the shaded area the admissible values for the call option price, c .






Upper Bounds for A Put Option Price

6

S

-

X e−r(T−t)

p

Figure 4: Bounds for a Put Option Price

1

A graphical representation of the upper and lower bounds on a putoption price, p, as function of the underlying asset price, S,

max{0, X e−r(T−t) − S} ≤ p(t,S ;T ,X ) ≤ X e−r(T−t).

In the shaded area the admissible values for the put option price,p.






American Options and Early Exercise

An important is issue is whether to exercise American options be-fore maturity. In this respect we have an important result:

Claim

American call options on assets which do not pay income shouldnever be exercised early.

This is because no income is forgone;

payments of the underlying asset is postponed;

the insurance element of the option is preserved.

Claim

American call options on assets which do not pay income have thesame value of European call options with the same characteristics.






American Options and Early Exercise

A similar results does not hold for American put options.

Claim

An American put option which is sufficiently deep in the moneyshould be exercised early.

Example: Consider an American put option with strike price $10on a stock whose price is virtually zero.

If exercised the holder makes an immediate gain of $10.

If she waits at most she can expect to gain $10.

Claim

American put options are worth more than European call optionswith the same characteristics.






Relations between Options

In synthesis using simple arbitrage arguments we have shown that:

For no-dividend-paying stocks corresponding American andEuropean call options are worth the same,

c(t,S ;T ,X ) = C (t,S ;T ,X ).

For no-dividend-paying stocks American put options are worthmore than corresponding European put options,

p(t,S ;T ,X ) ≤ P(t,S ;T ,X ).






The Put-Call Parity

Claim (Put-Call Parity)

Given two European call and put options with same characteristics,the following (put-call) parity holds,

S − c(t,S ;T ,X ) = X e−r(T−t) − p(t,S ;T ,X ).

To prove this result, which allows to derive prices for puts (calls)from their call (put) counter-parties, we consider the following:

Portfolio A: Buy 1 unit of the underlying asset + sell the calloption.

Portfolio B: Lend the present value of the strike price + sellthe put option.






The Put-Call Parity (proof)

Then, consider the following Table.

Position in A Investment Pay-off

ST ≤ X ST > X

Long 1 unit of the asset S ST ST

Short 1 call option −c 0 −(ST − X )

Total S − c ST X

Position in B Investment Pay-off

ST ≤ X ST > X

Lend the present value of X Xe−r(T−t) X XShort 1 put option −p −(X − ST ) 0

Total Xe−r(T−t) − p ST X

As the pay-offs are the same the parity must hold.





Speculation and Hedging Using OptionsHedges or Option-StocksBull, Bear and Butterfly SpreadsStraddles, Strips and Straps, Strangles

The Use of Options

Options, as other derivatives, can be used to:

Hedge risk: we refer in these cases to hedgers, who use theseinstruments to protect their investments against large swingsin asset prices.

Speculate: speculators magnify their position into risky assetsby using derivatives, as these require only a small initialinvestment.

Arbitrage: Some agents, also known as arbitrageurs, try toexploit small errors in securities prices to gain arbitrage profits.






The Use of Options: Example of Hedging

Assume that:

Ann possesses 2,000 shares of BA and £1,260 in cash.

Ann can buy European put options on BA stock with twoweeks maturity, strike price, X , 650p and price, p, 63p.

The current price of the underlying stock, S , is 630p.

In two weeks time either the price of the underlying stockjumps up to 800p or falls to 500p.






The Use of Options: Example of Hedging (cont.ed)

Ann can buy 2,000 put options (her position is covered).


ST = £8 ST = £5

2,000 shares of BA £12,600 £16,000 £10,0002,000 put options £1,260 0 £3,000

Total £13,860 £16,000 £13,000

Thus, if the probability that BA stock price falls is large the putoption provides Ann with some form of costly protection.






The Use of Options: Example of Speculation

If Ann is sure that in two weeks the stock price will fall, she can

1 sell her 2,000 shares of BA stock and

2 buy 22,000 put options (her position is naked).


ST = £8 ST = £5

22,000 put options £13,860 0 £33,000

If Ann is right she makes £33,000 with a profit of £19,140.

Otherwise she loses everything.






Types of Option Strategies

Trading strategies involving options can be classified in

Hedges: These combine an option position with a position inthe underlying asset (long stock, short call).

Spreads: Combine options of the same type (calls or puts),but with different strike prices or maturities (bull, bear, but-terfly spread).

Combinations: Combine options of different types, strikeprices and maturities (straddle, strip, strap and strangle).






Option Strategies: Hedges Using Call Options

6

-

6

-ST

ST

X

X

Long stock short call Short stock long call

Total profitsTotal profits

Figure 4: Stock and Call Options

1

In the left panel a long position in the stock is covered by wri-ting the call option.

In the right panel a short position in the stock is covered witha long position in the call option.






Option Strategies: Hedges Using Put Options

6

-

6

-ST

ST

X

X

Long stock long put Short stock short put


Figure 4: Stock and Put Options

1

In the left panel a long position in the stock is protected witha long position in the put option.

In the right panel a short position in the stock is protected bywriting the put option.






Option Strategies: Bull Spreads

These strategies are used when investors want to bet on bullmarkets with limited downside risk and upside potential.

Call bull spreads: You need to

1 buy 1 call with small strike price, X1,

2 and sell 1 call with large strike price, X2 (with X1 < X2).

Put bull spreads: You need to

1 buy 1 put with small strike price, X1,

2 and sell 1 put with large strike price, X2 (with X1 < X2).






Option Strategies: Bull Spread Using Call Options


Position Pay-off

ST ≤ X1 X1 < ST ≤ X2 ST > X2

Buy 1 call option at X1 0 ST − X1 ST − X1

Short 1 call option at X2 0 0 X2 − ST

Total 0 ST − X1 X2 − X1

The downside risk and upside potential are both limited.

? If ST < X1 the final pay-off is zero, but a loss is incurred asthe call option with the lower strike price is more expensive.

? If ST > X2 the final pay-off is just X2 − X1.






Option Strategies: Call and Put Bull Spreads

6

-

Total profits

ST

X1 X2

Call bull spread

−c(t, S, T, X1)

c(t, S; T, X2)

Figure 4: Bull Spread using Call Options

1

6

-

Total profits

ST

X1 X2

Put bull spread

−p(t, S; T, X1)

p(t, S; T, X2)

Figure 4: Bull Spread using Put Options

1

Here the total profits of the bull spreads are represented.

In the left panel a long position in a call option with smallstrike price and a short in a call option with large strike price.

In the right panel a long position in a put option with smallstrike price and a short in a put option with large strike price.






Option Strategies: Bear Spreads

These strategies are used when investors want to bet on bearmarkets with limited downside risk and upside potential.

Call bear spreads: You need to

1 sell 1 call with small strike price, X1,

2 and buy 1 call with large strike price, X2 (with X1 < X2).

Put bear spreads: You need to

1 sell 1 put with small strike price, X1,

2 and buy 1 put with large strike price, X2 (with X1 < X2).






Option Strategies: Bear Spread Using Call Options


Position Pay-off

ST ≤ X1 X1 < ST ≤ X2 ST > X2

Sell 1 call option at X1 0 X1 − ST X1 − ST

Buy 1 call option at X2 0 0 ST − X2

Total 0 X1 − ST X1 − X2

The downside risk and upside potential are both limited.

? If ST < X1 the final pay-off is zero, but a profit is gained asthe call option with the lower strike price is more expensive.

? If ST > X2 the final pay-off is negative as X1 − X2 < 0.






Option Strategies: Bear Spread Using Call Options6

-

Total profits

STX1 X2

Call bear spread

−c(t, S; T, X2)

c(t, S; T, X1)

Figure 4: Bear Spread using Call Options

1

Here the total profits of the bear spread using call options.

It is now evident that a small profit is gained if the stock pricefalls and a small loss is incurred if the stock price rises.






Option Strategies: Butterfly Spreads

These strategies are used when investors want to bet on low vo-latility with limited downside risk.

Call butterfly spreads: You need to:

1 buy 1 call with small strike price, X1,

2 sell 2 calls with medium strike price, X2,

3 buy 1 call with large strike price, X3 (with X1 < X2 < X3).

Put butterfly spreads: You need to:


2 sell 2 puts with medium strike price, X2,

3 buy 1 put with large strike price, X3 (with X1 < X2 < X3).






Option Strategies: Butterfly Spread Using Call Options

In the following Table we report the final pay-offs of a butterflyspread using call options under the assumption that,

X2 =1

2(X1 + X3) .

Position Pay-off

ST ≤ X1 X1 < ST ≤ X2 X2 < ST ≤ X3 ST > X3

Buy 1 call at X1 0 ST − X1 ST − X1 ST − X1

Short 2 calls at X2 0 0 2(X2 − ST ) 2(X2 − ST )Buy 1 call at X3 0 0 0 ST − X3

Total 0 ST − X1 X3 − ST 0






Option Strategies: Butterfly Spread Using Call Options6

-

Total profits

STX1 X2 X3

−c(t, S; T, X1)

−c(t, S; T, X3)

2c(t, S; T, X2)

Butterfly spread using call options

Figure 4: Butterfly Spread using Call Options

1

Here the total profits of the butterfly spread using call options.

It is clear that a small profit is gained if the stock priceremains in the interval between X1 and X3.






Option Strategies: Straddles

Straddles allow to bet on high volatility with

1 limited downside risk

2 and unlimited upside potential.

You need to

1 buy 1 call with strike price X , and

2 buy 1 put with the same strike price, X .






Option Strategies: Straddle6

-

Total profits

STX

Straddle

−c(t, S; T, X)

−p(t, S; T, X)

Figure 4: Straddle

1

Here the total profits of the straddle.

(Even large) profits are gained if the stock price wanders awayfrom the strike price.






Option Strategies: Strips and Straps

Strips and straps are similar to straddles, but these strategies beton downward or upward movement.

Strip: You need to

1 buy 1 call with strike price X , and

2 buy 2 puts with strike price, X .

Strap: You need to

1 buy 1 put with strike price X , and

2 buy 2 calls with strike price, X .






Option Strategies: Strips and Straps (cont.ed)

6

-

6

-ST STX X

Strip Strap


−c(t, S; T, X)

−2p(t, S; T, X)−2c(t, S; T, X)

−p(t, S; T, X)

Figure 4: Strip and Strap using Call and Put Options

1

Here the total profits of the strip and strap.

Profits are gained if the stock price wanders away from thestrike price.

Profits from the strip (strap) are large if the stock price falls(rises) substantially.






Option Strategies: Strangles

Strangles are similar to straddles, but with reduced downside risk.

You need to

1 buy 1 call with large strike price, X2,


where X1 < X2.






Option Strategies: Strangle Using Call Options6

-

Total profits

STX1 X2

Strangle using call and put options

−c(t, S; T, X2)

−p(t, S; T, X1)

Figure 4: Strangle using Call and Put Options

1

Here the total profits of the strangle using call and put options.

Profits are gained if the stock price wanders away from thestrike prices X1 and X2, with limited loss if it stays within.





The One-period Binomial ModelRisk Neutral ValuationRisk Neutral ProbabilitiesThe General Multi-period Binomial Model

The One-period Binomial Model

We now discuss the binomial model, from which we derive thefoundations of option pricing.

Example: Assume that

at time 0 a no-dividend-paying stock is priced at S = 100,

a European call option is offered with strike price X = 107and maturity T ,

the interest for the period from 0 to T is r = 7% (so that thegross rate of return on a risk-free investment is R = 1.07).

We wonder what is the proper price for the call option.






A Simple Stock Price Dynamics Modelt0 = 0 t1 = T

‘up’

‘down’

Stock price = 125

Stock price = 80

Stock price S = 100

Call price c = ?

Call price = 18

Call price = 0

Figure 4: Pricing a Call Option within a Binomial Model

1

Assume in the period between time 0 and T the stock price can

either move up by a factor u = 1.25 to uS = 125,

or move down by a factor d = 0.80 to dS = 80.

In the former event the option pays 18 in the latter it is worthless.






The Risk-Free Portfolio

To price this option we define a risk-free portfolio which must paythe gross rate of return R over the period from 0 to T .

To find it consider the following Table.


Up-Move Down-Move

Long ∆ shares of the stock 100∆ 125 ∆ 80 ∆Short 1 call option −c −18 0

Total 100 ∆− c 125 ∆− 18 80 ∆

We choose ∆ so that 125 ∆− 18 = 80 ∆, ie. ∆ = 0.40.

Remark: This portfolio is risk-free, as whatever the stock priceat time T , we obtain the same final pay-off of 32.






The No-arbitrage Price of the Option

Thus, we have derived the following portfolio:

Portfolio A: Buy 0.40 units of the stock for a price of 40 andsell 1 call option for a price of c .

Hence, the total investment is 40− c and will deliver at timeT a certain pay-off of 32.

Therefore, portfolio A presents a certain gross rate of return,RA, equal to,

RA =32

40− c.

To avoid arbitrage this must be equal to R = 1.07, so that,

32

40− c= 1.07 ie. c = 40 − 32

1.07= 10.09.






The Replicating Portfolio

To get to the same conclusion we can use a replicating portfolio,as indicated in the following Table.


Up-Move Down-Move

Long 0.40 shares of the stock 40 50 32Borrow the p.v. of 32 (32/R) −29.91 −32 −32

Total 10.09 18 0

Remark: This portfolio replicates the final pay-off of the call optionand must have the same value of the call at time 0.






Risk Neutral Probabilities

We now generalize our example assuming that:

at time 0 the stock price is S ;

a European call option is offered with strike price X and ma-turity T ;

the interest for the period from 0 to T is r (the risk-free grossrate of return is R = 1 + r),

over the period between time 0 and T the stock price can

? either move up by a factor u to uS with probability q,

? or move down by a factor d to dS , with probability 1− q,where d < R < u.






Risk Neutral Probabilities (cont.ed)

Let us indicate

? with cu the final pay-off of the call option when the price mo-ves up (cu = max [0, uS − X ]),

? and with cd the final pay-off of the call when the price movesdown (cd = max [0, dS − X ]).

We can build portfolio A by purchasing ∆ shares of the stockand shorting 1 call option.

Portfolio A will have final pay-off

? ∆ u S − cu if the stock price moves up to uS ,

? ∆ d S − cd if the stock price moves down to dS .







We take ∆ so that portfolio A is risk-free.

Then, to avoid an arbitrage opportunity its gross return mustbe equal to R, so that,

(∆S − c) R = ∆ u S − cu = ∆ d S − cd .

Since∆uS − cu = ∆dS − cd ,

for any value of q,

(∆S − c) R = q (∆ u S − cu) + (1− q) (∆ d S − cd) .







Arbitrary take the value of q which would prevail in a risk-neutral world.

In this world the stock price at time 0 equals the expectedprice prevailing at time T , discounted at the risk-free rate r .

In other words, in a risk-neutral world q is such that,

S =1

R

(q u S + (1− q) d S

).

Solving for q we find that the risk-neutral probability is

q =R − d

u − d.






The Call Option Price

Inserting the risk-neutral value of q into the equilibrium condi-tion for ∆ we obtain the equilibrium value for the current calloption price,

c =1

R

(q cu + (1− q) cd

).

This value corresponds to the expected final pay-off of the calloption discounted using the risk-free rate r .

This is the fair price of the option in a risk-neutral world.

Remark: This pricing technique can be used to price any derivati-ve whose pay-offs depend on a stock price movements.






The Call Option Price: An Example

Reconsider our example where u = 1.25, d = 0.80 and R = 1.07,

Using the formula for the risk-neutral probability we find that,

q =1.07 − 0.80

1.25 − 0.80= 0.6.

Considering that

? cu = max [0, uS − X ] = max [0, 125 - 107] = 18,

? cd = max [0, dS − X ] = max [0, 80 - 107] = 0,

Using the pricing formula for the call option we then find,

c =1

1.07

(0.60 · 18 + 0.40 · 0

)= 10.09.






The Two-period Binomial Model

We extend the one-period model assuming that:

the period between time 0 and T is divided in 2 equal inter-vals,

0 = t0 < t1 < t2 = T ;

a European call option exists at time 0 with strike price X andmaturity T ;

and in any interval j the interest rate for the period from tj−1

to tj is r2.






The Two-period Binomial Model (cont.ed)

Then, let us assume that for any single period we can replicate theone period model.

Suppose then that:

At time tj−1 (j = 1, 2) the stock price is Sj−1, with S0 = S ;

over the period between time tj−1 and tj the stock price can

? either move up by a factor u to uSj−1 with probability q

? or move down by a factor d to dSj−1, with probability 1− q,where d < R2 ≡ 1 + r2 < u.






The Two-period Binomial Model: An Example

S= 100

c= ?

Su = 111.80

cu = ?

Sd = 89.40

cd = 0

Suu = 125.00

cuu = 18

Sud = 100.00

cud = 0

Sdd = 80.00

cdd = 0

t0 = 0 t1 t2 = T

Figure 4: Binomial Tree with m = 2: Call Option and Stock Price Dynamics

1

Here the dynamics of the price of the stock, with S = 100,and the value of the call, with X = 107 and maturity T , for

R2 = 1.034, u = 1.118 and d = 0.894.

Thus,R2

2 ≈ 1.07, u2 ≈ 1.25 and d2 ≈ 0.80.






Recursive Pricing Formulae

To calculate the values of cu and c , just consider the risk-neutral probability,

q =R2 − d

u − d=

1.034 − 0.894

1.118 − 0.894= 0.625.

Then, using risk-neutral evaluation we have that,

cu =1

1.034

(0.625 · 18 + 0.375 · 0

)= 10.88.

Hence, proceeding backward, we find that,

c =1

1.034

(0.625 · 10.88 + 0.375 · 0

)= 6.58.






The Two-period Binomial Model: The General Case

c

cu

cd

cuu = max [0, u2S −X]

cud = max [0, udS −X]

cdd = max [0, d2S −X]

t0 = 0 t1 t2 = T

Figure 4: Binomial Tree with m = 2: Call Option Dynamics

1

Here we represent the generic dynamics of the call optionprice for R2, u and d arbitrary.

With a similar argument we can determine a generic expres-sion for the call option price.






Two-period Binomial Option Pricing Formula

Let us determine the call price one-period ahead, using thefamiliar formulas,

cu =1

R2

(q cuu + (1− q) cud

),

cd =1

R2

(q cud + (1− q) cdd

),

where,

q =R2 − d

u − d, cjk = max [0, jkS − X ], for j , k = d , u.






Two-period Binomial Option Pricing Formula (cont.ed)

Then, we can determine the current price of the call as follows,

c =1

R2

(q cu + (1− q) cd

),

=1

R22

(q2 cuu + 2 q (1− q) cud + (1− q)2 cdd

).

Clearly, this formula can be extended for any multi-periodformulation of the binomial model.





Pricing WarrantsPricing Convertibles

Warrants

Definition (Warrant)

A warrant gives the owner the right to buy a fixed number of sha-res at a specified price, by a given date.

A warrant is similar to a call option.

However, when a warrant is exercised new shares are created,which dilutes their value.






Convertible Bonds

Definition (Convertible Bonds)

A convertible bond pays coupons and principal and has priorityover equity in the event of bankruptcy. The owner is also entitledto exchange the convertible bond for a specified number of sharesof newly issued common stock, by a given date.

Differently from warrants, convertible bonds do not requirethe payment of an exercise price.

Typically the owner of a convertible bond can sell its warrantcomponent separately from the debt security component.






Pricing Warrants

Consider a corporation that has

1 n outstanding shares, and

2 m European warrants.

Assume that each warrant at maturity T can be converted atthe strike price X into 1 share of newly issued common stock.

Indicating with Wt and St the value at time t of respectivelya warrant and the stock price, the corresponding value of thecorporation is,

Vt = n St + m Wt .






Dilution and Exercise Decision

Assume that

1 at maturity VT indicates the value of the corporation

2 and that all the warrants are exercised.

Then, the value of the corporation increases to VT + mX witha new number of outstanding shares now equal to n + m.

The new stock price is thus,

ST =

(1

n + m

)(VT + m X ).

Then in case of exercise, the pay-off of the warrant holder is,(VT + m X

n + m− X

)=

n

n + m

(VT

n− X

).






Dilution and Exercise Decision (cont.ed)

The warrant-holder will exercise only if this value is positiveand hence his pay-off is,

n

n + mmax

(VT

n− X , 0

).

Thus, at time t the value of the warrant is equal to the valueof

n

n + m

regular call options on a fraction 1/n of the corporation, who-se leveraged value is currently Vt .






The Value of Stock and Warrants

Let us look at the problem from an aggregate point of view.

If the m warrants are exercised, we find that

1 the n pre-existing shares will be worth,

n ST =

(n

n + m

)(VT + mX ) ,

2 while the m warrants will be worth,

m WT =

(m

n + m

)(VT − nX ) .






Stock and Warrants Pay-offs at Maturity

(1)

nX VT

nST

nX

nX

(λ)

(0)

VT

00

(1− λ) mWT

Common stockWarrants

Figure 4: Stock and Warrants Pay-offs at Maturity

1

Here the final pay-offs of common stock and warrants whereλ ≡ m/(n + m) denotes the dilution factor of the stock.

One can prove that the value of the n shares and the mwarrants at time t is,

n St = Vt − λ c(t,Vt ;T , nX ),

m Wt = λ c(t,Vt ;T , nX ) .






Pricing Convertibles

Consider a corporation that has

1 n outstanding shares and

2 m convertible zero-coupon bonds.

Assume that

1 such bonds promise to pay an overall principal, P, at the ma-turity date T ,

2 and each of them can be converted into k shares of newly is-sued common stock immediately before maturity, T .

Indicating with Bt and St the value at time t of respectivelythe convertible bonds and the stock price, the correspondingvalue of the corporation is,

Vt = n St + Bt .






Stock and Convertible Bonds Pay-offs at Maturity

VT

(1)

nST BT

(1)

VT

(0)

00

(λ)

(1− λ)

(0)

P P/λ

P

PP/λ

Common stock Convertible bonds

Figure 4: Stock and Convertible Bonds Pay-offs at Maturity

1

In the graph the pay-offs at maturity of common stock andconvertible bonds.

If conversion is chosen, then the bondholders would own thefraction λ ≡ mk/(n + mk) of the firm.

Bondholders will therefore convert when λVT > P.






The Value of Stock and Convertible Bonds

Clearly most convertible bonds can be converted any time be-fore the maturity date.

However in the absence of anticipated dividend payments, asfor American call options, early conversion is never optimal.

The current value of the shares, and the convertible bonds arethen respectively,

n St = c(t,Vt ;T ,P) − λ c(t,Vt ;T ,P/λ),

Bt = Vt − c(t,Vt ;T ,P) + λ c(t,Vt ;T ,P/λ).


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