fi8000 option valuation i milind shrikhande. a call option a european call option gives the buyer of...

Fi8000Fi8000 Option Valuation I Option Valuation I

Milind ShrikhandeMilind Shrikhande

A Call OptionA Call Option

A European call option gives the buyer A European call option gives the buyer of the option of the option a right to purchasea right to purchase the the underlying asset, at the underlying asset, at the exerciseexercise priceprice on the on the expiration date.expiration date.

It is optimal to exercise the call option if the It is optimal to exercise the call option if the stock price exceeds the strike price:stock price exceeds the strike price:

CCTT = Max {S = Max {ST T – X, 0}– X, 0}

A Put OptionA Put Option

A European put option gives the buyer A European put option gives the buyer of the option of the option a right to sella right to sell the the underlying asset, at the underlying asset, at the exerciseexercise priceprice on the on the expiration date.expiration date.

It is optimal to exercise the put option if the It is optimal to exercise the put option if the stock price is below the strike price:stock price is below the strike price:

PPTT = Max {X - S = Max {X - ST T , 0}, 0}

The Put Call ParityThe Put Call Parity

If two portfolios have the same payoffs in If two portfolios have the same payoffs in every possible state and time in the future, every possible state and time in the future, their prices must be equal:their prices must be equal:

(1 )TX

C S Prf

Arbitrage – the Law of One PriceArbitrage – the Law of One Price

If two assets have the same payoffs in every If two assets have the same payoffs in every possible state in the future and their prices possible state in the future and their prices are not equal, there is an opportunity to are not equal, there is an opportunity to make an arbitrage profit.make an arbitrage profit.

We say that there exists an arbitrage We say that there exists an arbitrage opportunity if we identify that:opportunity if we identify that:

There is no initial investmentThere is no initial investment

There is no risk of lossThere is no risk of loss

There is a positive probability of profitThere is a positive probability of profit

Arbitrage – a Technical DefinitionArbitrage – a Technical Definition

Let CFLet CFtjtj be the cash flow of an investment be the cash flow of an investment

strategy at time t and state j. If the following strategy at time t and state j. If the following conditions are met this strategy generates an conditions are met this strategy generates an arbitrage profit.arbitrage profit.

(i)(i) all the possible cash flows in every all the possible cash flows in every possible state and time are positive or zero - possible state and time are positive or zero - CFCFtjtj ≥ 0 for every t and j. ≥ 0 for every t and j.

(ii)(ii) at least one cash flow is strictly positive - at least one cash flow is strictly positive - there exists a pair ( t , j ) for which CFthere exists a pair ( t , j ) for which CF tjtj > 0. > 0.

Arbitrage – an ExampleArbitrage – an ExampleIs there an arbitrage opportunity if the Is there an arbitrage opportunity if the following are the market prices of the assets:following are the market prices of the assets:

The price of one share of stock is $39;The price of one share of stock is $39;

The price of a call option on that stock, The price of a call option on that stock, which expires in one year and has an which expires in one year and has an exercise price of $40, is $7.25;exercise price of $40, is $7.25;

The price of a put option on that stock, The price of a put option on that stock, which expires in one year and has an which expires in one year and has an exercise price of $40, is $6.50;exercise price of $40, is $6.50;

The annual risk free rate is 6%.The annual risk free rate is 6%.

Arbitrage – an ExampleArbitrage – an Example

In this case we must check whether the put In this case we must check whether the put call parity holds. Since we can see that this call parity holds. Since we can see that this parity relation is violated, we will show that parity relation is violated, we will show that there is an arbitrage opportunity.there is an arbitrage opportunity.

1

$40$7.25 $44.986

(1 ) (1 0.06)

$39 $6.50 $45.5

T

XC

rf

S P

The Construction ofThe Construction ofan Arbitrage Transactionan Arbitrage Transaction

Constructing the arbitrage strategy:Constructing the arbitrage strategy:

1.1. Move all the terms to one side of the equation Move all the terms to one side of the equation so their sum will be positive;so their sum will be positive;

2.2. For each asset, use the sign as an indicator of For each asset, use the sign as an indicator of the appropriate investment in the asset. If the the appropriate investment in the asset. If the sign is negative then the cash flow at time t=0 is sign is negative then the cash flow at time t=0 is negative (which means that you buy the stock, negative (which means that you buy the stock, bond or option). If the sign is positive reverse bond or option). If the sign is positive reverse the position.the position.


In this case we move all terms to the LHS:In this case we move all terms to the LHS:

$45.5 $44.986 $0.514 0(1 )

. .

0(1 )

T

T

XS P C

rf

i e

XS P C

rf


Time: →Time: → t = 0t = 0 t = Tt = T

Strategy: ↓Strategy: ↓ State: →State: → SST T < X = 40< X = 40 SST T > X = 40> X = 40

Short stockShort stock +S=$39+S=$39 -S-STT -S-STT

Write putWrite put +P=$6.5+P=$6.5 -(X-S-(X-STT)) 00

Buy callBuy call -C=(-$7.25)-C=(-$7.25) 00 (S(STT-X)-X)

Buy bondBuy bond -X/(1+rf)=(-$37.736)-X/(1+rf)=(-$37.736) XX XX

Total CFTotal CF S+P-C-S+P-C-X/(1+rf) X/(1+rf)

= 0.514 > 0= 0.514 > 0

-S-ST T -(X-S-(X-STT)+X)+X

= 0= 0

-S-ST T -(X-S-(X-STT)+X)+X

= 0= 0

Valuation of OptionsValuation of Options

☺ Quantitative ValuationQuantitative Valuation☺ Binomial model (an algorithm)Binomial model (an algorithm)☺ Black-Scholes model (a formula)Black-Scholes model (a formula)

☺ Arbitrage Restrictions on the Values of OptionsArbitrage Restrictions on the Values of Options☺ European Call and Put optionsEuropean Call and Put options☺ American vs. European Call optionAmerican vs. European Call option☺ American vs. European Put optionAmerican vs. European Put option☺ The Put-Call ParityThe Put-Call Parity☺ The option price and the exercise priceThe option price and the exercise price☺ Option Price Convexity (options with the same Option Price Convexity (options with the same

expiration date but different exercise prices)expiration date but different exercise prices)

NotationNotation

S = the price of the underlying asset (stock)S = the price of the underlying asset (stock)

C = the price of a call optionC = the price of a call option

P = the price of a put optionP = the price of a put option

X or K = the exercise or strike priceX or K = the exercise or strike price

T = the expiration dateT = the expiration date

t = a time indext = a time index

The Value of a Call OptionThe Value of a Call Option

AssumptionsAssumptions::1. A European Call option on a stock1. A European Call option on a stock2. The stock pays no dividends before2. The stock pays no dividends before

expirationexpiration3. The stock is traded3. The stock is traded4. A risk free bond is traded4. A risk free bond is traded

Arbitrage restrictionsArbitrage restrictions::

Max{ S-PV(X) , 0 } ≤ CMax{ S-PV(X) , 0 } ≤ CEUEU < S < S


Max{ S-PV(X) , 0 } ≤ CMax{ S-PV(X) , 0 } ≤ CEUEU ≤ S ≤ S

☺ 0 ≤ C0 ≤ C : : the owner has a right but not an the owner has a right but not an obligation.obligation.

☺ S-PV(X) ≤ CS-PV(X) ≤ C : : arbitrage proof.arbitrage proof.

☺ C C ≤≤ S S : : you will not pay more than $S, the you will not pay more than $S, the market price of the stock, for an option to buy market price of the stock, for an option to buy that stock for $X. Buying the stock itself is that stock for $X. Buying the stock itself is always an alternative to buying the call option.always an alternative to buying the call option.


S S-PV(X)

S

C

PV(X)0


ExampleExample::☺ The current stock price is $83The current stock price is $83

☺ The stock will not pay dividends in the next six The stock will not pay dividends in the next six monthsmonths

☺ A call option on that stock is traded for $3A call option on that stock is traded for $3☺ The exercise price is $80The exercise price is $80☺ The expiration of the option is in 6 monthsThe expiration of the option is in 6 months

☺ The 6 months risk free rate is 5%The 6 months risk free rate is 5%

Is there an opportunity to make an arbitrage profit?Is there an opportunity to make an arbitrage profit?


ExampleExample::

☺Check for arbitrage opportunitiesCheck for arbitrage opportunities

☺Show the investment strategy and Show the investment strategy and calculate the arbitrage profitscalculate the arbitrage profits

☺Show the general proof for this caseShow the general proof for this case


AssumptionsAssumptions::1. A1. A Call option on a stockCall option on a stock2. 2. The stock pays no dividendsThe stock pays no dividends before expiration before expiration3. The stock is traded3. The stock is traded4. A risk free bond is traded4. A risk free bond is traded

Arbitrage restrictionArbitrage restriction::

C(European) = C(American)C(European) = C(American)

HintHint: compare the payoff from immediate exercise to the lower: compare the payoff from immediate exercise to the lower bound of the European call option price.bound of the European call option price.

The Value of a Put OptionThe Value of a Put Option

AssumptionsAssumptions::1. A European Put option on a stock1. A European Put option on a stock2. The stock pays no dividends before2. The stock pays no dividends before


Arbitrage restrictionsArbitrage restrictions::

Max{ PV(X)-S , 0 } ≤ PMax{ PV(X)-S , 0 } ≤ PEUEU < PV(X) < PV(X)


Max{ PV(X)-S , 0 } ≤ PMax{ PV(X)-S , 0 } ≤ PEUEU < PV(X) < PV(X)

☺ 0 ≤ P0 ≤ P : : the owner has a right but not an the owner has a right but not an obligation.obligation.

☺ PV(X)-S ≤ PPV(X)-S ≤ P : : arbitrage proof.arbitrage proof.

☺ P < PV(X)P < PV(X) : : the highest profit from the put the highest profit from the put option is realized when the stock price is zero. option is realized when the stock price is zero. That profit is $X realized on date T, which is That profit is $X realized on date T, which is PV($X) today.PV($X) today.


S

P

PV(X)0

PV(X)

PV(X)-S

PV(X)


ExampleExample::☺ The current stock price is $75The current stock price is $75

☺ The stock will not pay dividends in the next six The stock will not pay dividends in the next six monthsmonths

☺ A put option on that stock is traded for $1A put option on that stock is traded for $1☺ The exercise price is $80The exercise price is $80☺ The expiration of the option is in 6 monthsThe expiration of the option is in 6 months

☺ The 6 months risk free rate is 5%The 6 months risk free rate is 5%

Is there an opportunity to make an arbitrage profit?Is there an opportunity to make an arbitrage profit?


ExampleExample::

☺Check for arbitrage opportunitiesCheck for arbitrage opportunities

☺Show the investment strategy and Show the investment strategy and calculate the arbitrage profitscalculate the arbitrage profits

☺Show the general proof for this caseShow the general proof for this case


AssumptionsAssumptions::1. A1. A put option on a stockput option on a stock2. 2. The stock pays no dividendsThe stock pays no dividends before before


For a put option we always have:For a put option we always have:

P(European) < P(American)P(European) < P(American)Note: in this case an example is enough.Note: in this case an example is enough.

Exercise PricesExercise Prices

Let XLet X1 1 < X< X22 be two different exercise prices; be two different exercise prices;

CC11, C, C2 2 be the appropriate call option prices; andbe the appropriate call option prices; and

PP11, P, P2 2 be the appropriate put option pricesbe the appropriate put option prices

(we assume that all the options are European, on (we assume that all the options are European, on the same stock S that pays no dividends, with the the same stock S that pays no dividends, with the same expiration date T).same expiration date T).

Then,Then,

C(XC(X11) ≥ C(X) ≥ C(X22) and P(X) and P(X11) ≤ P(X) ≤ P(X22))

ExampleExample

Show that if there are two call options on the Show that if there are two call options on the same stock (that pays no dividends), and same stock (that pays no dividends), and both have the same expiration date but both have the same expiration date but different exercise prices as follows, there is different exercise prices as follows, there is an opportunity to make an arbitrage profit.an opportunity to make an arbitrage profit.

XX11= $40 and X= $40 and X22= $50= $50

CC11= $3 and C= $3 and C22= $4= $4

Option Price ConvexityOption Price ConvexityLet XLet X1 1 < X< X22 < X < X33 be three different exercise prices, be three different exercise prices, and Cand C11, C, C22, C, C33 be the appropriate call option prices, be the appropriate call option prices, and Pand P11, P, P22, P, P33 be the appropriate put option prices be the appropriate put option prices (all the options are European, on the same stock S (all the options are European, on the same stock S that pays no dividends, with the same expiration that pays no dividends, with the same expiration date T).date T).

2 1 3

2 1 3

2 1 3

If (1 )

Then (1 )

and (1 )

X X X

C C C

P P P

ExampleExample

Show that if there are three call options on Show that if there are three call options on the same stock (that pays no dividends), the same stock (that pays no dividends), and all three have the same expiration date and all three have the same expiration date but different exercise prices as follows, there but different exercise prices as follows, there is an opportunity to make an arbitrage profit.is an opportunity to make an arbitrage profit.

XX11= $40, X= $40, X22= $50 and X= $50 and X33= $60= $60

CC11= $4.6, C= $4.6, C22= $4 and C= $4 and C33= $3= $3

Binomial Option Pricing ModelBinomial Option Pricing Model

AssumptionsAssumptions::☺ A single periodA single period

☺ Two dates: time t=0 and time t=1Two dates: time t=0 and time t=1

☺ The future (time 1) stock price has only two The future (time 1) stock price has only two possible valuespossible values

☺ The price can go up or downThe price can go up or down

☺ The perfect market assumptionsThe perfect market assumptions☺ No transactions costs, borrowing and lending at the No transactions costs, borrowing and lending at the

risk free interest rate, no taxes…risk free interest rate, no taxes…

Binomial Option Pricing ModelBinomial Option Pricing ModelExampleExample

The stock priceThe stock price Assume S= $50,Assume S= $50,

u= 10% and d= (-3%)u= 10% and d= (-3%)

S

Su=S·(1+u)

Sd=S·(1+d)

S=$50

Su=$55

Sd=$48.5


The call option priceThe call option price Assume X= $50,Assume X= $50,

T= 1 year (T= 1 year (1 period1 period))

C

Cu= Max{Su-X,0}

Cd= Max{Sd-X,0}

C

Cu= $5 = Max{55-50,0}

Cd= $0 = Max{48.5-50,0}


The bond priceThe bond price Assume r= 6%Assume r= 6%

1

(1+r)

(1+r)

$1

$1.06

$1.06

Replicating PortfolioReplicating Portfolio

At time t=0, we can create a portfolio of At time t=0, we can create a portfolio of NN shares shares of the stock and an investment of of the stock and an investment of BB dollars in the dollars in the risk-free bond. The payoff of the portfolio will risk-free bond. The payoff of the portfolio will replicate the t=1 payoffs of the call option:replicate the t=1 payoffs of the call option:

NN·$55 + ·$55 + BB·$1.06 = $5·$1.06 = $5

NN·$48.5 + ·$48.5 + BB·$1.06 = $0·$1.06 = $0

Obviously, this portfolio should also have the same Obviously, this portfolio should also have the same price as the call option at t=0:price as the call option at t=0:

NN·$50 + ·$50 + BB·$1 = C·$1 = CWe get We get NN=0.7692, =0.7692, BB=(-35.1959) and the call option price is C=$3.2656.=(-35.1959) and the call option price is C=$3.2656.

A Different ReplicationA Different Replication

The price of $1 in the The price of $1 in the “up” state:“up” state:

The price of $1 in the The price of $1 in the “down” state:“down” state:

qu

$1

$0

qd

$0

$1

Replicating Portfolios Using the Replicating Portfolios Using the State PricesState Prices

We can replicate the t=1 payoffs of the stock and We can replicate the t=1 payoffs of the stock and the bond using the state prices:the bond using the state prices:

qquu·$55 + ·$55 + qqdd·$48.5 = $50·$48.5 = $50

qquu·$1.06 + ·$1.06 + qqdd·$1.06 = $1·$1.06 = $1

Obviously, once we solve for the two state prices Obviously, once we solve for the two state prices we can price any other asset such as the call we can price any other asset such as the call option:option:

qquu·$5 + ·$5 + qqdd·$0 = C·$0 = C

We get We get qquu=0.6531, =0.6531, qqdd =0.2903 and the call option price is C=$3.2656.=0.2903 and the call option price is C=$3.2656.


The put option priceThe put option price Assume X= $50,Assume X= $50,

T= 1 year (T= 1 year (1 period1 period))

P

Pu= Max{X-Su,0}

Pd= Max{X-Sd,0}

P

Pu= $0 = Max{50-55,0}

Pd= $1.5 = Max{50-48.5,0}

Replicating Portfolios Using the Replicating Portfolios Using the State PricesState Prices

We can replicate the t=1 payoffs of the stock and We can replicate the t=1 payoffs of the stock and the bond using the state prices:the bond using the state prices:

qquu·$55 + ·$55 + qqdd·$48.5 = $50·$48.5 = $50

qquu·$1.06 + ·$1.06 + qqdd·$1.06 = $1·$1.06 = $1

But the assets are exactly the same and so are the But the assets are exactly the same and so are the state prices. The put option price is:state prices. The put option price is:

qquu·$0 + ·$0 + qqdd·$1.5 = P·$1.5 = P

We get We get qquu=0.6531, =0.6531, qqdd =0.2903 and the put option price is P=$0.4354.=0.2903 and the put option price is P=$0.4354.

Two Period ExampleTwo Period Example

☺Assume that the current stock price is $50, Assume that the current stock price is $50, and it can either go up 10% or down 3% in and it can either go up 10% or down 3% in each period.each period.

☺The one period risk-free interest rate is The one period risk-free interest rate is 6%.6%.

☺What is the price of a European call option What is the price of a European call option on that stock, with an exercise price of $50 on that stock, with an exercise price of $50 and expiration in two periods?and expiration in two periods?

The Stock PriceThe Stock Price

S= $50, u= 10% and d= (-3%)S= $50, u= 10% and d= (-3%)

S=$50

Su=$55

Sd=$48.5

Suu=$60.5

Sud=Sdu=$53.35

Sdd=$47.05

The Bond PriceThe Bond Price

r= 6% (for each period)r= 6% (for each period)

$1

$1.06

$1.06

$1.1236

$1.1236

$1.1236

The Call Option PriceThe Call Option Price

X= $50 and T= 2 periodsX= $50 and T= 2 periods

C

Cu

Cd

Cuu=Max{60.5-50,0}=$11.5

Cud=Max{53.35-50,0}=$3.35

Cdd=Max{47.05-50,0}=$0

State Prices in the Two Period TreeState Prices in the Two Period Tree

We can replicate the t=1 payoffs of the stock and the bond We can replicate the t=1 payoffs of the stock and the bond using the state prices:using the state prices:

qquu·$55 + ·$55 + qqdd·$48.5 = $50·$48.5 = $50

qquu·$1.06 + ·$1.06 + qqdd·$1.06 = $1·$1.06 = $1

Note that if u, d and r are the same, our solution for the Note that if u, d and r are the same, our solution for the state prices will not change (regardless of the price levels state prices will not change (regardless of the price levels of the stock and the bond):of the stock and the bond):

qquu·S·(1+u) + ·S·(1+u) + qqdd·S ·(1+d) = S·S ·(1+d) = S

qqu u ·(1+r)·(1+r)tt + + qqd d ·(1+r)·(1+r)tt = (1+r) = (1+r)(t-1)(t-1)

Therefore, we can use the same state-prices in the two Therefore, we can use the same state-prices in the two period tree.period tree.

The Call Option PriceThe Call Option Price

qquu= 0.6531 and q= 0.6531 and qdd= 0.2903= 0.2903

C

Cu

Cd

Cuu=$11.5

Cud=$3.35

Cdd=$0


☺ What is the price of a European put option on What is the price of a European put option on that stock, with an exercise price of $50 and that stock, with an exercise price of $50 and expiration in two periods?expiration in two periods?

☺ What is the price of an American call option on What is the price of an American call option on that stock, with an exercise price of $50 and that stock, with an exercise price of $50 and expiration in two periods?expiration in two periods?

☺ What is the price of an American put option on What is the price of an American put option on that stock, with an exercise price of $50 and that stock, with an exercise price of $50 and expiration in two periods?expiration in two periods?


☺European put option - use the tree or the European put option - use the tree or the put-call parityput-call parity

☺What is the price of an American call What is the price of an American call option - if there are no dividends …option - if there are no dividends …

☺American put option – use the treeAmerican put option – use the tree

The European Put Option PriceThe European Put Option Price

qquu= 0.6531 and q= 0.6531 and qdd= 0.2903= 0.2903

P

Pu

Pd

Puu=$0

Pud=$0

Pdd=$2.955

Pu = 0.6531*0 + 0.2903*0 = $0

Pd = 0.6531*0 + 0.2903*2.955 = $0.8578

PEU = 0.6531*0 + 0.2903*0.8578 = $0.2490

The American Put Option PriceThe American Put Option Price

qquu= 0.6531 and q= 0.6531 and qdd= 0.2903= 0.2903

PAm

Pu

Pd

Puu=$0

Pud=$0

Pdd=$2.955

American Put OptionAmerican Put Option

Note that at time t=1 the option buyer will Note that at time t=1 the option buyer will decide whether to exercise the option or decide whether to exercise the option or keep it till expiration.keep it till expiration.

If the payoff from immediate exercise is If the payoff from immediate exercise is higher than the option value the optimal higher than the option value the optimal strategy is to exercise:strategy is to exercise:

If Max{ X-SIf Max{ X-Suu,0 } > P,0 } > Puu(European) => Exercise(European) => Exercise

The American Put Option PriceThe American Put Option Price

qquu= 0.6531 and q= 0.6531 and qdd= 0.2903= 0.2903

P

Pu

Pd

Puu=$0

Pud=$0

Pdd=$2.955

Pu = Max{ 0.6531*0 + 0.2903*0 , 50-55 } = $0

Pd = Max{ 0.6531*0 + 0.2903*2.955 , 50-48.5 } = 50-48.5 = $1.5

PAm = Max{ 0.6531*0 + 0.2903*1.5 , 50-50 } = $0.4354 > $0.2490 = PEu

Determinants of the ValuesDeterminants of the Valuesof Call and Put Optionsof Call and Put Options

VariableVariable C – Call ValueC – Call Value P – Put ValueP – Put Value

SS – – stock pricestock price IncreaseIncrease DecreaseDecrease

XX – – exercise priceexercise price DecreaseDecrease IncreaseIncrease

σσ – – stock price volatilitystock price volatility IncreaseIncrease IncreaseIncrease

TT – – time to expirationtime to expiration IncreaseIncrease IncreaseIncrease

rr – – risk-free interest raterisk-free interest rate IncreaseIncrease DecreaseDecrease

DivDiv – – dividend payoutsdividend payouts DecreaseDecrease IncreaseIncrease

Practice ProblemsPractice Problems

BKM Ch. 21: BKM Ch. 21:

End of chapter - 1, 2End of chapter - 1, 2

Example 21.1 and concept check Q #4 Example 21.1 and concept check Q #4 (pages 755-6)(pages 755-6)

Practice set: 17-24, 25-35Practice set: 17-24, 25-35

fi8000 option valuation i milind shrikhande. a call option a european call option gives the buyer of...

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