Fi8000 Valuation of Financial Assets

Fi8000Fi8000Valuation ofValuation of

Financial AssetsFinancial Assets

Spring Semester 2010Spring Semester 2010

Dr. Isabel TkatchDr. Isabel TkatchAssistant Professor of FinanceAssistant Professor of Finance

Valuation of OptionsValuation of Options

☺Arbitrage Restrictions on the Values of Arbitrage Restrictions on the Values of OptionsOptions

☺Quantitative Pricing ModelsQuantitative Pricing Models

☺ Binomial modelBinomial model☺A formula in the simple caseA formula in the simple case☺An algorithm in the general caseAn algorithm in the general case

☺ Black-Scholes model (a formula)Black-Scholes model (a formula)

Binomial Option Pricing ModelBinomial Option Pricing Model

AssumptionsAssumptions::☺ A single periodA single period

☺ Two dates: time t=0 and time t=1 (expiration)Two dates: time t=0 and time t=1 (expiration)

☺ 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 (expirationexpiration))

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.


The weights of Bonds and Stocks in the replicating portfolio The weights of Bonds and Stocks in the replicating portfolio are:are:

ZZBB= (-$35.1959) / $3.2656 = -10.78= (-$35.1959) / $3.2656 = -10.78

ZZSS= (0.7692 * $50) / $3.2656 = 11.78= (0.7692 * $50) / $3.2656 = 11.78

Say you invest $100. The two equivalent investment Say you invest $100. The two equivalent investment strategies are:strategies are:

1.1. Buy call options for $100 Buy call options for $100 (i.e., buy $100 / $3.2656 = 30.62 call options) (i.e., buy $100 / $3.2656 = 30.62 call options)

2.2. Sell bonds for 10.78 * $100 = $1,078Sell bonds for 10.78 * $100 = $1,078

Buy stocks for 11.78 * $100 = $1,178Buy stocks for 11.78 * $100 = $1,178

Binomial Option Pricing ModelBinomial Option Pricing ModelExample ContinuedExample Continued

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


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 PortfolioReplicating Portfoliofor the Put Optionfor the Put Option

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 put option:replicate the t=1 payoffs of the put option:

NN·$55 + ·$55 + BB·$1.06 = $0·$1.06 = $0

NN·$48.5 + ·$48.5 + BB·$1.06 = $1.5·$1.06 = $1.5

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

NN·$50 + ·$50 + BB·$1 = P·$1 = PWe get We get NN=(-0.2308), =(-0.2308), BB=11.9739 and the put option price is P=$0.4354.=11.9739 and the put option price is P=$0.4354.


The weights of Bonds and Stocks in the replicating portfolio The weights of Bonds and Stocks in the replicating portfolio are:are:

ZZBB= ($11.9739) / $0.4354 = 27.5= ($11.9739) / $0.4354 = 27.5

ZZSS= (-0.2308 * $50) / $0.4354 = -26.5= (-0.2308 * $50) / $0.4354 = -26.5

Say you invest in one put options contract (i.e. 100 Say you invest in one put options contract (i.e. 100 options). The two equivalent investment strategies are:options). The two equivalent investment strategies are:

1.1. Buy one put options contract for $0.4354*100 = $43.54Buy one put options contract for $0.4354*100 = $43.54

2.2. Buy bonds for 27.5 * $43.54 = $1,197.35Buy bonds for 27.5 * $43.54 = $1,197.35

Sell stocks for 26.5 * $43.54 = $1,153.81Sell stocks for 26.5 * $43.54 = $1,153.81

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 in that economy. In we can price any other asset in that economy. In particular we can price the call option:particular we can price the call 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,


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}=$10.5

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 every part Therefore, we can use the same state-prices in every part of the tree.of the 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=$10.5

Cud=$3.35

Cdd=$0

Cu = 0.6531*$10.5 + 0.2903*$3.35 = $7.83

Cd = 0.6531*$3.35 + 0.2903*$0.00 = $2.19

C = 0.6531*$7.83 + 0.2903*$2.19 = $5.75


☺ What is the price of a What is the price of a European put optionEuropean put option on 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 What is the price of an American call optionAmerican call option on 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 What is the price of an American put optionAmerican put option on 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 = $0.858

PEU = 0.6531*$0 + 0.2903*$0.858 = $0.25

The European Put Option PriceThe European Put Option Price

Another way to calculate the European put option Another way to calculate the European put option price is to use the put-call-parity restriction:price is to use the put-call-parity restriction:

2

( )

( )

50 5.75 50

(1 .06)

$0.25

C PV X S P

P C PV X S

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

If the payoff from immediate exercise is higher If the payoff from immediate exercise is higher than the value of keeping the option for one more than the value of keeping the option for one more period (“European”), then the optimal strategy is to period (“European”), then the optimal strategy is to exercise: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

Black-Scholes ModelBlack-Scholes Model

☺Developed around 1970Developed around 1970

☺Closed form, analytical pricing modelClosed form, analytical pricing model☺ An equationAn equation☺ Can be calculated easily and quickly (using a Can be calculated easily and quickly (using a

computer or even a calculator)computer or even a calculator)☺ The limit of the binomial model if we are making the The limit of the binomial model if we are making the

number of periods infinitely large and every period number of periods infinitely large and every period very small – continuous timevery small – continuous time

☺Crucial assumptionsCrucial assumptions☺ The risk free interest rate and the stock price volatility The risk free interest rate and the stock price volatility

are constant over the life of the option.are constant over the life of the option.


C – call premiumC – call premium

S – stock priceS – stock price

X – exercise priceX – exercise price

T – time to expirationT – time to expiration

r – the interest rater – the interest rate

σσ – std of stock returns – std of stock returns

ln(z) – natural log of zln(z) – natural log of z

ee-rT-rT – exp{-rT} = (2.7183) – exp{-rT} = (2.7183)-rT-rT

N(z) – standard normalN(z) – standard normal

cumulative probability cumulative probability

1 2

2

1

2 1

( ) ( )

1ln

2,

rTC S N d Xe N d

Where

ST r

Xd

T

d d T

The N(0,1) DistributionThe N(0,1) Distribution

μz=0

pdf(z)

z

N(z)

Black-Scholes exampleBlack-Scholes example

C – ?C – ?

S – $47.50S – $47.50

X – $50X – $50

T – 0.25 yearsT – 0.25 years

r – 0.05 r – 0.05 (5% annual rate (5% annual rate

compoundedcompounded

continuously)continuously)

σσ – 0.30 (or 30%) – 0.30 (or 30%)

1 2

2

1

2 1

( ) ( )

1ln

2,

rTC S N d Xe N d

Where

ST r

Xd

T

d d T

Black-Scholes ExampleBlack-Scholes Example

C – ?C – ?

S – $47.50S – $47.50

X – $50X – $50





σσ – 0.30 (or 30%) – 0.30 (or 30%)

1

1

2

2

( 0.1836)

( ) 0.4272

( 0.3336)

( ) 0.3693

$2.0526

d

N d

d

N d

and

C


☺Continuous time and therefore continuous Continuous time and therefore continuous compoundingcompounding

☺N(d) – loosely speaking, N(d) is the “risk N(d) – loosely speaking, N(d) is the “risk adjusted” probability that the call option adjusted” probability that the call option will expire in the money (check the pricing will expire in the money (check the pricing for the extreme cases: 0 and 1)for the extreme cases: 0 and 1)

☺ ln(S/X) – approximately, a percentage ln(S/X) – approximately, a percentage measure of option “moneyness” measure of option “moneyness”


P – Put premiumP – Put premiumS – stock priceS – stock priceX – exercise priceX – exercise priceT – time to expirationT – time to expirationr – the interest rater – the interest rateσσ – std of stock returns – std of stock returnsln(z) – natural log of zln(z) – natural log of zee-rT-rT – exp{-rT} = (2.7183) – exp{-rT} = (2.7183)-rT-rT

N(z) – standard normalN(z) – standard normal cumulativecumulative probability probability

2 1

2

1

2 1

1 ( ) 1 ( )

1ln

2,

rTP Xe N d S N d

Where

ST r

Xd

T

d d T

Black-Scholes ExampleBlack-Scholes Example

P – ?P – ?

S – $47.50S – $47.50

X – $50X – $50





σσ – 0.30 (or 30%) – 0.30 (or 30%)

1

1

2

2

( 0.1836)

( ) 0.4272

( 0.3336)

( ) 0.3693

$3.9315

d

N d

d

N d

and

P

The Put Call ParityThe Put Call Parity

The continuous time version (continuous The continuous time version (continuous compounding):compounding):

0.05 0.25

( )

$2.0526 $50 ? $47.5 $3.9315

rT

C PV X S P

C Xe S P

e

Stock Return VolatilityStock Return VolatilityOne approachOne approach::

Calculate an estimate of the volatility using the Calculate an estimate of the volatility using the historical stock returns and plug it in the option historical stock returns and plug it in the option formula to get pricingformula to get pricing

2

1

1

1( )

1

ln

n

tt

tt

t

Est return average returnn

Where

Sreturn S

Stock Return VolatilityStock Return VolatilityAnother approachAnother approach::

Calculate the stock return volatility implied by Calculate the stock return volatility implied by the option price observed in the marketthe option price observed in the market

(a trial and error algorithm)(a trial and error algorithm)

1 2

2

1 2 1

( ) ( )

1ln

2 and

rTC S N d Xe N d

Where

ST r

Xd d d T

T

Option Price and VolatilityOption Price and VolatilityLet Let σσ1 1 < < σσ 22 be two possible, yet different return volatilities; be two possible, yet different return volatilities; CC11, C, C2 2 be the appropriate call option prices; andbe the appropriate call option prices; andPP11, P, P2 2 be the appropriate put option prices.be the appropriate put option prices.

We assume that the options are European, on the same We assume that the options are European, on the same stock S that pays no dividends, with the same expiration stock S that pays no dividends, with the same expiration date T.date T.

Note that our estimate of the stock return volatility changes. Note that our estimate of the stock return volatility changes. The two different prices are of the same option, and can The two different prices are of the same option, and can not exist at the same time!not exist at the same time!

Then,Then,

CC11 ≤ C ≤ C2 2 and P and P11 ≤ P ≤ P2 2

Implied Volatility - exampleImplied Volatility - example

C – $2.5C – $2.5

S – $47.50S – $47.50

X – $50X – $50


r – 0.05r – 0.05 (5% annual rate (5% annual rate



σσ – ? – ?

1 2

2

1

2 1

( ) ( )

1ln

2,

rTC S N d Xe N d

Where

ST r

Xd

T

d d T


C – ?C – ?

S – $47.50S – $47.50

X – $50X – $50





σσ – 0.30 (or 30%) – 0.30 (or 30%)

1

1

2

2

( 0.1836)

( ) 0.4272

( 0.3336)

( ) 0.3693

$2.0526 $2.5

0.3 or 0.3 ?

d

N d

d

N d

and

C


C – ?C – ?

S – $47.50S – $47.50

X – $50X – $50





σσ – 0.40 (or 40%) – 0.40 (or 40%)

1

1

2

2

( 0.0940)

( ) 0.4626

( 0.2940)

( ) 0.3844

$2.9911 $2.5

d

N d

d

N d

and

C


C – ?C – ?

S – $47.50S – $47.50

X – $50X – $50





σσ – 0.35 (or 35%) – 0.35 (or 35%)

1

1

2

2

( 0.1342)

( ) 0.4466

( 0.3092)

( ) 0.3786

$2.5205

d

N d

d

N d

and

C

Application: Portfolio InsuranceApplication: Portfolio Insurance

Options can be used to guarantee minimum Options can be used to guarantee minimum returns from an investment in stocks.returns from an investment in stocks.

Purchasing portfolio insurance (protective put Purchasing portfolio insurance (protective put strategy):strategy):

Long one stock;Long one stock;

Buy a put option on one stock;Buy a put option on one stock;

If no put option exists, use a stock andIf no put option exists, use a stock and

a bond to replicate the put option payoffs.a bond to replicate the put option payoffs.

Portfolio Insurance ExamplePortfolio Insurance Example

You decide to invest in one share of General Pills (GP) You decide to invest in one share of General Pills (GP) stock, which is currently traded for $56. The stock pays no stock, which is currently traded for $56. The stock pays no dividends.dividends.

You worry that the stock’s price may decline and decide to You worry that the stock’s price may decline and decide to purchase a European put option on GPs stock. The put purchase a European put option on GPs stock. The put allows you to sell the stock at the end of one year for $50.allows you to sell the stock at the end of one year for $50.

If the std of the stock price is If the std of the stock price is σσ=0.3 (30%) and rf=0.08 (8% =0.3 (30%) and rf=0.08 (8% compounded continuously), what is the price of the put compounded continuously), what is the price of the put option?option?

What is the CF from your strategy at time t=0? What is the CF from your strategy at time t=0? What is the CF at time t=1 as a function of 0<SWhat is the CF at time t=1 as a function of 0<STT<100?<100?


What if there is no put option on the stock that What if there is no put option on the stock that you wish to insure? - Use the B&S formula to you wish to insure? - Use the B&S formula to replicate the protective put strategy.replicate the protective put strategy.

What is your insurance strategy?What is your insurance strategy?What is the CF from your strategy at time t=0? What is the CF from your strategy at time t=0? Suppose that one week later, the price of the Suppose that one week later, the price of the stock increased to $60, what is the value of the stock increased to $60, what is the value of the stocks and bonds in your portfolio?stocks and bonds in your portfolio?How should you rebalance the portfolio to keep How should you rebalance the portfolio to keep the insurance?the insurance?


The B&S formula for the put option:The B&S formula for the put option:

-P = -Xe-P = -Xe-rT-rT[1-N(d[1-N(d22)]+S[1-N(d)]+S[1-N(d11)])]

Therefore the insurance strategy (Original Therefore the insurance strategy (Original portfolio + synthetic put) is:portfolio + synthetic put) is:

Long one share of stockLong one share of stock

Long X·[1-N(dLong X·[1-N(d22)] bonds)] bonds

Short [1-N(dShort [1-N(d11)] stocks)] stocks


The total time t=0 The total time t=0 CFCF of the protective put of the protective put (insured portfolio) is:(insured portfolio) is:

CFCF00 = -S = -S00-P-P00

= -S= -S00-Xe-Xe-rT-rT[1-N(d[1-N(d22)]+S)]+S00[1-N(d[1-N(d11)])]

= -S= -S00[N(d[N(d11)])] -Xe-Xe-rT-rT[1-N(d[1-N(d22)])]

And the proportion invested in the stock is:And the proportion invested in the stock is:

wwstockstock=-S=-S00[N(d[N(d11)])] /{-S/{-S00[N(d[N(d11)])] -Xe-Xe-rT-rT[1-N(d[1-N(d22)]})]}


The proportion invested in the stock is:The proportion invested in the stock is:

wwstock stock = S= S00[N(d[N(d11)])] /{S/{S00[N(d[N(d11)])] +Xe+Xe-rT-rT[1-N(d[1-N(d22)]})]}

Or, if we remember the original (protective Or, if we remember the original (protective put) strategy:put) strategy:

wwstock stock = S= S00[N(d[N(d11)])] /{S/{S0 0 + P+ P00}}

Finally, the proportion invested in the bond is:Finally, the proportion invested in the bond is:

wwbondbond = 1-w = 1-wstockstock


Say you invest $1,000 in the portfolio today (t=0)Say you invest $1,000 in the portfolio today (t=0)

Time t = 0:Time t = 0:

wwstockstock= 56·0.7865/(56+2.38) = 75.45%= 56·0.7865/(56+2.38) = 75.45%

Stock value = 0.7545*$1,000 = $754.5Stock value = 0.7545*$1,000 = $754.5Bond value = 0.2455 *$1,000 = $245.5Bond value = 0.2455 *$1,000 = $245.5

End of week 1 (we assumed that the stock price increased End of week 1 (we assumed that the stock price increased to $60):to $60):

Stock value = ($60/$56)·$754.5 = $808.39Stock value = ($60/$56)·$754.5 = $808.39Bond value = $245.5·eBond value = $245.5·e0.08·(1/52)0.08·(1/52) = $245.88 = $245.88Portfolio value = $808.39+$245.88 = $1,054.27Portfolio value = $808.39+$245.88 = $1,054.27


Now you have a $1,054.27 portfolio Now you have a $1,054.27 portfolio

Time t = 1 (beginning of week 2):Time t = 1 (beginning of week 2):

wwstockstock= 60·0.8476/(60+1.63) = 82.53%= 60·0.8476/(60+1.63) = 82.53%Stock value = 0.8253*$1,054.27 = $870.06Stock value = 0.8253*$1,054.27 = $870.06Bond value = 0.1747 *$1,054.27 = $184.21Bond value = 0.1747 *$1,054.27 = $184.21

I.e. you should rebalance your portfolio (increase the I.e. you should rebalance your portfolio (increase the proportion of stocks to 82.53% and decrease the proportion proportion of stocks to 82.53% and decrease the proportion of bonds to 17.47%).of bonds to 17.47%).

Why should we rebalance the portfolio? Should we Why should we rebalance the portfolio? Should we rebalance the portfolio if we use the protective put strategy rebalance the portfolio if we use the protective put strategy with a real put option?with a real put option?

Practice ProblemsPractice Problems

BKM Ch. 21BKM Ch. 21

7th Ed. : 7-10, 17,187th Ed. : 7-10, 17,18

8th Ed. : 11-14, 17,188th Ed. : 11-14, 17,18

Practice set: 36-42.Practice set: 36-42.

Fi8000 Valuation of Financial Assets

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