Chapter 18 DERIVATIVES: ANALYSIS AND VALUATION. Chapter 18 Questions How are spot and futures prices related?How are spot and futures prices related?

Chapter 18Chapter 18

DERIVATIVES: ANALYSIS AND DERIVATIVES: ANALYSIS AND VALUATIONVALUATION

Chapter 18 QuestionsChapter 18 Questions

• How are spot and futures prices related?How are spot and futures prices related?• What is basis risk?What is basis risk?• What is program trading and stock index What is program trading and stock index

arbitrage? How can futures be used to arbitrage? How can futures be used to hedge or speculate on changes in yield curve hedge or speculate on changes in yield curve spreads?spreads?

• Why would investors want to invest in an Why would investors want to invest in an option on a futures contract?option on a futures contract?

Chapter 18 QuestionsChapter 18 Questions

• What factors influence the price of an option?What factors influence the price of an option?• How do we compute the price of a call option How do we compute the price of a call option

using the binomial option pricing model?using the binomial option pricing model?• How does one use the Black-Scholes option-How does one use the Black-Scholes option-

pricing model?pricing model?• How do option-like features affect the price of How do option-like features affect the price of

bonds?bonds?

Futures Valuation IssuesFutures Valuation Issues

Cost of Carry ModelCost of Carry Model• Suppose that you needed some commodity Suppose that you needed some commodity

in three months. You have at least the in three months. You have at least the following two options:following two options:– Purchase the commodity now at the current spot Purchase the commodity now at the current spot

market price (Smarket price (S00) and “carry” the commodity for 3 ) and “carry” the commodity for 3 monthsmonths

– Buy a futures contract for delivery of the Buy a futures contract for delivery of the commodity in 3 months for the current futures commodity in 3 months for the current futures price (Fprice (F0,30,3))


Cost of Carry ModelCost of Carry Model• The futures prices and spot prices must be The futures prices and spot prices must be

related to one another in order for there to be related to one another in order for there to be no arbitrage opportunities for investors.no arbitrage opportunities for investors.

• If the carrying cost only amounts to forgone If the carrying cost only amounts to forgone interest at a risk-free rate (rinterest at a risk-free rate (r ff) for T time ) for T time periods, then the following relationship must periods, then the following relationship must hold:hold:

FF0,T0,T = S = S00 (1+r (1+rff))TT


Cost of Carry Model Example: Suppose that you Cost of Carry Model Example: Suppose that you can buy gold in the spot market for $300. The can buy gold in the spot market for $300. The monthly risk-free is .25%. You need the gold in monthly risk-free is .25%. You need the gold in three months.three months.

• What should be the current futures price?What should be the current futures price?FF0,T0,T = 300 (1+.0025) = 300 (1+.0025)33 = 302.26 = 302.26

• What if the futures price is $305?What if the futures price is $305?– You have a risk-less profit opportunity. Buy gold at You have a risk-less profit opportunity. Buy gold at

$300, sell futures at $305. In three months, delivery $300, sell futures at $305. In three months, delivery the gold, pay the known interest, pocket the the gold, pay the known interest, pocket the difference.difference.

Futures Valuations IssuesFutures Valuations Issues

• Similar futures-spot price relationships can Similar futures-spot price relationships can be derived when there are “market be derived when there are “market imperfections” involved with carrying the imperfections” involved with carrying the commodity or financial assetcommodity or financial asset

• Incorporating storage and insurance costs as Incorporating storage and insurance costs as a percentage of contract value (SI):a percentage of contract value (SI):

FF0,T0,T = S = S00 (1+r (1+rff +SI) +SI)TT

• Incorporating ownership benefits lost with a Incorporating ownership benefits lost with a futures position, especially dividends(d):futures position, especially dividends(d):

FF0,T0,T = S = S00 (1+r (1+rff +SI -d) +SI -d)TT


• BasisBasis– Basis is the difference between the spot and Basis is the difference between the spot and

futures prices.futures prices.– For a contract expiring at time T, the basis at time For a contract expiring at time T, the basis at time

t is:t is:BBt,Tt,T = S = Stt – F – Ft,Tt,T

– Over time, the spot and futures prices converge, Over time, the spot and futures prices converge, and basis becomes zero at expirationand basis becomes zero at expiration

– Between time t and expiration, basis can change Between time t and expiration, basis can change as the difference between spot and futures prices as the difference between spot and futures prices vary (known as basis risk)vary (known as basis risk)

Advanced Applications of Advanced Applications of Financial FuturesFinancial Futures

• Stock Index ArbitrageStock Index Arbitrage– An example of a program trading strategy An example of a program trading strategy

designed to take advantage of temporarily designed to take advantage of temporarily “mis-pricing” of securities“mis-pricing” of securities

– Monitor the parity condition (one period):Monitor the parity condition (one period):

FF0,T0,T = S = S00 + S + S00 (r (rff - d) - d)

– If it does not hold, construct a risk-free If it does not hold, construct a risk-free position to take advantage of the situation.position to take advantage of the situation.

Advanced Applications of Advanced Applications of Financial FuturesFinancial Futures

• T-Bond/T-Note Futures SpreadT-Bond/T-Note Futures Spread– ““Note over bond” (NOB) spreadNote over bond” (NOB) spread– Strategies based on speculating the Strategies based on speculating the

changing slope of the yield curvechanging slope of the yield curve

Options on FuturesOptions on Futures

• Also known as Futures OptionsAlso known as Futures Options

• Options on Stock Index FuturesOptions on Stock Index Futures– Gives the owner the right to buy (call) or Gives the owner the right to buy (call) or

sell (put) a stock futures contractsell (put) a stock futures contract

• Options on Treasury Bond FuturesOptions on Treasury Bond Futures– Gives the owner the right to buy (call) or Gives the owner the right to buy (call) or

sell (put) a Treasury bond futures contractsell (put) a Treasury bond futures contract

Options on FuturesOptions on Futures

• Why would they be attractive?Why would they be attractive?– If exercised, it would seem to have been better to If exercised, it would seem to have been better to

simply buy a futures contract instead (no option simply buy a futures contract instead (no option premium to pay)premium to pay)

– One primary advantage can be found when looking One primary advantage can be found when looking at all the potential price movementsat all the potential price movements

• Futures contracts used for hedging offset portfolio value Futures contracts used for hedging offset portfolio value changes; thus, advantageous price movements for a changes; thus, advantageous price movements for a portfolio are offset by the futures positionportfolio are offset by the futures position

• Options give the right (but not the obligation) to purchase Options give the right (but not the obligation) to purchase the futures contract; thus, favorable price movements will the futures contract; thus, favorable price movements will be offset only by the option premium rather than by a be offset only by the option premium rather than by a corresponding loss on the futures positioncorresponding loss on the futures position

Valuation of OptionsValuation of Options

• Factors influencing the value of a call option:Factors influencing the value of a call option:– Stock price (+)Stock price (+)

• For a given exercise price, the higher the stock price, the For a given exercise price, the higher the stock price, the greater the intrinsic value of the option (or at least the greater the intrinsic value of the option (or at least the closer to being in-the-money)closer to being in-the-money)

– Exercise price (-)Exercise price (-)• The lower the price at which you can buy, the more The lower the price at which you can buy, the more

valuevalue

– Time to expiration (+)Time to expiration (+)• The longer the time to expiration, the more likely the The longer the time to expiration, the more likely the

option will be valuableoption will be valuable


• Factors influencing the value of a call Factors influencing the value of a call option:option:– Interest rate (+)Interest rate (+)

• Options involve less money to invest, lower Options involve less money to invest, lower opportunity costsopportunity costs

– Volatility of underlying stock price (+)Volatility of underlying stock price (+)• The greater the volatility of the underlying The greater the volatility of the underlying

stock, the more likely that the option position stock, the more likely that the option position will be valuablewill be valuable


• Factors influencing the value of a put option:Factors influencing the value of a put option:– The same listed, but different directions for The same listed, but different directions for

several items.several items.– Stock price (-)Stock price (-)– Exercise price (+)Exercise price (+)– Time to expiration (+)Time to expiration (+)– Interest rate (-)Interest rate (-)– Volatility of underlying stock price (+)Volatility of underlying stock price (+)

The Binomial Option-The Binomial Option-Pricing ModelPricing Model

• Derives an option price using the Derives an option price using the principle of no riskless profitprinciple of no riskless profit

• Find a portfolio of stock and call options Find a portfolio of stock and call options that gives the same payoff in the future that gives the same payoff in the future regardless of whether the stock goes up regardless of whether the stock goes up or downor down– Called a hedge portfolioCalled a hedge portfolio

The Binomial Option-The Binomial Option-Pricing ModelPricing Model

• If the hedged portfolio offers a risk-free return, we can determine the portfolio’s current value by discounting this return at the risk-free rate

• Once we know the value of the portfolio, we can separate this value into two components– The value of the stock– The value of the option

Binomial Option Pricing Binomial Option Pricing Model: Example 1Model: Example 1

Stock price now

Price in one year

$50

$65

$40Assume: rf = 8%

Want the price of a call option (C0) with X = $52.50

Calculating Binomial Calculating Binomial Option PricesOption Prices

• We will look at a three step procedure:We will look at a three step procedure:

Step 1: Estimate the number of call Step 1: Estimate the number of call options neededoptions needed

Step 2: Determine the present value of Step 2: Determine the present value of the hedge portfoliothe hedge portfolio

Step 3: Compute the price of a call optionStep 3: Compute the price of a call option

Calculating Binomial Calculating Binomial Option Prices: Step 1Option Prices: Step 1

• Calculate the option’s payoffs for each Calculate the option’s payoffs for each possible future stock pricepossible future stock price– If stock goes to $65, option pays off $12.50If stock goes to $65, option pays off $12.50– If stock goes to $40, option pays off $0If stock goes to $40, option pays off $0


• Determine the composition of the hedge Determine the composition of the hedge portfolioportfolio– It contains one share of stock and “n” call It contains one share of stock and “n” call

optionsoptions

• Portfolio value = 1 share + n optionsPortfolio value = 1 share + n options– If stock goes up, portfolio will pay:If stock goes up, portfolio will pay:

$65 + [n x $12.50]$65 + [n x $12.50]– If stock goes down, portfolio will pay:If stock goes down, portfolio will pay:

$40 + [n x $0]$40 + [n x $0]


• To determine the composition of the To determine the composition of the hedge portfolio, find the number of hedge portfolio, find the number of options that equates the payoffsoptions that equates the payoffs

• $65 + $12.50n = $40 + $0n$65 + $12.50n = $40 + $0n– Implies n = -2Implies n = -2– Hedge portfolio is long one share of stock Hedge portfolio is long one share of stock

and short two call optionsand short two call options


• Value of hedge portfolio today:Value of hedge portfolio today:$50 - 2.00(C$50 - 2.00(C00))


• Next, we must determine the today’s Next, we must determine the today’s value of the hedge portfoliovalue of the hedge portfolio

• We know the portfolio will pay $40 in We know the portfolio will pay $40 in one year with certaintyone year with certainty

• Thus the value of that portfolio right Thus the value of that portfolio right now isnow is

40/1.08 = $37.0440/1.08 = $37.04


• Finally, separate the current value of Finally, separate the current value of the portfolio into its component partsthe portfolio into its component parts

• The portfolio is worth $37.04 right nowThe portfolio is worth $37.04 right now– $50 of this value is the current price of the $50 of this value is the current price of the

stockstock– The difference between $37.04 and $50 is The difference between $37.04 and $50 is

the revenue you would have received if the revenue you would have received if you sold the two call optionsyou sold the two call options


• Value of the Call Option:Value of the Call Option:$50 – 2 C$50 – 2 C00 x = $37.04 x = $37.04

Call price = $6.48Call price = $6.48

Black-Scholes Option Black-Scholes Option Pricing ModelPricing Model

• Model for determining the value of Model for determining the value of American call optionsAmerican call options

• This work warranted the awarding of This work warranted the awarding of the 1997 Nobel Prize in Economics!the 1997 Nobel Prize in Economics!

Black-Scholes Option Black-Scholes Option Pricing FormulaPricing Formula

PP00 = P = PSS[N(d[N(d11)] - X[e)] - X[e-rt-rt][N(d][N(d22)])]where:where:

PP00 = market value of call option = market value of call option

PPSS = current market price of underlying stock = current market price of underlying stock

N(dN(d11) = cumulative density function of d) = cumulative density function of d11 as defined later as defined later

X = exercise price of call optionX = exercise price of call optionr = current annualized market interest rate for prime commercial r = current annualized market interest rate for prime commercial

paperpapert = time remaining before expiration (in years)t = time remaining before expiration (in years)

N(dN(d22) = cumulative density function of d) = cumulative density function of d22 as defined later as defined later

Black-Scholes Option Black-Scholes Option Pricing FormulaPricing Formula

PP00 = P = PSS[N(d[N(d11)] - X[e)] - X[e-rt-rt][N(d][N(d22)])]

The cumulative density functions are defined as:The cumulative density functions are defined as:

21

12

21

2

1

)(

)(

)5.0(/ln(

tdd

t

trXPd S

Where:

ln(PS/X) = natural logarithm of (Ps/X)

S = standard deviation of annual rate of return on underlying stock

Using the Black-Scholes Using the Black-Scholes FormulaFormula

• Besides mathematical values, there are five Besides mathematical values, there are five inputs needed to use this model:inputs needed to use this model:– Current stock price (PCurrent stock price (Pss))– Exercise price (X)Exercise price (X)– Market interest rate (r)Market interest rate (r)– Time to expiration (t)Time to expiration (t)– Standard deviation of annual returns (Standard deviation of annual returns ())

• Of these, only the last in not observableOf these, only the last in not observable

• Also, using the put/call parity, we can value Also, using the put/call parity, we can value put options as well after calculating call valueput options as well after calculating call value

Option-like SecuritiesOption-like Securities

• Several types of securities contain Several types of securities contain embedded options:embedded options:– Callable and Putable BondsCallable and Putable Bonds– WarrantsWarrants– Convertible SecuritiesConvertible Securities

Callable and Putable Callable and Putable BondsBonds

• Callable Bonds contain a “call provision”Callable Bonds contain a “call provision”– The issuer has the option of buying the bonds The issuer has the option of buying the bonds

back at the call (exercise) price rather than having back at the call (exercise) price rather than having to wait until maturityto wait until maturity

– Attractive option for issuers if interest rates fall, Attractive option for issuers if interest rates fall, since they can purchase back old bonds and since they can purchase back old bonds and refinance (refunding) with new, lower interest refinance (refunding) with new, lower interest bondsbonds

– Typically will trade at no more than the call price, Typically will trade at no more than the call price, since call becomes likely at that pointsince call becomes likely at that point

Callable and Putable Callable and Putable BondsBonds

• Putable Bonds contain a “put provision”Putable Bonds contain a “put provision”– Investors may resell the bonds back to the Investors may resell the bonds back to the

issuer prior to maturity at the put (exercise) issuer prior to maturity at the put (exercise) price, often par valueprice, often par value

– Puts can generally be exercised only when Puts can generally be exercised only when designated events take placedesignated events take place

WarrantsWarrants

• Warrant is an option to buy a stated Warrant is an option to buy a stated number of shares of common stock at a number of shares of common stock at a specified price at any time during the specified price at any time during the life of the warrantlife of the warrant

• Similar to a call option, but usually with Similar to a call option, but usually with a much longer lifea much longer life

• Issued by the company whose stock Issued by the company whose stock the warrant is forthe warrant is for

WarrantsWarrants

• Intrinsic value is the difference between the Intrinsic value is the difference between the market price of the common stock and the market price of the common stock and the warrant exercise pricewarrant exercise price

Intrinsic Value = (Stock Price – Exercise Price) Intrinsic Value = (Stock Price – Exercise Price) x Number of Sharex Number of Share

• Speculative value is the value of the warrant Speculative value is the value of the warrant above its intrinsic valueabove its intrinsic value– Like other options, the value is higher than Like other options, the value is higher than

intrinsic value, except at maturityintrinsic value, except at maturity

Convertible SecuritiesConvertible Securities

• Allows the holder to convert one type of Allows the holder to convert one type of security into a stipulated amount of another security into a stipulated amount of another type (usually common stock) at the investor’s type (usually common stock) at the investor’s discretiondiscretion

• With convertible securities, value depends With convertible securities, value depends both on the value of the original asset and both on the value of the original asset and the value if conversion takes placethe value if conversion takes place– Value cannot fall below the greater of the two Value cannot fall below the greater of the two

valuesvalues


Convertible BondsConvertible Bonds

• Advantages to issuing firmsAdvantages to issuing firms– Lower interest rate on debtLower interest rate on debt– Debt represents potential common stockDebt represents potential common stock

• Advantages to investorsAdvantages to investors– Upside potential of common stockUpside potential of common stock– Downside protection of a bondDownside protection of a bond


Convertible bondsConvertible bonds– Conversion ratio = number of shares Conversion ratio = number of shares

obtained if convertedobtained if converted– Conversion price = Face Value/Number of Conversion price = Face Value/Number of

sharesshares

• Valuation of convertible bondsValuation of convertible bonds– Combination value of stock and bondCombination value of stock and bond– Two step process to determine minimum Two step process to determine minimum

valuevalue


Convertible BondsConvertible Bonds• Value of a convertible as a bondValue of a convertible as a bond

– Determine the bond’s value as if it had no Determine the bond’s value as if it had no conversion featureconversion feature

– This is the convertible’s investment value or floor This is the convertible’s investment value or floor valuevalue

• Value of a convertible as stockValue of a convertible as stock– Compute the value of the common stock received Compute the value of the common stock received

on conversionon conversion– This is the conversion valueThis is the conversion value


Convertible BondsConvertible Bonds• Minimum Value = Max (Bond Value, Minimum Value = Max (Bond Value,

Conversion Value)Conversion Value)• Like other options, including embedded Like other options, including embedded

options, they typically only sell at their options, they typically only sell at their minimum, intrinsic value only at maturity.minimum, intrinsic value only at maturity.– Conversion Premium = (Market Price – Minimum Conversion Premium = (Market Price – Minimum

Value)/Minimum ValueValue)/Minimum Value


Convertible BondsConvertible Bonds• Conversion Parity Price = Market Conversion Parity Price = Market

Price/Conversion RatioPrice/Conversion Ratio– An risk-free profit opportunity would exist if the An risk-free profit opportunity would exist if the

price of the convertible below this price, since price of the convertible below this price, since immediate conversion of the bond and then selling immediate conversion of the bond and then selling the stock would yield a profitthe stock would yield a profit

• Conversion ArbitrageConversion Arbitrage– An attempt to take advantage of mis-priced An attempt to take advantage of mis-priced

convertible bonds relative to the conversion ratioconvertible bonds relative to the conversion ratio

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Chapter 18 DERIVATIVES: ANALYSIS AND VALUATION. Chapter 18 Questions How are spot and futures prices related?How are spot and futures prices related?