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Derivatives Securities:Derivatives Securities: OPTIONS CONTRACTOPTIONS CONTRACT

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Definition of DerivativesDefinition of Derivatives

AA derivativederivative isis aa financialfinancial instrumentinstrument whosewhose valuevalue isis derivedderived fromfrom thethe priceprice ofof aa moremore basicbasic assetasset calledcalled thethe underlyingunderlying assetasset..

ExamplesExamples ofof underlyingunderlying assetsassets:: shares,shares, commodities,commodities, currencies,currencies, credits,credits, stockstock marketmarket indices,indices, weatherweather temperatures,temperatures, resultsresults ofof sportsport matchesmatches oror elections,elections, etcetc..

ExamplesExamples ofof derivativesderivatives areare:: OptionsOptions –– putput andand callcall options,options, forwards,forwards, futures,futures, andand swapsswaps

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Derivative MarketsDerivative Markets

Derivative markets are markets for contractual Derivative markets are markets for contractual instruments whose performance is determined instruments whose performance is determined by how another instrument or asset performsby how another instrument or asset performs

–– Cash market or spot market Cash market or spot market maximum delivery two maximum delivery two working daysworking days

–– Forward markets Forward markets related to forward and/or futures related to forward and/or futures contractcontract

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The Role of Derivative MarketsThe Role of Derivative Markets

1. 1. risk management risk management hedginghedging

Because derivative prices are related to the prices of the Because derivative prices are related to the prices of the underlying spot market goods, they can be used to reduce underlying spot market goods, they can be used to reduce or increase the risk of investing in the spot itemsor increase the risk of investing in the spot items

2. Price discovery2. Price discovery

futures and forward markets are an important means of futures and forward markets are an important means of obtaining information about investors’ expectations of obtaining information about investors’ expectations of future pricesfuture prices

3. Operational advantages3. Operational advantages

Lower transaction cost, have greater liquidity than spot Lower transaction cost, have greater liquidity than spot markets (futures & options), allow investors to sell short markets (futures & options), allow investors to sell short more easilymore easily

4. Market efficiency4. Market efficiency

5. Speculation5. Speculation

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Types of Derivative Securities:

Options ContractOptions Contract

Forward ContractForward Contract

Futures ContractFutures Contract

SwapSwap

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Definition of OptionsDefinition of Options

Arrangement or agreement between the Arrangement or agreement between the

seller and the buyer in which the buyer seller and the buyer in which the buyer

has the right to buy (call option) or sell (put has the right to buy (call option) or sell (put

option) an underlying assets at some time option) an underlying assets at some time

in the future at a price stipulated at in the future at a price stipulated at

present. present.

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Option Terminology

Buy Buy -- Long Long

Sell Sell -- ShortShort

CallCall

Put Put

Key ElementsKey Elements

–– Exercise or Strike PriceExercise or Strike Price

–– Premium or PricePremium or Price

–– Maturity or ExpirationMaturity or Expiration

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Market and Exercise Price

Relationships

In the Money In the Money -- exercise of the option would be exercise of the option would be profitableprofitable

Call: market price>exercise priceCall: market price>exercise price

Put: exercise price>market pricePut: exercise price>market price

Out of the Money Out of the Money -- exercise of the option would exercise of the option would not be profitablenot be profitable

Call: market price<exercise priceCall: market price<exercise price

Put: exercise price<market pricePut: exercise price<market price

At the Money At the Money -- exercise price and asset price are exercise price and asset price are equalequal

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American vs European Options

American American -- the option can be exercised at the option can be exercised at

any time before expiration or maturityany time before expiration or maturity

European European -- the option can only be the option can only be

exercised on the expiration or maturity exercised on the expiration or maturity

datedate

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Different Types of Options

Stock OptionsStock Options

Index OptionsIndex Options

Futures OptionsFutures Options

Foreign Currency OptionsForeign Currency Options

Interest Rate OptionsInterest Rate Options

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Payoffs and Profits on Options at

Expiration - Calls

NotationNotation

Stock Price = SStock Price = ST T Exercise Price = XExercise Price = X

Payoff to Call Holder Payoff to Call Holder

((SST T -- X) X) if if SSTT >X>X

00 if if SST T << XX

Profit to Call HolderProfit to Call Holder

Payoff Payoff -- Purchase PricePurchase Price

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Payoffs and Profits on Options at

Expiration - Calls

Payoff to Call Writer Payoff to Call Writer

-- ((SST T -- X) X) if if SSTT >X>X

00 if if SST T << XX

Profit to Call WriterProfit to Call Writer

Payoff + PremiumPayoff + Premium

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Figure Payoff and Profit to

Call at Expiration

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Figure Payoff and Profit to

Call Writers at Expiration

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Payoffs and Profits at Expiration - Puts

Payoffs to Put HolderPayoffs to Put Holder

00 if Sif STT >> XX

(X (X -- SSTT) ) if Sif STT < X< X

Profit to Put Holder Profit to Put Holder

Payoff Payoff -- PremiumPremium

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Payoffs and Profits at Expiration - Puts

Payoffs to Put WriterPayoffs to Put Writer

00 if Sif ST T >> XX

--(X (X -- SSTT)) if Sif STT < X< X

Profits to Put WriterProfits to Put Writer

Payoff + PremiumPayoff + Premium

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Figure Payoff and Profit to

Put Option at Expiration

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Optionlike Securities

Callable BondsCallable Bonds

Convertible SecuritiesConvertible Securities

WarrantsWarrants

Collateralized LoansCollateralized Loans

Levered Equity and Risky DebtLevered Equity and Risky Debt

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Option Values

Intrinsic value Intrinsic value -- profit that could be made if profit that could be made if

the option was immediately exercisedthe option was immediately exercised

–– Call: stock price Call: stock price -- exercise priceexercise price

–– Put: exercise price Put: exercise price -- stock price stock price

Time value Time value -- the difference between the the difference between the

option price and the intrinsic valueoption price and the intrinsic value

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Figure Call Option Value

Before Expiration

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Factors Influencing Option

Values: Calls

FactorFactor Effect on valueEffect on value

Stock price Stock price increasesincreases

Exercise price Exercise price decreasesdecreases

Volatility of stock price Volatility of stock price increasesincreases

Time to expirationTime to expiration increasesincreases

Interest rate Interest rate increasesincreases

Dividend RateDividend Rate decreasesdecreases

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Black-Scholes Option Valuation

CCoo = S= Sooee--ddTTN(dN(d11) ) -- XeXe--rTrTN(dN(d22))

dd11 = [ln(S= [ln(Soo/X) + (r /X) + (r –– dd + + ss22/2)T] / (/2)T] / (s s TT1/21/2))

dd22 = d= d11 -- ((s s TT1/21/2))

wherewhere

CCo o = Current call option value.= Current call option value.

SSoo = Current stock price= Current stock price

N(d) = probability that a random draw from a N(d) = probability that a random draw from a

normal dist. will be less than d.normal dist. will be less than d.

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Black-Scholes Option Valuation

X = Exercise price.X = Exercise price.

dd = Annual dividend yield of underlying stock= Annual dividend yield of underlying stock

e = 2.71828, the base of the natural loge = 2.71828, the base of the natural log

r = Riskr = Risk--free interest rate (annualizes continuously free interest rate (annualizes continuously compounded with the same maturity as the compounded with the same maturity as the option.option.

T = time to maturity of the option in years.T = time to maturity of the option in years.

ln = Natural log functionln = Natural log function

s = s = Standard deviation of annualized cont. Standard deviation of annualized cont. compounded rate of return on the stockcompounded rate of return on the stock

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Figure 15-3 A Standard Normal Curve

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Call Option Example

SSoo = 100= 100 X = 95X = 95

r = .10r = .10 T = .25 (quarter)T = .25 (quarter)

s s = .50= .50 d d = 0= 0

dd11 = [ln(100/95)+(.10= [ln(100/95)+(.10--0+(0+(..5 5 22/2))]/(/2))]/(..55 .25.251/21/2))

= .43= .43

dd22 = .43 = .43 -- ((((..55)( )( .25).25)1/21/2

= .18= .18

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Probabilities from Normal Dist.

N (.43) = .6664N (.43) = .6664

Table 17.2Table 17.2

dd N(d)N(d)

.42.42 .6628.6628

.43.43 .6664 Interpolation.6664 Interpolation

.44.44 .6700.6700

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Probabilities from Normal Dist.

N (.18) = .5714N (.18) = .5714

Table 17.2Table 17.2

dd N(d)N(d)

.16.16 .5636.5636

.18.18 .5714.5714

.20.20 .5793.5793

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Call Option Value

CCoo = S= Sooee--ddTTN(dN(d11) ) -- XeXe--rTrTN(dN(d22))

CCo o = 100 X .6664 = 100 X .6664 -- 95 e95 e-- .10 X .25.10 X .25 X .5714 X .5714

CCo o = 13.70= 13.70

Implied VolatilityImplied Volatility

Using BlackUsing Black--Scholes and the actual price of Scholes and the actual price of

the option, solve for volatility.the option, solve for volatility.

Is the implied volatility consistent with the Is the implied volatility consistent with the

stock?stock?

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Put Option Value: Black-Scholes

P=XeP=Xe--rT rT [1[1--N(dN(d22)] )] -- SS00ee--ddTT [1[1--N(dN(d11)] )]

Using the sample dataUsing the sample data

P = $95eP = $95e((--.10X.25).10X.25)(1(1--.5714) .5714) -- $100 (1$100 (1--.6664).6664)

P = $6.35P = $6.35

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Put Option Valuation:

Using Put-Call Parity

P = C + PV (X) P = C + PV (X) -- SSoo

= C + Xe= C + Xe--rTrT -- SSoo

Using the example dataUsing the example data

C = 13.70C = 13.70 X = 95X = 95 S = 100S = 100

r = .10r = .10 T = .25T = .25

P = 13.70 + 95 eP = 13.70 + 95 e --.10 X .25.10 X .25 -- 100100

P = 6.35P = 6.35

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Using the Black-Scholes Formula

Hedging: Hedge ratio or deltaHedging: Hedge ratio or delta

The number of stocks required to hedge against the The number of stocks required to hedge against the

price risk of holding one optionprice risk of holding one option

Call = N (dCall = N (d11))

Put = N (dPut = N (d11) ) -- 11

Option ElasticityOption Elasticity

Percentage change in the option’s value given a Percentage change in the option’s value given a

1% change in the value of the underlying stock1% change in the value of the underlying stock

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Figure Call Option Value

and Hedge Ratio

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Portfolio Insurance - Protecting Against

Declines in Stock Value

Buying Puts Buying Puts -- results in downside results in downside

protection with unlimited upside potentialprotection with unlimited upside potential

Limitations Limitations

–– Tracking errors if indexes are used for the Tracking errors if indexes are used for the

putsputs

–– Maturity of puts may be too shortMaturity of puts may be too short

–– Hedge ratios or deltas change as stock Hedge ratios or deltas change as stock

values changevalues change

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Empirical Tests of Black-Scholes

Option Pricing

Implied volatility varies with exercise priceImplied volatility varies with exercise price

–– Lower exercise price leads to higher option Lower exercise price leads to higher option

pricingpricing

–– If the model was complete implied variability If the model was complete implied variability

should be the same for different exercise should be the same for different exercise

pricesprices