Applications of Option Applications of Option Methods in Corporate Methods in Corporate

FinanceFinance

Timothy A. ThompsonTimothy A. Thompson

Financial DecisionsFinancial Decisions

Goals of option valuationGoals of option valuation

Purpose is not to be derivatives tradersPurpose is not to be derivatives traders• We want to understand what options are We want to understand what options are

present in financial contractspresent in financial contracts• We want to understand what the economic We want to understand what the economic

function these options have in financial function these options have in financial contractingcontracting

• We are going to talk about simple option We are going to talk about simple option pricing models (Black Scholes/binomial) to get pricing models (Black Scholes/binomial) to get a ballpark of the value of imbedded optionsa ballpark of the value of imbedded options

• We want to be able to understand when the We want to be able to understand when the “ballpark” is big or small“ballpark” is big or small

What are options?What are options?

A call/put option represents a right, not an A call/put option represents a right, not an obligation, for the “holder” of the option to obligation, for the “holder” of the option to buy/sell an underlying assets for a fixed buy/sell an underlying assets for a fixed price (exercise price or strike price) on or price (exercise price or strike price) on or before a specified future date (expiration before a specified future date (expiration date)date)• American vs. EuropeanAmerican vs. European

ExamplesExamples• Calls and puts on CBOECalls and puts on CBOE• WarrantsWarrants• Caps and FloorsCaps and Floors

Options, options, everywhere…Options, options, everywhere…

Warrants, convertibles, callablesWarrants, convertibles, callables Embedded options in PERCS, LYONS, Embedded options in PERCS, LYONS,

etc.etc. Real asset optionsReal asset options

• Option to waitOption to wait• Option for follow up investmentsOption for follow up investments• Flexibility optionsFlexibility options• Abandonment optionsAbandonment options

Determinants of option pricesDeterminants of option prices

Parameters of call and put prices, CParameters of call and put prices, Ctt and Pand Ptt

• Price of the underlying asset (stock, Price of the underlying asset (stock, etc.), Setc.), Stt

• Time to maturity, Time to maturity, , T - t, T - t• Strike price (exercise price), XStrike price (exercise price), X• Risk free interest rate, rRisk free interest rate, r• Volatility (std dev of ror on underlying), Volatility (std dev of ror on underlying), σσ• Dividend yield on underlying assetDividend yield on underlying asset

Interesting what parameters are not Interesting what parameters are not therethere

Expected return on the underlyingExpected return on the underlying Expected risk premium on stocks over risk Expected risk premium on stocks over risk

freesfrees Risk aversion of investorsRisk aversion of investors Why aren’t these there?Why aren’t these there?

• Because they are there: they are in the stock Because they are there: they are in the stock priceprice

• Options are derivative assets: they derive their Options are derivative assets: they derive their value from the value of the underlying assetvalue from the value of the underlying asset

Black Scholes modelBlack Scholes model The mathematics behind the Black Scholes model is difficultThe mathematics behind the Black Scholes model is difficult But for our purposes the model is like a black (no pun But for our purposes the model is like a black (no pun

intended) boxintended) box• We put in parametersWe put in parameters• We get an answerWe get an answer

We want to know how the answer depends on the We want to know how the answer depends on the parametersparameters

We want to know whether the model will get us in the We want to know whether the model will get us in the ballparkballpark

If the model really is not appropriate for an application, we If the model really is not appropriate for an application, we would go to a model that could be modified for the would go to a model that could be modified for the applicationapplication• Like the binomial method or numerical estimation methodsLike the binomial method or numerical estimation methods

Assumptions of Black Scholes Assumptions of Black Scholes modelmodel

Perfect marketsPerfect markets No taxes/transactions costs, information costsNo taxes/transactions costs, information costs

Option is EuropeanOption is European This is crucial. Next slide.This is crucial. Next slide.

Stock follows a diffusion processStock follows a diffusion process People can borrow or lend at rPeople can borrow or lend at r r, r, and and σσ are known constants are known constants X and T are known constantsX and T are known constants

Black Scholes EquationBlack Scholes Equation

12

1

21

2

ln

)()(

dd

and

XeSe

d

where

dNXedNeSC

r

rtt

When will a European model “work” When will a European model “work” when pricing American options?when pricing American options?

Generally, it won’tGenerally, it won’t• An American option is always worth at least as much as An American option is always worth at least as much as

its European counterpartits European counterpart Because you can do anything with an American option that Because you can do anything with an American option that

you can do with an European option andyou can do with an European option and You can exercise it prior to maturity. This right can’t have You can exercise it prior to maturity. This right can’t have

negative value.negative value. Important no-arbitrage result from optionsImportant no-arbitrage result from options

An American call option on a non-dividend paying An American call option on a non-dividend paying underlying asset will never be optimally exercised prior to underlying asset will never be optimally exercised prior to maturitymaturity

If the option we need to value can be characterized as a If the option we need to value can be characterized as a call option on a non-dividend paying stock, then BS will be call option on a non-dividend paying stock, then BS will be reasonablereasonable

As a practical matter, as long as dividends aren’t large As a practical matter, as long as dividends aren’t large enough to induce early exercise, then BS will be reasonableenough to induce early exercise, then BS will be reasonable

Luckily the computer does the math Luckily the computer does the math for us!for us!

InputsStock Price 40 Call Put

Exercise Price 40 Price 2.784733 1.99268Volatility 0.3 Delta 0.582516 -0.41748

Risk-free interest rate 0.08 Gamma 0.065063 0.065063Time to Expiration 0.25 Vega 0.078076 0.078076

Dividend Yield 0 Theta -0.01733 -0.00874# Binomial steps 10 Rho 0.05129 -0.04673

Type (0=Eur, 1=Amer) 0Implied Volatility 32.76% 30.09%

Observed Call Price 3Observed Put Price 2 Call Put

Price 2.725865 1.933812Delta 0.580522 -0.41948

Gamma 0.07071 0.07071Theta -0.01844 -0.00983

For Implied Volatility

Black-Scholes

Binomial (CRR)

These option pricing functions are intended for educational use only.

© 1994-2000 Robert McDonald, Kellogg School, Northwestern University

Call option valueCall option valueCall option value

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Call value and volatilityCall value and volatilityCall value and volatility

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vol = 30%

vol = 40%

vol = 50%

vol = 60%

Call value and maturityCall value and maturity

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Estimating parameters for traded Estimating parameters for traded call optionscall options

Time to expirationTime to expiration• Calendar time to expirationCalendar time to expiration

Risk free interest rateRisk free interest rate• Nearest Treasury strip to maturity of optionNearest Treasury strip to maturity of option• Annualized and restated to be continuously compoundedAnnualized and restated to be continuously compounded

Exercise price (strike price)Exercise price (strike price) Stock priceStock price

• Current market price of underlying assetCurrent market price of underlying asset DividendsDividends

• Annualized dividend to price ratio and cont. comp.Annualized dividend to price ratio and cont. comp.• Or subtract present value of dividends from stock priceOr subtract present value of dividends from stock price

VolatilityVolatility• Standard deviation of the rate of return on the underlying Standard deviation of the rate of return on the underlying

assetasset

Volatility estimationVolatility estimation

Historical sample standard deviationHistorical sample standard deviation

Implied volatilityImplied volatility• Estimate all the B/S parameters except Estimate all the B/S parameters except

for volatilityfor volatility• Using the market price of an option, Using the market price of an option,

back into the value of volatility back into the value of volatility parameter that equates the B/S value of parameter that equates the B/S value of the option to its market pricethe option to its market price

Assumptions behind historical and Assumptions behind historical and implied volatilityimplied volatility

Historical volatilityHistorical volatility Assuming that historical volatility is a Assuming that historical volatility is a

reasonable forecast of future volatilityreasonable forecast of future volatility Same as many other issues we face (betas, Same as many other issues we face (betas,

etc.)etc.)

Implied volatilityImplied volatility Assuming that the option is priced correctly Assuming that the option is priced correctly

by the Black Scholes modelby the Black Scholes model Assuming that the option price and underlying Assuming that the option price and underlying

asset price are efficiently priced and available asset price are efficiently priced and available at the same timeat the same time

WarrantsWarrants What is a warrant?What is a warrant?

Security giving the holder the right to purchase the Security giving the holder the right to purchase the underlying stock for a fixed price and given duration of underlying stock for a fixed price and given duration of time.time.

Sounds just like an American call optionSounds just like an American call option Differences between warrants and callsDifferences between warrants and calls

Warrant is a primary market instrument for firmWarrant is a primary market instrument for firm• Issued for cash or consideration, which is cash inflow to the Issued for cash or consideration, which is cash inflow to the

firm when issuedfirm when issued• If warrants exercised, the exercise funds are cash inflow to the If warrants exercised, the exercise funds are cash inflow to the

firm and there are more shares outstanding (dilution)firm and there are more shares outstanding (dilution)• Executive stock options are warrants in this sense.Executive stock options are warrants in this sense.

Warrants typically have longer maturities than callsWarrants typically have longer maturities than calls Can have much more flexible terms than exchange traded Can have much more flexible terms than exchange traded

optionsoptions

Applying Black Scholes model to Applying Black Scholes model to value warrantsvalue warrants

Addiitonal notation:Addiitonal notation: W = Warrant valueW = Warrant value N = Number of shares of stock outstanding N = Number of shares of stock outstanding

before exercise of warrantsbefore exercise of warrants M = number of warrant shares outstandingM = number of warrant shares outstanding

AssumptionsAssumptions The warrants being valued are the only The warrants being valued are the only

securities convertible into common stocksecurities convertible into common stock Assume all warrants would be exercised only Assume all warrants would be exercised only

at maturityat maturity

Warrants and common are Warrants and common are “options” on total firm equity value“options” on total firm equity value

The value of a European warrant is The value of a European warrant is equivalent to the value of a European call equivalent to the value of a European call option on the stock of on an option on the stock of on an otherwise otherwise identical firmidentical firm with no warrants outstanding with no warrants outstanding

Same number of shares outstanding, NSame number of shares outstanding, N Multiplied by dilution factor M/(N+M)Multiplied by dilution factor M/(N+M)

The value of the total equity of the The value of the total equity of the “identical” firm is NS*, equal to“identical” firm is NS*, equal to

The value of the total equity of this firm = The value of the total equity of this firm = NS + MW, so S* = S + (M/N)WNS + MW, so S* = S + (M/N)W

Firm with equity and warrantsFirm with equity and warrants

Equity

Warrants

““Black Scholes” Warrant ModelBlack Scholes” Warrant Model

WN

MSS

and

ddXeeS

d

where

MN

NdNXedNeSW

r

rtt

*

*12

*

*

*

1

21*

,2

ln

)()(

Debt and equity as optionsDebt and equity as options AssumptionsAssumptions

• Company has only one debt issue (Face value = F, Zero Company has only one debt issue (Face value = F, Zero coupon, Maturing in T years) and equity outstandingcoupon, Maturing in T years) and equity outstanding

• Company pays no dividends on commonCompany pays no dividends on common• Bankruptcy costs are zero and absolute priority will be Bankruptcy costs are zero and absolute priority will be

observedobserved

At maturity (date T), the value of the equity is At maturity (date T), the value of the equity is given by Egiven by ETT = max[0, V = max[0, VTT – F] – F]

Value of the debt at maturity (date T) is given by Value of the debt at maturity (date T) is given by DDTT = min[V = min[VTT, F], F]

Equity payoff is identical to the payoff on a call Equity payoff is identical to the payoff on a call option written on the assets (value) of the firm with option written on the assets (value) of the firm with a strike price equal to the face value of the debt and a strike price equal to the face value of the debt and maturity equal to the maturity of the debtmaturity equal to the maturity of the debt..

Debt and Equity as Options

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lue

at m

atu

rity

Firm value

Equity value

Debt value

Risky debt is riskless debt minus a Risky debt is riskless debt minus a put optionput option

• From above we have EFrom above we have Ett = Call = Calltt• The options are European here and there The options are European here and there

are no dividends (or coupons on debt) so are no dividends (or coupons on debt) so we can use put-call-parity formula (PCP):we can use put-call-parity formula (PCP):

• EEtt = V = Vtt – PV(F) + Put – PV(F) + Puttt

• Using the balance sheet constraint, D = Using the balance sheet constraint, D = V-EV-E

• DDtt = V = Vtt – V – Vtt + PV(F) – Put + PV(F) – Puttt, or, or

• DDtt = PV(F) - Put = PV(F) - Puttt

Value of loan guarantee as a put Value of loan guarantee as a put optionoption

Suppose the government were to Suppose the government were to guarantee a firm’s debtguarantee a firm’s debt• If the firm were to default, the If the firm were to default, the

government pays the bondholders their government pays the bondholders their promised paymentspromised payments

• Bonds become like riskless debtBonds become like riskless debt• Put option from the last slide is Put option from the last slide is

contingent liability that the government contingent liability that the government assumes.assumes.

Loan Guarantee is a Put

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Va

lue

at

ma

turi

ty Firm value

Equity value

Risky debt value

Implied Put Value

Riskless debt