Applications of Option Methods in Corporate Finance Timothy A. Thompson Financial Decisions

Goals of option valuation Purpose is not to be derivatives traders Purpose is not to be derivatives traders We want to understand what options are present in financial contractsWe want to understand what options are present in financial contracts We want to understand what the economic function these options have in financial contractingWe want to understand what the economic function these options have in financial contracting We are going to talk about simple option pricing models (Black Scholes/binomial) to get a ballpark of the value of imbedded optionsWe are going to talk about simple option pricing models (Black Scholes/binomial) to get a ballpark of the value of imbedded options We want to be able to understand when the “ballpark” is big or smallWe want to be able to understand when the “ballpark” is big or small

What are options? A call/put option represents a right, not an obligation, for the “holder” of the option to buy/sell an underlying assets for a fixed price (exercise price or strike price) on or before a specified future date (expiration date) A call/put option represents a right, not an obligation, for the “holder” of the option to buy/sell an underlying assets for a fixed price (exercise price or strike price) on or before a specified future date (expiration date) American vs. EuropeanAmerican vs. European Examples Examples Calls and puts on CBOECalls and puts on CBOE WarrantsWarrants Caps and FloorsCaps and Floors

Options, options, everywhere… Warrants, convertibles, callables Warrants, convertibles, callables Embedded options in PERCS, LYONS, etc. Embedded options in PERCS, LYONS, etc. Real asset options Real 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 prices Parameters of call and put prices, C t and P t Parameters of call and put prices, C t and P t Price of the underlying asset (stock, etc.), S tPrice of the underlying asset (stock, etc.), S t Time to maturity, , T - tTime to maturity, , 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), σVolatility (std dev of ror on underlying), σ Dividend yield on underlying assetDividend yield on underlying asset

Interesting what parameters are not there Expected return on the underlying Expected return on the underlying Expected risk premium on stocks over risk frees Expected risk premium on stocks over risk frees Risk aversion of investors Risk aversion of investors Why aren’t these there? Why aren’t these there? Because they are there: they are in the stock priceBecause they are there: they are in the stock price Options are derivative assets: they derive their value from the value of the underlying assetOptions are derivative assets: they derive their value from the value of the underlying asset

Black Scholes model The mathematics behind the Black Scholes model is difficult The mathematics behind the Black Scholes model is difficult But for our purposes the model is like a black (no pun intended) box But for our purposes the model is like a black (no pun intended) 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 parameters We want to know how the answer depends on the parameters We want to know whether the model will get us in the ballpark We want to know whether the model will get us in the ballpark If the model really is not appropriate for an application, we would go to a model that could be modified for the application If the model really is not appropriate for an application, we would go to a model that could be modified for the application Like the binomial method or numerical estimation methodsLike the binomial method or numerical estimation methods

Assumptions of Black Scholes model Perfect markets Perfect markets No taxes/transactions costs, information costs No taxes/transactions costs, information costs Option is European Option is European This is crucial. Next slide. This is crucial. Next slide. Stock follows a diffusion process Stock follows a diffusion process People can borrow or lend at r People can borrow or lend at r r,  and σ are known constants r,  and σ are known constants X and T are known constants X and T are known constants

Black Scholes Equation

When will a European model “work” when pricing American options? Generally, it won’t Generally, it won’t An American option is always worth at least as much as its European counterpartAn American option is always worth at least as much as its European counterpart Because you can do anything with an American option that you can do with an European option and Because you can do anything with an American option that you can do with an European option and You can exercise it prior to maturity. This right can’t have negative value. You can exercise it prior to maturity. This right can’t have negative value. Important no-arbitrage result from options Important no-arbitrage result from options An American call option on a non-dividend paying underlying asset will never be optimally exercised prior to maturity An American call option on a non-dividend paying underlying asset will never be optimally exercised prior to maturity If the option we need to value can be characterized as a call option on a non-dividend paying stock, then BS will be reasonable If the option we need to value can be characterized as a call option on a non-dividend paying stock, then BS will be reasonable As a practical matter, as long as dividends aren’t large enough to induce early exercise, then BS will be reasonable As a practical matter, as long as dividends aren’t large enough to induce early exercise, then BS will be reasonable

Luckily the computer does the math for us!

Call option value

Call value and volatility

Call value and maturity

Estimating parameters for traded call options Time to expiration Time to expiration Calendar time to expirationCalendar time to expiration Risk free interest rate Risk 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 price Stock price Current market price of underlying assetCurrent market price of underlying asset Dividends Dividends 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 Volatility Volatility Standard deviation of the rate of return on the underlying assetStandard deviation of the rate of return on the underlying asset

Volatility estimation Historical sample standard deviation Historical sample standard deviation Implied volatility Implied volatility Estimate all the B/S parameters except for volatilityEstimate all the B/S parameters except for volatility Using the market price of an option, back into the value of volatility parameter that equates the B/S value of the option to its market priceUsing the market price of an option, back into the value of volatility parameter that equates the B/S value of the option to its market price

Assumptions behind historical and implied volatility Historical volatility Historical volatility Assuming that historical volatility is a reasonable forecast of future volatility Assuming that historical volatility is a reasonable forecast of future volatility Same as many other issues we face (betas, etc.) Same as many other issues we face (betas, etc.) Implied volatility Implied volatility Assuming that the option is priced correctly by the Black Scholes model Assuming that the option is priced correctly by the Black Scholes model Assuming that the option price and underlying asset price are efficiently priced and available at the same time Assuming that the option price and underlying asset price are efficiently priced and available at the same time

Warrants What is a warrant? What is a warrant? Security giving the holder the right to purchase the underlying stock for a fixed price and given duration of time. Security giving the holder the right to purchase the underlying stock for a fixed price and given duration of time. Sounds just like an American call option Sounds just like an American call option Differences between warrants and calls Differences between warrants and calls Warrant is a primary market instrument for firm Warrant is a primary market instrument for firm Issued for cash or consideration, which is cash inflow to the firm when issuedIssued for cash or consideration, which is cash inflow to the firm when issued If warrants exercised, the exercise funds are cash inflow to the firm and there are more shares outstanding (dilution)If warrants exercised, the exercise funds are cash inflow to the 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 calls Warrants typically have longer maturities than calls Can have much more flexible terms than exchange traded options Can have much more flexible terms than exchange traded options

Applying Black Scholes model to value warrants Addiitonal notation: Addiitonal notation: W = Warrant value W = Warrant value N = Number of shares of stock outstanding before exercise of warrants N = Number of shares of stock outstanding before exercise of warrants M = number of warrant shares outstanding M = number of warrant shares outstanding Assumptions Assumptions The warrants being valued are the only securities convertible into common stock The warrants being valued are the only securities convertible into common stock Assume all warrants would be exercised only at maturity Assume all warrants would be exercised only at maturity

Warrants and common are “options” on total firm equity value The value of a European warrant is equivalent to the value of a European call option on the stock of on an otherwise identical firm with no warrants outstanding The value of a European warrant is equivalent to the value of a European call option on the stock of on an otherwise identical firm with no warrants outstanding Same number of shares outstanding, N Same 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 “identical” firm is NS*, equal to The value of the total equity of the “identical” firm is NS*, equal to The value of the total equity of this firm = NS + MW, so S* = S + (M/N)W The value of the total equity of this firm = NS + MW, so S* = S + (M/N)W

Firm with equity and warrants

“Black Scholes” Warrant Model

Debt and equity as options Assumptions Assumptions Company has only one debt issue (Face value = F, Zero coupon, Maturing in T years) and equity outstandingCompany has only one debt issue (Face value = F, Zero coupon, 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 observedBankruptcy costs are zero and absolute priority will be observed At maturity (date T), the value of the equity is given by E T = max[0, V T – F] At maturity (date T), the value of the equity is given by E T = max[0, V T – F] Value of the debt at maturity (date T) is given by D T = min[V T, F] Value of the debt at maturity (date T) is given by D T = min[V T, F] Equity payoff is identical to the payoff on a call option written on the assets (value) of the firm with a strike price equal to the face value of the debt and maturity equal to the maturity of the debt. Equity payoff is identical to the payoff on a call option written on the assets (value) of the firm with a strike price equal to the face value of the debt and maturity equal to the maturity of the debt.

Risky debt is riskless debt minus a put option From above we have E t = Call tFrom above we have E t = Call t The options are European here and there are no dividends (or coupons on debt) so we can use put-call-parity formula (PCP):The options are European here and there are no dividends (or coupons on debt) so we can use put-call-parity formula (PCP): E t = V t – PV(F) + Put tE t = V t – PV(F) + Put t Using the balance sheet constraint, D = V-EUsing the balance sheet constraint, D = V-E D t = V t – V t + PV(F) – Put t, orD t = V t – V t + PV(F) – Put t, or D t = PV(F) - Put tD t = PV(F) - Put t

Value of loan guarantee as a put option Suppose the government were to guarantee a firm’s debt Suppose the government were to guarantee a firm’s debt If the firm were to default, the government pays the bondholders their promised paymentsIf the firm were to default, the government pays the bondholders their promised payments Bonds become like riskless debtBonds become like riskless debt Put option from the last slide is contingent liability that the government assumes.Put option from the last slide is contingent liability that the government assumes.