Options and Their Applications

Options

  • Many corporate securities have similar characteristics to stock options traded on exchanges.
  • Almost all corporate stocks and bonds have option features.
  • Capital structure and capital budgeting decisions can be analyzed using options.

Options: Ubiquitous Nature

  • Options are found everywhere.
  • Traded options are commonly available.
  • Non-traded options such as employee stock options exist.
  • Investment opportunities can be viewed as real options.
  • Embedded options are part of securities like convertible bonds.
  • Equity and debt contain option features.

Option Contracts: Preliminaries

  • An option grants the holder the right, but not the obligation, to buy or sell an asset at a price agreed upon today, on or before a specific date.
  • Calls vs. Puts
    • Call options provide the right, without the obligation, to buy an asset at a future date at a price agreed upon today.
      • Exercising a call option means you "call in" the asset.
    • Put options provide the right, without the obligation, to sell an asset at a future date at a price agreed upon today.
      • Exercising a put option means you "put" the asset to someone.

Option Contracts: Key Definitions

  • Exercising the Option
    • The act of buying or selling the underlying asset via the option contract.
  • Strike Price or Exercise Price
    • The fixed price in the option contract at which the holder can buy or sell the underlying asset.
  • Expiry
    • The maturity date of the option, also known as the expiration date.
  • European vs. American Options
    • European options can only be exercised at expiry.
    • American options can be exercised at any time up to expiry.

Option Contracts: Moneyness and Intrinsic Value

  • In-the-Money
    • For a call option, the exercise price is less than the spot price of the underlying asset.
    • For a put option, the exercise price is more than the spot price of the underlying asset.
  • At-the-Money
    • The exercise price equals the spot price of the underlying asset.
  • Out-of-the-Money
    • For a call option, the exercise price is greater than the spot price of the underlying asset.
    • For a put option, the exercise price is less than the spot price of the underlying asset.
  • Intrinsic Value
    • The value of the option if exercised immediately.

Call Option Payoffs

  • Buying a Call
    • The payoff increases as the stock price rises above the exercise price.
    • If the stock price is below the exercise price, the payoff is zero.
      *Exercise price = $50
      *Graph illustrating payoffs for various stock prices, where the payoff increases linearly above $50.

Call Option Payoffs (Selling)

  • Selling a Call
    • The payoff decreases as the stock price rises above the exercise price.
    • If the stock price is below the exercise price, the payoff is zero.
      *Exercise price = $50
      *Graph illustrating payoffs for various stock prices, where the payoff decreases linearly above $50.

Put Option Pricing Relationships at Expiry

  • At expiry, an American put option has the same value as a European option with the same characteristics.
  • If the put is in-the-money, its value is the exercise price (E) minus the stock price at expiry (ST): E–STE – S_T. E > ST
  • If the put is out-of-the-money, it is worthless.
  • Put Option Value Formula: P=Max[E–ST,0]P = Max[E – S_T, 0]

Put Option Payoffs (Buying)

  • Buying a Put
    • The payoff increases as the stock price falls below the exercise price.
    • If the stock price is above the exercise price, the payoff is zero.
      *Exercise price = $50
      *Graph illustrating payoffs for various stock prices, where the payoff increases linearly below $50.

Put Option Payoffs (Selling)

  • Selling a Put
    • The payoff decreases as the stock price falls below the exercise price.
    • If the stock price is above the exercise price, the payoff is zero.
      *Exercise price = $50
      *Graph illustrating payoffs for various stock prices, where the payoff decreases linearly below $50.

Option Value Determinants

  • Call Option
    • Stock price: Positive relationship (+)
    • Exercise price: Negative relationship (-)
    • Interest rate: Positive relationship (+)
    • Volatility in the stock price: Positive relationship (+)
    • Expiration date: Positive relationship (+)
  • Put Option
    • Stock price: Negative relationship (-)
    • Exercise price: Positive relationship (+)
    • Interest rate: Negative relationship (-)
    • Volatility in the stock price: Positive relationship (+)
    • Expiration date: Positive relationship (+)
  • Call Option Value Bounds: max (S0 – E, 0) < C0 < S_0
    • C0C_0 represents the value of a call option.
    • S0S_0 represents the current stock price.
    • EE represents the exercise price.
  • The exact option value will depend on the factors listed above.

The Black-Scholes Model

The Black-Scholes Model allows for the valuation of options in the real world.

  • C0C_0 = the value of a European option at time t = 0
  • rr = the risk-free interest rate.
  • N(d)N(d) = Probability that a standardized, normally distributed, random variable will be less than or equal to d.

Black-Scholes Model Example

  • Consider a six-month call option on Microsoft.
    • Exercise price = $150
    • Current stock price = $160
    • Risk-free interest rate = 5%
    • Time to maturity = 0.5 years
    • Volatility = 30% per annum
  • Intrinsic value of the option is $10 (since $160 - $150 = $10), so the answer must be at least that amount.

Black-Scholes Model: Calculation

*To value a call option with the Black-Scholes model, you need to calculate d<em>1d<em>1 and d</em>2d</em>2:
*C<em>1=S×N(d</em>1)−Ee−rT×N(d<em>2)C<em>1 = S × N(d</em>1) - Ee^{-rT} × N(d<em>2) *d</em>1=0.5282d</em>1 = 0.5282
*d<em>2=0.31602d<em>2 = 0.31602 *N(d</em>1)=N(0.52815)=0.7013N(d</em>1) = N(0.52815) = 0.7013
*N(d<em>2)=N(0.31602)=0.62401N(d<em>2) = N(0.31602) = 0.62401 *C1 = $160 × 0.7013 - 150e^{-.05 × 0.5} × 0.62401
*C_1 = $20.92

Stocks and Bonds as Options

  • Levered Equity as a Call Option
    • The underlying asset is the firm's assets.
    • The strike price is the payoff of the bond.
    • If, at debt maturity, the firm's assets exceed the debt, shareholders exercise their call option and pay off the bondholders.
    • If the firm's assets are less than the debt, shareholders default and let the call option expire.

Stocks and Bonds as Options (Continued)

  • Levered Equity as a Put Option
    • The underlying asset is the firm's assets.
    • The strike price is the payoff of the bond.
    • If, at debt maturity, the firm's assets are less than the debt, shareholders have an in-the-money put and effectively "put" the firm to the bondholders.
    • If the firm's assets exceed the debt, shareholders do not exercise the put option (do not declare bankruptcy) and let it expire.

Put-Call Parity

  • It all comes down to put-call parity.
  • Value of a call on the firm + Value of a risk-free bond = Value of a put on the firm + Value of the firm
  • Stockholder's position in terms of call options
  • Stockholder’s position in terms of put options
  • c<em>0=S</em>0+p0–(1+r)−TEc<em>0 = S</em>0 + p_0 – (1+ r)^{-T} E

Applying Option Pricing Theory

  • Consider a firm with a single outstanding bond.
    • 1-year maturity
    • Zero Coupon
    • Face Value (F) = $5 million
  • Let VF∗V_F^* be the value of the firm's assets at debt expiration.
  • Two possibilities:
    • V_F^* < $5 million: shareholders default.
    • V_F^* > $5 million: shareholders pay bondholders and retain the residual value.
  • Payoff functions resemble common option forms.
  • Option pricing theory helps:
    • Assign values to debt and equity.
    • Understand how firm decisions affect relative values.

Debt and Equity Claims as Options: Payoffs

  • Payoff to Debtholders at Maturity
    • V<em>D∗=FV<em>D^* = F if VF^* > F
    • V<em>D∗=V</em>F<em>V<em>D^* = V</em>F^<em> if V_F^ < F
  • Payoff to Shareholders at Maturity
    • V<em>E∗=V</em>F∗−FV<em>E^* = V</em>F^* - F if V_F^* > F
    • V<em>E∗=0V<em>E^* = 0 if VF^* < F

Straight Bond Value

*Graph illustrating the relationship between firm value and bond value. The bond value is capped at a certain level corresponding to face value and is below the firm value.

Conversion value

*Graph illustrating the conversion value to firm value.

Total Value of Convertible Bonds

*Graph of Total value of convertible bonds showing straight bond, conversion and option values.

Reasons for Issuing Warrants and Convertibles

  • Matching cash flows
  • Risk synergy
  • Reduction of agency costs

Advantages to Companies

  • Sell stock at a premium to the current price
  • Lower cash requirements (lower coupon interest)
  • Flexible capital
  • Treated as partial equity
  • Tax deductibility of coupon payments
  • Access to market
  • Facilitate future financing

Advantages to Investors

  • Components of growth and income
  • Enhanced equity yield from coupon payment
  • Hedge funds
  • Fixed income buyers who want upside potential

Signaling Impact

  • 2% negative abnormal returns, but there is a wide variation
  • Best reaction: Companies with high post-issue capital expenditures, high market-to-book ratios, low credit ratings, high D/E ratios, never paid a dividend
  • Worst reaction: Issuers who paid high dividends and then suspended them