>Finance

Created 7/1/1996
Go to Brad De Long's Home Page

Business Administration 130:

Options I

http://www.j-bradford-delong.net/Intro_Finance/BAonethirty20.html

Basics:

• Present value of a perpetuity: C/r
• Present value of a growing (or shrinking) perpetuity: C/(r-g)
• Present value of C dollars t years from now: C/[(1+r)^t]
• Present value of a C-dollar t-year annuity: C[(1/r)-(1/[r(1+r)^t])
• "Rule of 72": (1+r)^t = 2 (approximately) whenever rt=.72
• beta = [E((r₁-r*₁)(r_m-r*_m)]/[(r_m-r*_m)²]
• r*_i = r*_f + beta_i(r*_m-r_f)
• r*_a=(D/V)r*_d+(E/V)r*_e
• Expected return of a portfolio with N securities, a share 1/N invested in each security:
  
• Standard deviation of a portfolio with N securities, a share 1/N invested in each security:
  

Interactions of Investment and Financing:

• after-tax weighted-averaged cost of capital: =(D/V)(1-T_c)r*_d+(E/V)r*_e
• adjusted present value: APV = base NPV + PV of financing decisions; APV calculated as if the project is but one simple equity-financed firm.
• Discount safe, nominal cash flows at the after-tax borrowing rate (why? because of the tax shield that you get; no reason not to borow 100% of the financing...)

Options:

• Chicago Board of Trade Options Exchange was founded in 1973; an immediate success.
• Buy options (if you are a firm) to offset idiosyncratic risk that may lead to financial distress.
• Buy options (if you are an individual) if you need psychiatric help.
• Options pricing theory also helps value growth opportunities. "Disguised" options.

Calls, Puts, and Shares:

• A call option gives its owner the right to buy stock at a specified exercise or strike price on or before a specified exercise date. European options--only on the particular date; American options--on or before that date.
• A put option gives its owner the right to sell stock at a specified exercise or strike price on or before a specified exercise date. European options--only on the particular date; American options--on or before that date.

Intel Options Prices in July 1995; Stock Trading at $65 a Share

Exercise Date	Exercise Price	Price of Put	Price of Call
10/95	$65	$6.25	$4.625
1/96	$65	$8	$5.875
1/96	$70	$5.875	$8.5

Value of call at expiration = max(price of share - exercise price, 0)

Value of put at expiration = max(exercise price - price of share, 0)

Bachelier diagrams//payoffs to owners/payoffs to writers

[buy call, invest PV of exercise price in safe asset] has the same payoff as [buy put, buy share]

V[call] + PV[exercise price] = V[put]+[share price]

[buy call, sell put] has the same payoff as [buy share, borrow PV of exercise price]

Synthetic Option:

Buy put = buy call + sell share + invest PV of exercise price

Bankruptcy as shareholders' exercise of a put option

What determines option values?

Value of call is less than share price; value of call is greater than payoff if exercised immediately

• When the stock is worthless, the option is worthless
• When the stock price is very large, option price approaches stock price minus PV of exercise price. [thus the value of an option increases with the rate of interest and the time to maturity]--buying on credit
• The option price exceeds its minimum value--higher by an amount that depends on the variance

Why DCF Doesn't Work for Options:

Because the riskiness of an option changes every time the stock price moves.

Valuing Options:

Price options by constructing a synthetic option.

Suppose we have our $65 Intel stock, and buy a call option with a strike price of $65 and an expiration date six months from now. r of 5% per year. If Intel stock can only (a) fall by 20% to $52 or rise by 25% to $81.25, then

Option value = 0 in bad case; $16.25 in good case. Spread=5/9 spread of stock price. Suppose you bought 5/9 of a share and borrowed the PV of 5/9 of a share in the bad case from the bank--borrow $28.18, the PV of $28.89.

Then you have the same payoffs as the option. Value of 5/9 of a share today is $36.11, minus $28.18 = $7.93. We have just valued our option. The number of shares to replicate the spread from an option is the hedge ratio or option delta. (If the option sells for more than $7.93, you have a money machine by selling options and covering.

Value of put option--option delta = -4/9; payoff = +$13 in low state; = 0 in high state; sell 4/9 of a share and lend out $35.23 (collect $36.11 in six months). $35.23 - 4/9 x $65 = $6.34.

V[call] + PV[exercise price] = V[put]+[share price]

$7.93 +$65/1.025 = $6.34 + $65

Summary:

Our authors have gotten punchy....

Black-Scholes

Send e-mail to Brad De Long

Return to Brad DeLong's Home Page

Return to Business Administration 130 Home Page