Created 7/1/1996

Go to Brad De
Long's Home Page

**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((r
_{1}-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

ÿ