>Finance

Created 7/1/1996
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Business Administration 130:

Options II

http://www.j-bradford-delong.net/Intro_Finance/BAonethirty21.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:
  

Qualitative Basics of Option Pricing:

• 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:

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)

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]

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 from above. [thus the value of an option increases with the rate of interest and the time to maturity; kind of like buying stock with an interest-free loan]
• When the stock price is low (below the exercise price), the option price is positive--is above the value at immediate exercise.
• The option price must be less than the price of the stock.

Effects of variables:

• Increase in the stock price increases the value of the call option
• Increase in the exercise price decreases the value of the call option
• Increases in the risk-free rate r increase the value of the call option.
• Increases in the time to expiration increase the value of the call option.
• Increases in the per-period volatility of the stock price increases the value of the call option.

A Restatement:

1. In order to exercise an option you have the pay the exercise price. Other things being equal, the less you are obliged to pay, the better. Therefore the value of an option increases as the ratio of the underlying asset price to the exercise price increases.
2. You do not have to pay the exercise price until you decide to exercise the option. Yet once you have bought the option you are "invested" in the underlying asset. Thus buying an option is a little bit like being offered an interest free loan of the amount of the strike price. The higher is the rate of interest and the longer is the time to maturity, the higher is the option value.
3. If the price of an asset falls short of the exercise price on the exercise date, you lose 100 percent of your investment--but no more. On the other hand, the more the price rises above the strike price, the more profit you will make. Therefore the option holder does not lose from increased volatility if things go wrong, but does gain if things go right. The value of the option increases with the varaince per period of the stock return multiplied by the number of periods to maturity.

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. Let the risk-free interest rate r be 5% per year.

Suppose, further, that Intel stock can only (a) fall by 20% to $52 or rise by 25% to $81.25, then

• Option value = 0 in bad case
• Option value = $16.25 in good case.
• Spread=5/9 spread of stock price between good and bad case.
• 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, which is 5/9 of $52.

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 then covering by (a) buying 5/9 times as many shares as you sold options, and (b) funding your purchase by borrowing $28.18 at the risk-free rate for each option you sold. The $28.18+$7.93 will pay for the 5/9 share of stock--and you are perfectly hedged. Anything more than $7.93 for the option that you sold is pure gravy.

Did that go by too fast? Let's value the put option

Value of put option payoff

• = +$13 in low state;
• = 0 in high state;
• difference = -4/9 times the spread in the stock price between the bad and good states.

Therefore:

• sell 4/9 of a share;
• lend out $35.23 (to collect $36.11--the price of 4/9 of a share in six months in the good state). $35.23 - 4/9 x $65 = $6.34.

And we know that:

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

$7.93 +$65/1.025 = $6.34 + $65, which checks out...

Let's suppose we start with a stock worth $553 dollars; a risk-free interest rate of 5%; a 60% chance that it will appreciate 33% to $738 in one year; a 40% chance that it will lose 25% of its value down to $415 a share in one year.

Let's value a put option with a strike price of $500 (and an expiration date of 1 year). Value of put option = $85 in bad state; =$0 in good state. Option delta = 85/323 means that you go short -0.263 of a share and loan out $184.85 in order to replicate the option portfolio. Cost of the replicated portfolio = $39 is the value of the option.

Suppose there are more than two possible outcomes? Suppose that in each six-month period the value of the stock could either rise by 22.6% or fall by 18.4% (values chosen so that the stock price a year hence has the same proportional standard deviation). Stock price could then rise to $832, remain unchanged at $553, or fall to $368.

Suppose stock price rises over the next six months--so that after a year the stock price might be $832, might be $553. In either case the value of the option six months from now is zero. So if the stock price rises over the next six months, the option value falls to zero.

Suppose the stock price falls over the next six months to $451. Then after a year the stock price might be $553 (in which case the option is worht zero) or might be $368 (in which case the option value at expiration would be $132). Option delta = 132/185 = -.7135 is the number of shares you sell short; and you loan out $384.95 so that the return in six months in the "good" state of the world of $394.57 will buy the .7135 shares you need to cover your short sale.

Cost of replicating portfolio (six months hence) is $63; hence value of option in "bad" state six months from now (sotkc price of $451) is $63.

Now let's sit in today and consider the next six months. If the stock price rises (to $679) the option is worth zero; if the stock price falls (to $451) over the next six months, the value of the option then will be $63.

Let's construct a replicating portfolio for the next six months. Option delta = 63/228 = -.276; amount you need to loan out is $183.04; proceeds from selling short .276 of a share today are $152.80; cost of replicating portfolio today--hence the value of the option today--is $30; and change.

Take a deep breath

In the case where we consider six month changes--where instead of two ultiamte stock prices (738 and 415) there are three ultimate stock prices with the same standard deviation and expected value (368, 553, 832) you construct the replicating portfolio by:

(a) selling short today -.276 shares of the underlying, and investing $183.04 at the riskless rate (a $30 investment)

(b) wait six months;

(c) if the stock price rises over the next six months, liquidate your position for a gross return of zero (on your original $30 investment).

(d) if the stock price falls over the next six months, extend your short position by selling an additional .438 of a share short, so your total position is .714 of a share short; loan out the proceeds from this second short sale (they are just enough to boost your total lending to $385).

(e) if the stock price rises over the second six months, liquidate your portfolio for a gross return of zero.

(f) if the stock price falls over the second six months, liquidate your portfolio for a gross return of $132.

Voila. You have replicated the option.

We can extend this analysis--make it more realistic--by considering finer and finer divisions of the year and smaller and smaller moves in the stock price. Don't try this at home: each time the price moves, you have to buy (or sell) more stock in order to construct the proper replicating portfolio for the next period. And calculating what these replicating portfolios are at every stage is not that easy.

Intervals in a Year	Upside Change	Downside Change	Calculated Option Value
1	33.3%	25.0%	$39.40
2	22.6%	18.4%	$30.20
3	18.2%	15.3%	$29.80
4	15.5%	13.4%	$30.40
12	8.7%	8.0%	$29.10
52	4.1%	3.9%	$28.30
		Black-Scholes:	$28.20

Black Scholes: Value of call option = N(d1) x P - N(d2) x PV(EX)

where:

N() is the cumulative unit normal probability density function.

EX is the exercise price of the option.

PV() is the present value discounted at the continuously-compounded risk-free interest rate.

d1 = [log(P/PV(EX))/(sigma x root(t)) + (sigma x root(t))/2]

d2=d1 - sigma x root(t)

root(t) is the square root of the number of periods until the exercise date

sigma is the per period standard deviation of the return on the stock

P is the price of the stock now

Thus the Black-Scholes formula tells us that the value of a call is equal to the value of an investment (today) of N(d1) in the common stock, less borrowing of N(d2) x PV(EX).

Tables 6 and 7 give the Black-Scholes formula values fror a range of P/PV(EX) and sigma x root(t) values. Table six gives the price (as a percent of P); table 7 gives the option delta or hedge ratio.

If you take any more finance courses; you will see this a lot; you may see this on an exam--either asked to identify what it is, or (perhaps) asked to apply it (in which case I will give it to you). But from now on we are going to look only at simple two-case binomial options.

But the real world ain't binomial; Black-Scholes plays a pretty big part in it.

Embedded Options:

Lots of embedded options

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