Created 7/1/1996
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Long's Home Page
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Basics:
Qualitative Basics of Option Pricing:
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
Effects of variables:
A Restatement:
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 riskfree 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
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.
Did that go by too fast? Let's value the put option
Value of put option payoff
Therefore:
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 riskfree 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 sixmonth 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 monthsso 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 todayhence the value of the option todayis $30; and change.
Take a deep breath
In the case where we consider six month changeswhere 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 analysismake it more realisticby
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 


BlackScholes: 
$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 continuouslycompounded riskfree 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 BlackScholes 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 BlackScholes 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 exameither 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 twocase binomial options.
But the real world ain't binomial; BlackScholes plays a pretty big part in it.
Embedded Options:
Lots of embedded options