Transcript Puts and calls

Monty Hall and options
Demonstration: Monty Hall
 A prize
is behind one of three doors.
 Contestant chooses one.
 Host opens a door that is not the
chosen door and not the one
concealing the prize. (He knows where
the prize is.)
 Contestant is allowed to switch doors.
Solution
 The
contestant should always switch.
 Why? Because the host has information
that is revealed by his action.
Representation
switch and win
or
stay and lose
Nature’s move,
plus the contestant’s
guess.
switch and lose
or
stay and win
Definition of a call option
 A call
option is the right but not the
obligation to buy 100 shares of the
stock at a stated exercise price on or
before a stated expiration date.
 The price of the option is not the
exercise price.
Example
 A share
of IBM sells for 75.
 The call has an exercise price of 76.
 The value of the call seems to be zero.
 In fact, it is positive and in one example
equal to 2.
t=0
t=1
S = 80, call = 4
S = 75
S = 70, call = 0
Value of call = .5 x 4 = 2
Definition of a put option
 A put
option is the right but not the
obligation to sell 100 shares of the
stock at a stated exercise price on or
before a stated expiration date.
 The price of the option is not the
exercise price.
Example
 A share
of IBM sells for 75.
 The put has an exercise price of 76.
 The value of the put seems to be 1.
 In fact, it is more than 1 and in our
example equal to 3.
t=0
t=1
S = 80, put = 0
S = 75
S = 70, put = 6
Value of put = .5 x 6 = 3
Put-call parity
S
+ P = X*exp(-r(T-t)) + C at any time t.
 s + p = X + c at expiration
 In the previous examples, interest was
zero or T-t was negligible.
 Thus S + P=X+C
 75+3=76+2
 If not true, there is a money pump.
Puts and calls as random
variables
 The
exercise price is always X.
 s, p, c, are cash values of stock, put,
and call, all at expiration.
 p = max(X-s,0)
 c = max(s-X,0)
 They are random variables as viewed
from a time t before expiration T.
 X is a trivial random variable.
Puts and calls before expiration
 S,
P, and C are the market values at
time t before expiration T.
 Xe-r(T-t) is the market value at time t of
the exercise money to be paid at T
 Traders tend to ignore r(T-t) because it
is small relative to the bid-ask spreads.
Put call parity at expiration
 Equivalence
at expiration (time T)
s+p=X+c
 Values at time t in caps:
S + P = Xe-r(T-t) + C
No arbitrage pricing implies
put call parity in market prices
 Put
call parity holds at expiration.
 It also holds before expiration.
 Otherwise, a risk-free arbitrage is
available.
Money pump one
S + P = Xe-r(T-t) + C + e
 S and P are overpriced.
 Sell short the stock.
 Sell the put.
 Buy the call.
 “Buy” the bond. For instance deposit Xe-r(T-t)
in the bank.
 The remaining e is profit.
 The position is riskless because at expiration
s + p = X + c. i.e.,
 If
Money pump two
S + P + e = Xe-r(T-t) + C
 S and P are underpriced.
 “Sell” the bond. That is, borrow Xe-r(T-t).
 Sell the call.
 Buy the stock and the put.
 You have + e in immediate arbitrage
profit.
 The position is riskless because at
expiration s + p = X + c. i.e.,
 If
Money pump either way
 If
the prices persist, do the same thing
over and over – a MONEY PUMP.
 The existence of the e violates noarbitrage pricing.
Measuring risk
Rocket science
Rate of return =
 (price
increase + dividend)/purchase
price.
Pt 1  Pt  divt 1
Rj 
Pt
Sample average
Year
Rate of return
on common stocks
1926
1927
1928
1929
11.62
37.49
43.61
-8.42
Sample average
11.62  37.49  43.61  8.42
R
4
 21.075
Sample versus population
 A sample
is a series of random draws
from a population.
 Sample is inferential. For instance the
sample average.
 Population: model: For instance the
probabilities in the problem set.
Population mean
 The
value to which the sample average
tends in a very long time.
 Each sample average is an estimate,
more or less accurate, of the population
mean.
Abstraction of finance
 Theory
works for the expected values.
 In practice one uses sample means.
Deviations
Rate of return
on common stocks 11.62
37.49
43.61
sample average
21.075
21.075
21.075
deviation
-9.455
16.415
22.535
deviation squared
89.39703 269.4522 507.8262
sample variance
578.8768
standard deviation 24.05986
-8.42
21.075
-29.495
869.955
Explanation
 Square
deviations to measure both
types of risk.
 Take square root of variance to get
comparable units.
 Its still an estimate of true population
risk.
Why divide by 3 not 4?
 Sample
deviations are probably too
small …
 because the sample average minimizes
them.
 Correction needed.
 Divide by T-1 instead of T.
Derivation of sample average
as an estimate of population
mean.
Select m to min imize
(11.62  m) 2  (37.49  m) 2  (43.62  m) 2  (8.420  m) 2
Solution
 2(11.62  m)  2(37.49  m)  2(43.62  m)
 2(8.420  m)  0
4m  11.62  37.49  43.62  8.42
11.62  37.49  43.62  8.42
m
4
Rough interpretation of
standard deviation
 The
usual amount by which returns
miss the population mean.
 Sample standard deviation is an
estimate of that amount.
 About 2/3 of observations are within
one standard deviation of the mean.
 About 95% are within two S.D.’s.
Estimated risk and return
1926-1999
Sample average Sample sigma Sample Premium
T-Bills
3.8
3.2
0
Common stocks
13.3
20.1
9.5
Small cap stocks
17.6
33.6
13.8
LT Corp bonds
5.9
8.7
2.1
Inflation
3.2
4.5
-0.6
Review question
 What
is the difference between the
population mean and the sample
average?
Answer
 Take
a sample of T observations drawn
from the population
 The sample average is
(sum of the rates)/T
 The sample average tends to the
population mean as the number of
observations T becomes large.