Options and risk measurement

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 = 75

S = 80, call = 4

S = 70, call = 0Pr. = .5

Pr. = .5

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 = 75

S = 80, put = 0

S = 70, put = 6Pr. = .5

Pr. = .5

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 impliesput 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

If S + P = Xe-r(T-t) + C + 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 is profit. The position is riskless because at expiration

s + p = X + c. i.e.,

Money pump two

If S + P + = 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 + in immediate arbitrage

profit. The position is riskless because at

expiration s + p = X + c. i.e.,

Money pump either way

If the prices persist, do the same thing over and over – a MONEY PUMP.

The existence of the violates no-arbitrage pricing.

Measuring risk

Rocket science

Rate of return =

(price increase + dividend)/purchase price.

t

tttj P

divPPR 11

Sample average

Year 1926 1927 1928 1929Rate of returnon common stocks 11.62 37.49 43.61 -8.42

Sample average

075.214

42.861.4349.3762.11

R

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 returnon common stocks 11.62 37.49 43.61 -8.42sample average 21.075 21.075 21.075 21.075deviation -9.455 16.415 22.535 -29.495deviation squared 89.39703 269.4522 507.8262 869.955sample variance 578.8768standard deviation 24.05986

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.

2222 )420.8()62.43()49.37()62.11(

min

mmmm

imizetomSelect

0)420.8(2

)62.43(2)49.37(2)62.11(2

m

mmm

Solution

42.862.4349.3762.114 m

4

42.862.4349.3762.11 m

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 PremiumT-Bills 3.8 3.2 0Common stocks 13.3 20.1 9.5Small cap stocks 17.6 33.6 13.8LT Corp bonds 5.9 8.7 2.1Inflation 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.