measurment risk & return

8/2/2019 Measurment Risk & Return

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Options and riskmeasurement


2/28

Definition of a call option

A call option is the right but not theobligation to buy 100 shares of the

stock at a stated exercise price on orbefore a stated expiration date.

The price of the option is not theexercise price.


3/28

Example

A share of IBM sells for 75.

The call has an exercise price of 76.

The value of the call seems to bezero.

In fact, it is positive and in one

example equal to 2.


4/28

t = 0 t = 1

S = 75

S = 80, call = 4

S = 70, call = 0

Value of call = .5 x 4 = 2


5/28

Definition of a put option

A put option is the right but not theobligation to sell 100 shares of the

stock at a stated exercise price on orbefore a stated expiration date.

The price of the option is not theexercise price.


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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 ourexample equal to 3.


7/28

t = 0 t = 1

S = 75

S = 80, put = 0

S = 70, put = 6

Value of put = .5 x 6 = 3


8/28

Put-call parity

S + P = X*exp(-r(T-t)) + C at any timet.

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.


9/28

Puts and calls as randomvariables

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 viewedfrom a time t before expiration T.

X is a trivial random variable.


10/28

Puts and calls before expiration

S, P, and C are the market values attime t before expiration T.

Xe-r(T-t) is the market value at time t ofthe exercise money to be paid at T

Traders tend to ignore r(T-t) because

it is small relative to the bid-askspreads.


11/28

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


12/28

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 isavailable.


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Money pump one

If 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.,


14/28

Money pump two

If 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.,


15/28

Money pump either way

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

The existence of the e violates no-arbitrage pricing.


16/28

Measuring risk


17/28

Rate of return =

(price increase + dividend)/purchaseprice.

t

ttt

jP

divPP

R11


18/28

Sample average

Year 1926 1927 1928 1929

Rate of returnon common stocks 11.62 37.49 43.61 -8.42


19/28

Sample average

075.21

4

42.861.4349.3762.11

R


20/28

Sample versus population

A sample is a series of random drawsfrom a population.

Sample is inferential. For instance thesample average.

Population: model: For instance the

probabilities in the problem set.


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Population mean

The value to which the sampleaverage tends in a very long time.

Each sample average is an estimate,more or less accurate, of thepopulation mean.


22/28

Abstraction of finance

Theory works for the expected values.

In practice one uses sample means.


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Deviations

Rate of return

on common stocks 11.62 37.49 43.61 -8.42sample average 21.075 21.075 21.075 21.075

deviation -9.455 16.415 22.535 -29.495

deviation squared 89.39703 269.4522 507.8262 869.955

sample variance 578.8768standard deviation 24.05986


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Explanation

Square deviations to measure bothtypes of risk.

Take square root of variance to getcomparable units.

Its still an estimate of true population

risk.


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Why divide by 3 not 4?

Sample deviations are probably toosmall

because the sample averageminimizes them.

Correction needed.

Divide by T-1 instead of T.


26/28

Derivation of sample average

as an estimate of populationmean.

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


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Rough interpretation ofstandard deviation

The usual amount by which returnsmiss the population mean.

Sample standard deviation is anestimate of that amount.

About 2/3 of observations are within

one standard deviation of the mean.About 95% are within two S.D.s.


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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 stock 17.6 33.6 13.8

LT Corp bonds 5.9 8.7 2.1Inflation 3.2 4.5 -0.6

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