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 = 0 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 = 6 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 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  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.

Sample average

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

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.

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

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.