Solution from Summer 1997

The CBOE digital option

Let p denote the probability that one period before the maturity of the option, the price of Asset A is greater than that of Asset B. Since for either company the probability of the stock price increase is the same, the probability that Asset B's price one period before the maturity is greater than A's is also p, and the probability that the prices are equal is 1 - 2p. Then the probability that the digital option will pay out N T-shirts at maturity is:

p + (1 - 2p) ½ = ½

and the option is therefore worth N/2 T-shirts.

The Banach option

Let _pN denote the probability that the option will expire after N months. Then the price of one of the stocks at expiration is R and that of the other is N - R, and the payout is 2R - N. The number of ways that this can happen is:

while the total number of ways in which the two stock prices can evolve over an N-month period is 2^N. The probability _pN can therefore be obtained:

To value the option, one has to sum all the possible payouts weighted by their probability of occurrence. This is equivalent to evaluating the following sum:

P = S (2R-N)_PN

Knockout options

Suppose that, after some time, the stock on which the first type of knockout option has been written has exhibited k distinct values of return. Let M_k denote the expected number of months before the return on the stock is one which has not yet been observed. We can write:

M_k = (1)(M - k) / M +(1+M_k)(k/M)

Solving for M_k, we get:

M_k = M / (M - k)

Now the expected life of the option in months, and hence the fair value of the first type of knockout option in T-shirts, can be obtained:

P = S M_k = M(1 + 1/2 + ... + 1/M).

For the second type of knockout option, let M_k denote the expected life of the option—i.e., M_k is the expected number of months before k identical consecutive returns are observed. After the first run of k-1 consecutive identical returns, the option will expire in the next month with probability 1/M, and with probability (1-1/M) one will expect to wait for another M_k months for the option to expire:

M_k = M_k - 1 + 1/M(1) + (1 - 1/M)(M_k)

From this, we get: