Binomial Model

Question 1

Values of S and c for one period

Answer 1

Question 2

u & d

Answer 2

• S⁺/S
• S^-/S

Question 3

n & H

Answer 3

• n is the number of unit of the underlying per option sold

Question 4

Suppose the underlying is a non-dividend-paying stock currently valued at $50. It can either go up by 25 percent or go down by 20 percent. Thus, u = 1.25 and d = 0.80

• S⁺ = Su = 50(1.25) = 62.50
• S^– = Sd = 50(0.80) = 40

Assume that the call option has an exercise price of 50 and the risk-free rate is 7 percent. Thus, the option values one period later will be

• c⁺ = Max(0,S⁺ – X) = Max(0,62.50 – 50) = 12.50
• c^– = Max(0,S^– – X) = Max(0,40 – 50) = 0

Answer 4

First, we calculate π:

• π = (1 + r − d) / (u − d) = (1.07 − 0.80) / (1.25 − 0.80) = 0.6
• and, hence, 1 − π = 0.4

Now, we can directly calculate the option price:

• c = [0.6 (12.50) + 0.4 (0)] / 1.07 = 7.01

Thus, the option should sell for $7.01

Question 5

Suppose the option is selling for $8. If the option should be selling for $7.01 and it is selling for $8, it is overpriced. Investors would exploit this opportunity by selling the option and buying the underlying. The number of units of the underlying purchased for each option sold would be the value n:

• n = (c⁺−c⁻) / (S⁺−S⁻) = (12.50 − 0) / (62.50 − 40) = 0.556

Answer 5

Thus, for every option sold, we would buy 0.556 units of the underlying. Suppose we sell 1,000 calls and buy 556 units of the underlying. Doing so would require an initial outlay of H = 556 ($50) − 1,000 ($8) = $19,800. One period later, the portfolio value will be either

• H⁺ = nS⁺ – c⁺ = 556 ($62.60) – 1,000 ($12.50) = $22,250, or
• H^– = nS^– – c^– = 556 ($40) – 1,000($0) = $22,240

Question 6

These two values are not exactly the same, but the difference is due only to rounding the hedge ratio, n. We shall use the $22,250 value. If we invest $19,800 and end up with $22,250, the return is

• $22,250 / $19,800 − 1 = 0.1237

that is, a risk-free return of more than 12 percent in contrast to the actual risk-free rate of 7 percent. Thus, we could borrow $19,800 at 7 percent to finance the initial net cash outflow, capturing a risk-free profit of (0.1237 – 0.07) × $19,800 = $1,063 (to the nearest dollar) without any net investment of money. Other investors will recognize this opportunity and begin selling the option, which will drive down its price. When the option sells for $7.01, the initial outlay would be

• H = 556 ($50) − 1,000 ($7.01) = $20,790

Answer 6

The payoffs at expiration would still be $22,250. This transaction would generate a return of

• $22,250 / $20,790 − 1 ≈ 0.07

Thus, when the option is trading at the price given by the model, a hedge portfolio would earn the risk-free rate, which is appropriate because the portfolio would be risk-free.

If the option sells for less than $7.01, investors would buy the option and sell short the underlying, which would generate cash up front. At expiration, the investor would have to pay back an amount less than 7 percent. All investors would perform this transaction, generating a demand for the option that would push its price back up to $7.01

Question 7

Values of S and c for two period

Answer 7

Question 8

• S^{+ +} = S⁺u = Suu = Su²
• S^{+ –} = S⁺d = Sud

Answer 8

• S^{– +} = S^–u = Sdd = Sd²
• S^{– –} = S^–d = Sdu

Question 9

Answer 9