The question you've asked is about the Cox-Ross-Rubinstein (CRR) model, which is a method used to price European and American options. Here's how you can solve it:

(a) The parameters u and d are calculated as follows:

u = Su1/S0 = 120/100 = 1.2
d = Sd1/S0 = 90/100 = 0.9

The stock price at time T=2 can be calculated by multiplying the initial stock price by the up and down factors. Since there are two periods, there are three possible stock prices at time T=2: Su2 = Su1 * u = 120 * 1.2 = 144, Sd2 = Sd1 * d = 90 * 0.9 = 81, and Sdu2 = Su1 * d = 120 * 0.9 = 108.

The martingale measure ˜P is a probability measure under which the price process of a financial asset or derivative is a martingale. Since the interest rate r = 0, the risk-neutral probability measure is ˜p = (1 - d) / (u - d) = (1 - 0.9) / (1.2 - 0.9) = 1/3.

(b) The price process P a of the option can be computed using the recursive relationship given in the question. The terminal condition is P a2 = (L2 - S2)+.

P a2 = max{(L2 - S2)+, 0} = max{(111 - 144)+, (111 - 108)+, (111 - 81)+, 0} = max{-33, 3, 30, 0} = 0, 3, 30.

Then, we can calculate P a1:

P au1 = max{(L1 - Su1)+, (1 + r)−1(˜pP au2 + (1 − ˜p)P adu2)} = max{(116 - 120)+, (1/3 * 0 + 2/3 * 3)} = max{-4, 2} = 2
P ad1 = max{(L1 - Sd1)+, (1 + r)−1(˜pP adu2 + (1 − ˜p)P ad2)} = max{(116 - 90)+, (1/3 * 3 + 2/3 * 30)} = max{26, 21} = 26

Finally, we can calculate P a0:

P a0 = max{(L0 - S0)+, (1 + r)−1(˜pP au1 + (1 − ˜p)P ad1)} = max{(105 - 100)+, (1/3 * 2 + 2/3 * 26)} = max{5, 18} = 18

(c) The holder’s rational exercise time τ ∗0 is the first time t at which the holder should exercise the option, i.e., the first time t at which the payoff from exercising the option is greater than the expected payoff from continuing to hold the option. In this case, τ ∗0 = 1, because at time t = 1, the payoff from exercising the option (26 or 2) is greater than the expected payoff from continuing to hold the option (18).

Question

The question you've asked is about the Cox-Ross-Rubinstein (CRR) model, which is a method used to price European and American options. Here's how you can solve it:

(a) The parameters u and d are calculated as follows:

u = Su1/S0 = 120/100 = 1.2
d = Sd1/S0 = 90/100 = 0.9

The stock price at time T=2 can be calculated by multiplying the initial stock price by the up and down factors. Since there are two periods, there are three possible stock prices at time T=2: Su2 = Su1 * u = 120 * 1.2 = 144, Sd2 = Sd1 * d = 90 * 0.9 = 81, and Sdu2 = Su1 * d = 120 * 0.9 = 108.

The martingale measure ˜P is a probability measure under which the price process of a financial asset or derivative is a martingale. Since the interest rate r = 0, the risk-neutral probability measure is ˜p = (1 - d) / (u - d) = (1 - 0.9) / (1.2 - 0.9) = 1/3.

(b) The price process P a of the option can be computed using the recursive relationship given in the question. The terminal condition is P a2 = (L2 - S2)+.

P a2 = max{(L2 - S2)+, 0} = max{(111 - 144)+, (111 - 108)+, (111 - 81)+, 0} = max{-33, 3, 30, 0} = 0, 3, 30.

Then, we can calculate P a1:

P au1 = max{(L1 - Su1)+, (1 + r)−1(˜pP au2 + (1 − ˜p)P adu2)} = max{(116 - 120)+, (1/3 * 0 + 2/3 * 3)} = max{-4, 2} = 2
P ad1 = max{(L1 - Sd1)+, (1 + r)−1(˜pP adu2 + (1 − ˜p)P ad2)} = max{(116 - 90)+, (1/3 * 3 + 2/3 * 30)} = max{26, 21} = 26

Finally, we can calculate P a0:

P a0 = max{(L0 - S0)+, (1 + r)−1(˜pP au1 + (1 − ˜p)P ad1)} = max{(105 - 100)+, (1/3 * 2 + 2/3 * 26)} = max{5, 18} = 18

(c) The holder’s rational exercise time τ ∗0 is the first time t at which the holder should exercise the option, i.e., the first time t at which the payoff from exercising the option is greater than the expected payoff from continuing to hold the option. In this case, τ ∗0 = 1, because at time t = 1, the payoff from exercising the option (26 or 2) is greater than the expected payoff from continuing to hold the option (18).

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