A stock is currently trading at $100. In each month, the stock will either increase in price by a factor of $u = 1.05$ or fall by a factor of $d = 0.90$ . The risk-free rate of interest per month is 0.1668% in simple terms, i.e., an investment of $1 at the risk-free rate returns $1.001668 after one month. What is the price of a 100-strike, two-month European put option?
A) $2.36
B) $3.36
C) $4.36
D) $5.36

Question

Quizplus · Accepted Answer

To price a European put option, we can use the binomial option pricing model. The model involves calculating the expected payoff of the option at expiration and then discounting it back to the present value using the risk-free rate.1. Calculate the stock price in two months:   - If the stock goes up twice: $S_{uu} = 100 \times 1.05 \times 1.05 = 110.25$   - If the stock goes up then down or vice versa (it doesn't matter the order for this calculation): $S_{ud} = 100 \times 1.05 \times 0.90 = 94.5$   - If the stock goes down twice: $S_{dd} = 100 \times 0.90 \times 0.90 = 81$2. Calculate the payoff of the put option at expiration:   - If the stock goes up twice: $P_{uu} = \max(100 - S_{uu}, 0) = \max(100 - 110.25, 0) = 0$   - If the stock goes up then down or vice versa: $P_{ud} = \max(100 - S_{ud}, 0) = \max(100 - 94.5, 0) = 5.5$   - If the stock goes down twice: $P_{dd} = \max(100 - S_{dd}, 0) = \max(100 - 81, 0) = 19$3. Calculate the risk-neutral probabilities:   - The risk-neutral probability of an up move, $p$, is given by $(e^{r\Delta t} - d) / (u - d)$, where $r$ is the risk-free rate, and $\Delta t$ is the time step (1 month in this case).   - $p = (e^{0.001668} - 0.90) / (1.05 - 0.90) = (1.001668 - 0.90) / (0.15) \approx 0.6778$   - The probability of a down move is $1 - p = 0.3222$.4. Calculate the expected payoff:   - Expected payoff = $p^2 \times P_{uu} + 2p(1-p) \times P_{ud} + (1-p)^2 \times P_{dd}$   - Expected payoff = $0.6778^2 \times 0 + 2 \times 0.6778 \times 0.3222 \times 5.5 + 0.3222^2 \times 19$   - Expected payoff = $0 + 2 \times 0.6778 \times 0.3222 \times 5.5 + 0.3222^2 \times 19 \approx 4.36$5. Discount the expected payoff back to the present value using the risk-free rate:   - The present value of the expected payoff = $4.36 / (1 + 0.001668)^2 \approx 4.36$ (since the risk-free rate is very small, the discounting effect is minimal over two months).Therefore, the price of the 100-strike, two-month European put option is approximately $4.36.

A Stock Is Currently Trading at $100 u=1.05u = 1.05u=1.05 Or Fall by a Factor Of

A Stock Is Currently Trading at $100 $u = 1.05$ Or Fall by a Factor Of