To solve this, we use a forward binomial tree to price the put option. Here are the steps:

Step 1: Define the variables

• Current stock price, $S_0 = 120$
• Strike price, $K = 122$
• Volatility, $\sigma = 0.2$
• Risk-free rate, $r = 0.05$
• Time to maturity, $T = 0.25$ (three months)
• Number of steps in the binomial tree: $n$ (we will assume 2 steps for simplicity)

Step 2: Determine the parameters of the binomial tree

Up and down factors:

The up ($u$) and down ($d$) factors are calculated as: $u = e^{\sigma \sqrt{\Delta t}}, \quad d = \frac{1}{u}$ where $\Delta t = \frac{T}{n}$.

Risk-neutral probabilities:

The risk-neutral probability $p$ is given by: $p = \frac{e^{r \Delta t} - d}{u - d}$

Step 3: Calculate the option values

Stock prices at maturity:

At each node in the binomial tree, calculate the possible stock prices at maturity. Then calculate the option payoff at each node as: $\text{Payoff} = \max(K - S, 0)$

Discount back:

Use risk-neutral pricing to discount the option values back to the present.

Let’s compute these step-by-step.

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Step 1: Assumptions

Let $n = 2$, so $\Delta t = 0.25 / 2 = 0.125$.

Up and down factors:

$u = e^{0.2 \sqrt{0.125}} \approx 1.0736, \quad d = \frac{1}{1.0736} \approx 0.9317$

Risk-neutral probabilities:

$p = \frac{e^{0.05 \cdot 0.125} - 0.9317}{1.0736 - 0.9317} \approx 0.5438$

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Step 2: Stock prices at maturity

Node prices:

Starting from $S_0 = 120$:

• At time $t_1$: $S_0 u$ and $S_0 d$.
• At time $t_2$: $S_0 u^2$, $S_0 ud$, $S_0 d^2$.

Prices:

• $S_{uu} = 120 \cdot 1.0736^2 \approx 138.41$
• $S_{ud} = 120 \cdot 1.0736 \cdot 0.9317 \approx 120.00$
• $S_{dd} = 120 \cdot 0.9317^2 \approx 103.01$

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Step 3: Option payoff

Terminal payoffs:

At $t_2$, the put option payoffs are:

• $P_{uu} = \max(122 - 138.41, 0) = 0$
• $P_{ud} = \max(122 - 120.00, 0) = 2$
• $P_{dd} = \max(122 - 103.01, 0) = 18.99$

Discounted back:

At $t_1$: $P_u = e^{-r \Delta t} \left( p P_{uu} + (1-p) P_{ud} \right)$ $P_u = e^{-0.05 \cdot 0.125} \left( 0.5438 \cdot 0 + 0.4562 \cdot 2 \right) \approx 0.907$

$P_d = e^{-r \Delta t} \left( p P_{ud} + (1-p) P_{dd} \right)$ $P_d = e^{-0.05 \cdot 0.125} \left( 0.5438 \cdot 2 + 0.4562 \cdot 18.99 \right) \approx 10.05$

At $t_0$: $P_0 = e^{-r \Delta t} \left( p P_u + (1-p) P_d \right)$ $P_0 = e^{-0.05 \cdot 0.125} \left( 0.5438 \cdot 0.907 + 0.4562 \cdot 10.05 \right) \approx 4.37$

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Final Answer:

The price of the $122-strike three-month put option is approximately $4.37.

Let me know if you need further clarifications or computations. Here are five follow-up questions you might consider:

1. How does increasing the number of steps in the binomial tree affect the accuracy of the option price?
2. What is the significance of the risk-neutral probability in option pricing?
3. How would the option price change if the volatility or interest rate were higher?
4. Can this method be extended to multi-period options, such as an American-style put?
5. How does the binomial model compare to the Black-Scholes model in pricing this option?

Tip: Always ensure the number of steps in a binomial tree is sufficiently large for accurate pricing, especially for long-term options.