You are attempting to value a put option with an exercise price of $150 and one year to expiration. The underlying stock pays no dividends, its current price is $150, and you believe it has a 50% chan

Question

You are attempting to value a put option with an exercise price of $150 and one year to expiration. The underlying stock pays no dividends, its current price is $150, and you believe it has a 50% chance of increasing to $200 and a 50% chance of decreasing to $100. The risk-free rate of interest is 10%.
a. What will be the payoff to the put, Pu, if the stock goes up?
b. What will be the payoff, Po, if the stock price falls?
c. What is the value of the put using the risk-neutral shortcut? (Do not round intermediate calculations. Round your answer to 3 decimal places.)
expected value=?

VIPSolutions · Accepted Answer

# Valuing a Put Option: Binomial Model

Given:

- **Exercise price (K):** $  
- **Stock price now (S₀):** $150  
- **Up state (Sᵤ):** $200  
- **Down state (S_d):** $100  
- **Probability (real-world):** 50% up, 50% down  
- **Risk-free rate (r):** 10% (use as .10, continuous compounding not specified)  
- **Time to expiration (T):** 1 year  
- **Dividends:** None

---

## a. Put Payoff if Stock Goes Up ($ P_u $)

The payoff for a put option is $\max(K - S, )$:

- If the stock goes **up to $200**:
  $$
  P_u = \max(150 - 200, ) = 
  $$

---

## b. Put Payoff if Stock Goes Down ($ P_d $)

- If the stock goes **down to $100**:
  $$
  P_d = \max(150 - 100, ) = 50
  $$

---

## c. Risk-Neutral Valuation of the Put

### 1. Calculate Risk-Neutral Probability ($ p $):

$$
p = \frac{e^{rT} S_ - S_d}{S_u - S_d}
$$

Since $ r = .10 $ and $ T = 1 $:

$$
e^{rT} = e^{.10} \approx 1.10517
$$
$$
p = \frac{1.10517 \times 150 - 100}{200 - 100} = \frac{165.7755 - 100}{100} = \frac{65.7755}{100} = .657755
$$

### 2. Expected Payoff (risk-neutral):

$$
\text{Expected Payoff} = p \times P_u + (1 - p) \times P_d
$$
$$
= .657755 \times  + (1 - .657755) \times 50
$$
$$
=  + .342245 \times 50 = 17.11225
$$

### 3. Discount Expected Payoff at Risk-Free Rate:

$$
\text{Present Value} = \frac{\text{Expected Payoff}}{e^{rT}}
$$
$$
= \frac{17.11225}{1.10517} = 15.486
$$

---

## Summary Table

| Outcome        | Stock Price | Put Payoff |
|----------------|-------------|------------|
| Stock goes up  | $200        | $         |
| Stock goes down| $100        | $50        |

---

## Answers

**a.** $ P_u =  $

**b.** $ P_d = 50 $

**c.** **Put Value (risk-neutral):**
$$
\boxed{15.486}
$$

(Rounded to 3 decimal places.)

---

**Expected value:** $ 15.486 $

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