Act like a helpful tutor and exlain me :Problem 12

Draw the profit and payoff diagrams for a cap, where you purchase a stock for $100, and short
(sold) a call option, with strike K = 115, and call

Question

Act like a helpful tutor and exlain me :Problem 12

Draw the profit and payoff diagrams for a cap, where you purchase a stock for $100, and short
(sold) a call option, with strike K = 115, and call premium C' = 12%. Assume the continuously
compounded risk-free interest rate is r = 5%.

VIPSolutions · Accepted Answer

## Problem 12 Explanation

You are asked to **draw the profit and payoff diagrams** for a "cap" strategy, which consists of:
- **Buying a stock at $100**
- **Selling (shorting) a call option** with:  
  - Strike price $ K = 115 $  
  - Call premium $ C = 12e^{-.05} $  
- The risk-free rate $ r = 5\% $ (for discounting)

Let's break down each component and then combine them.

---

### 1. **Payoff of Each Position at Expiry**

#### **A. Stock Position**
- **Payoff:** $ S_T $ (where $ S_T $ is the stock price at expiry)

#### **B. Short Call Option**
- **Payoff:** You sold the call, so your payoff is the negative of the call's payoff:
  - Call payoff: $ \max(, S_T - K) $
  - Your payoff: $ -\max(, S_T - K) $

#### **C. Call Premium Received**
- You receive $ C = 12e^{-.05} $ at time .  
  To make the profit diagram at expiry, you should **compound this forward** to expiry:
  $$
  C_{\text{future}} = 12e^{-.05} \cdot e^{.05} = 12
  $$
  So, at expiry, you keep the full $12.

---

### 2. **Total Payoff and Profit Formulas**

#### **A. Total Payoff (ignoring initial costs)**
$$
\text{Total Payoff} = S_T - \max(, S_T - 115)
$$

- For $ S_T \leq 115 $: $ \text{Payoff} = S_T $
- For $ S_T > 115 $: $ \text{Payoff} = S_T - (S_T - 115) = 115 $

So, **the cap "flattens out" above $115**.

#### **B. Total Profit**
$$
\text{Profit} = \text{Payoff} + \text{Call Premium} - \text{Initial Stock Cost}
$$
$$
\text{Profit} = (\text{Payoff as above}) + 12 - 100
$$

---

### 3. **Summary Table**

| $ S_T $ Range        | Payoff         | Profit                |
|------------------------|----------------|-----------------------|
| $ S_T \leq 115 $     | $ S_T $      | $ S_T + 12 - 100 $  |
| $ S_T > 115 $        | 115            | $ 115 + 12 - 100 = 27 $ |

---

### 4. **Diagrams**

#### **A. Payoff Diagram**
- **X-axis:** Stock price at expiry $ S_T $
- **Y-axis:** Payoff
- **Line:**  
  - For $ S_T \leq 115 $: Line with slope 1, starts at  when $ S_T= $, reaches 115 at $ S_T=115 $.
  - For $ S_T > 115 $: Flat at 115.

#### **B. Profit Diagram**
- **X-axis:** Stock price at expiry $ S_T $
- **Y-axis:** Profit
- **Line:**  
  - For $ S_T \leq 115 $: Line with slope 1, intercepts Y-axis at $ -88 $ (since $  + 12 - 100 = -88 $), reaches 27 at $ S_T=115 $.
  - For $ S_T > 115 $: Flat at 27.

#### **Profit Diagram Sketch:**
```
Profit ($)
  ^
  |
27|---------------------------- (flat for S_T > 115)
  |                   /
  |                /
  |             /
-88|---------/
  +-----------------------------> S_T
             100    115
```

---

## **Summary**

- The **cap strategy** gives unlimited profit up to $ S_T = 115 $ and then flatlines at $27 profit.
- **Payoff**: Flat at $115 for $ S_T > 115 $, linear below.
- **Profit**: Flat at $27 for $ S_T > 115 $, linear below.

---

**Let me know if you'd like a matplotlib code or further explanation!**

✨ VIPSolutions

Act like a helpful tutor and exlain me :Problem 12 Draw the profit and payoff diagrams for a cap, where you purchase a stock for $100, and short (sold) a call option, with strike K = 115, and call premium C' = 12%. Assume the continuously compounded risk-free interest rate is r = 5%.

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