Options Trading: Payoffs, Strategies, and Valuation

1. Option Payoffs:

• Call Option: (S_T − K)⁺ = max(S_T − K, 0)
• Put Option: (K − S_T)⁺ = max(K − S_T, 0)
• Call Binary Option: 1 or Q$ when S_T > K, 0 otherwise.

2. Profit and Loss (P&L) Diagrams:

• An asset.
• A Future.
• Call option.
• Put option.
• A Binary option.
• An asset (current value 50 euros) and a Put option on the same asset with Strike 30. This strategy is used to cover potential losses, limiting them to 20 euros plus the option fee.
• Long Call + Put Option with the same Strike and Maturity. This strategy is used when a significant price movement is expected, but the direction is uncertain.
• Long position in a Call option with Strike 50 and Short Position in a Call option with Strike 60. Selling the call option reduces the overall cost of the strategy, but limits potential gains.

3. Option Moneyness:

Company XY shares are at 25 euros.

4. Call Option with Strike 50 and 1 year to Maturity:

• Out of the money. The intrinsic value is zero.

5. Put Option with Strike 50 and 1 year to Maturity:

• In the money. The intrinsic value is 25.

6. Option Pricing and Volatility:

Shares A and B have the same current value. Volatility is 20% and 30%, respectively. Both do not pay dividends. For Call Options with 1 year to maturity:

• Option B will be more expensive than option A. Higher volatility leads to a higher option price.

7. Call-Put Parity and Interest Rate:

S₀ = 100. A European call option with K = 100 and T = 1/2 is priced at 5, while a European put option with the same K and maturity is priced at 3. Volatility is 10%. The continuously compounded interest rate is:

Using call-put parity: 5 − 3 = 100 − 100e^−r/2. Therefore, r = 4.04%.

8. Call-Put Parity and Put Price:

Non-dividend paying stock: S₀ = 100, r = 1% (continuously compounded), T = 1, and K = 100. If the call option price is 10, the corresponding put price is:

Using call-put parity: 10 − p = 100 − 100e^−0.01. Therefore, p = 9.