You take a short position in one European put option contract, with strike price

Question

You take a short position in one European put option contract, with strike price

100
and maturity six months, on a stock that is trading at
100
. The annual volatility of the stock is constant and equal to
25
%
. The dividend rate is zero. The annual (continuously compounded) risk-free interest rate is constant and equal to
5
%
. Suppose that you sold the option at a premium of
6
%
over the Black-Scholes price, that is, for
1.06
times the Black-Scholes price. You hedge your portfolio with the underlying stock and the risk-free asset. The hedge is rebalanced monthly. After two months the portfolio is liquidated (you buy the option and undo the hedge).

Enter the final overall profit or loss, if the price of the stock is
101
at the end of the first month and
99
at the end of the second month, and assume that the option is traded at exactly the Black-Scholes price at the end of the first month and at the end of the second month:

Answer 1

To determine the final overall profit or loss, we need to calculate the profit/loss from the short position in the European put option, as well as the profit/loss from the hedge using the underlying stock and the risk-free asset.

Here are the steps to calculate the final overall profit or loss:

1. Calculate the Black-Scholes price of the European put option at the beginning of the first month.
- Use the Black-Scholes formula to calculate the option price.
- The formula is:
𝑃 = 𝑆𝑡 × 𝑁(−𝑑₁) − 𝑋𝑒^(−𝑟𝑡) × 𝑁(−𝑑₂)
Where:
𝑃 = Option price
𝑆𝑡 = Stock price
𝑋 = Strike price
𝑟 = Risk-free interest rate
𝑡 = Time to maturity
𝑑₁ = (ln(𝑆𝑡/𝑋) + (𝑟 + 𝜎²/2)𝑡) / (𝜎 √𝑡)
𝑑₂ = 𝑑₁ - 𝜎 √𝑡
𝜎 = Volatility

2. Calculate the premium you received for selling the option at 1.06 times the Black-Scholes price.
- Multiply the Black-Scholes price by 1.06.

3. Calculate the change in the value of the option over the first month.
- The change in the option value is given by the difference between the Black-Scholes price at the beginning of the first month and the Black-Scholes price at the end of the first month.

4. Calculate the change in the value of the stock over the first month.
- The change in the stock value is given by the difference between the stock price at the end of the first month and the stock price at the beginning of the first month.

5. Calculate the change in the value of the option over the second month.
- The change in the option value is given by the difference between the Black-Scholes price at the end of the first month and the Black-Scholes price at the end of the second month.

6. Calculate the change in the value of the stock over the second month.
- The change in the stock value is given by the difference between the stock price at the end of the second month and the stock price at the end of the first month.

7. Calculate the overall profit or loss from the short position in the option.
- The profit or loss is given by the premium received minus the change in the option value over the first and second months.

8. Calculate the profit or loss from the hedge using the underlying stock and the risk-free asset.
- The profit or loss is given by the change in the value of the stock over the first and second months plus the interest earned on the risk-free asset.

9. Calculate the final overall profit or loss.
- The final overall profit or loss is the sum of the profit or loss from the short position in the option and the profit or loss from the hedge.

Note: The calculations involve complex formulas and assumptions. It is recommended to use a financial calculator or a specialized software to perform these calculations accurately.