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
To determine the final overall profit or loss, we need to go through the process step by step and calculate the values at the end of each month.
1. Find the option price at the end of the first month using the Black-Scholes formula:
- Current stock price = 101
- Strike price = 100
- Time to maturity = 4/12 (as the hedge is rebalanced monthly)
- Volatility = 25%
- Risk-free interest rate = 5%
Using these values in the Black-Scholes formula, you can calculate the option price at the end of the first month.
2. Determine the profit or loss from the stock position at the end of the first month:
- You initially shorted the stock at 100, and the stock price is now 101. As a short position, this results in a loss.
3. Calculate the total value of the portfolio at the end of the first month by summing the option price and the value of the stock position.
4. Rebalance the hedge by adjusting the stock and risk-free asset quantities according to the portfolio delta. This ensures that the portfolio remains delta-neutral.
5. Repeat steps 1-4 for the end of the second month using the updated parameters and stock price of 99.
6. Calculate the overall profit or loss by subtracting the initial premium (6% over the Black-Scholes price) from the final portfolio value.
By following these steps, you can determine the final overall profit or loss in this scenario.