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 calculate the final overall profit or loss, we need to go through a series of steps. Let's break it down:
Step 1: Calculate the initial premium received:
The premium received is 1.06 times the Black-Scholes price of the European put option. However, we need to calculate the Black-Scholes price first.
Step 2: Calculate the Black-Scholes price of the European put option:
The Black-Scholes formula for a European put option is given by:
Put Price = X * e^(-rT) * N(-d2) - S * N(-d1)
Where:
- S is the stock price
- X is the strike price
- r is the risk-free interest rate
- T is the time to maturity
- N() represents the cumulative standard normal distribution function
- d1 = (ln(S/X) + (r + (σ^2)/2) * T) / (σ * sqrt(T))
- d2 = d1 - σ * sqrt(T)
Plug in the given values and calculate d1 and d2 using the formula.
Step 3: Calculate the premium received:
Multiply the Black-Scholes price by 1.06 to get the premium received.
Step 4: Calculate the profit or loss after the first month:
In the first month, the stock price increases to 101. Since the hedge is rebalanced monthly, we need to adjust the hedge.
To maintain a delta-neutral position, we need to buy an additional amount of shares to compensate for the increase in the stock price. The amount of shares to buy can be calculated using the delta of the put option.
Delta = -N(-d1)
Delta represents the sensitivity of the option price to changes in the stock price. Multiply Delta by the stock price increase (101-100) to get the number of shares to buy.
Step 5: Calculate the profit or loss after the second month:
In the second month, the stock price decreases to 99. Again, we need to adjust the hedge.
Use the same process as in Step 4 to calculate the number of shares to buy or sell to maintain a delta-neutral position.
Step 6: Calculate the final overall profit or loss:
To calculate the final overall profit or loss, we need to consider the changes in the stock price, the premium received, and the adjustments made during the hedge.
Subtract the premium received from the sum of the changes in the stock price (101-100, 99-101) and the adjustments made during the hedge.
The resulting value will give you the final overall profit or loss.
Please note that these steps provide the general approach to solving the problem, and the exact calculations may require additional mathematical inputs and formulas.