Assume the Black-Scholes framework. You are given:

(i) The current price of a stock is $50.

(ii) The stock pays no dividends.

(iii) The stock’s volatility is 25%.

(iv) The continuously compounded risk-free interest rate is 5%.

Suppose you have just sold 1,000 1-year 50-strike European call options.

(a) Determine the numbers of units of a 1-year 60-strike European call option and the stock you should buy or sell in order to both delta-hedge and gamma-hedge your position in the 50-strike European calls.

(b) You are further given:

(v) The original (i.e., time-0) prices of the 1-year 50-strike call and 1-year 60-strike call are 6.1680 and 2.5127, respectively.

(vi) If the one-month stock price remains unchanged at $50, then the one-month prices of the 50-strike call and 60-strike call (both of which will expire in 11 months) are 5.8611 and 2.2591, respectively.

Calculate your profit after one month if the delta-gamma-hedging strategy in part (a) is implemented and the one-month stock price remains unchanged at $50.