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2. Suppose a stock is trading at $50. Further suppose that we have the follow- ing information on the Black-Scholes prices of three European call options with 3 months to expiration: K T 45 50 55 6.95 4.10 2.22 0.74 0.55 0.0323 0.0396 0.0373 0.36 (a) Write down the equations which one needs to solve to make a zero- cost, Delta-neutral portfolio with a convexity of 0.05 using the three types of calls. Note: You do not need to solve these equations. (b) What is the sign of the portfolio's Theta? Please briefly discuss the intuition behind your answer. (c) If the stocks volatility is 40%, and the continuously compounded risk- free rate is 2% annualized, what is each of these call options' thetas ? Please calculate these using the Black-Scholes PDE (the equation that prices a derivative as a "portfolio of Greeks). (d) If you bought 1,000 butterflies using these options, how much would you make or lose in one day if the underlyi didn't change? What if the underlying went down $1, or up $2? 2. Suppose a stock is trading at $50. Further suppose that we have the follow- ing information on the Black-Scholes prices of three European call options with 3 months to expiration: K T 45 50 55 6.95 4.10 2.22 0.74 0.55 0.0323 0.0396 0.0373 0.36 (a) Write down the equations which one needs to solve to make a zero- cost, Delta-neutral portfolio with a convexity of 0.05 using the three types of calls. Note: You do not need to solve these equations. (b) What is the sign of the portfolio's Theta? Please briefly discuss the intuition behind your answer. (c) If the stocks volatility is 40%, and the continuously compounded risk- free rate is 2% annualized, what is each of these call options' thetas ? Please calculate these using the Black-Scholes PDE (the equation that prices a derivative as a "portfolio of Greeks). (d) If you bought 1,000 butterflies using these options, how much would you make or lose in one day if the underlyi didn't change? What if the underlying went down $1, or up $2

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