Consider a call price on the S&P 500 bought on March 23 expiring in one month on

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Question:

Consider a call price on the S&P 500 bought on March 23 expiring in one month on April 23rd (T = 1/12 years) that is struck at K = $2, 200 (i.e. the call is at the money.) On April 23rd, the S&P 500 closed at around $2800. If you bought the call in March, what was the payoff when it expired?

b) What would your return (in percent) be if you bought the S&P 500 at $2,200 and sold April 23rd at $2,800?

c) On March 23, 2020, the implied VIX was at 61.6%. Using the provided Excel sheet, find the Black-Scholes model price of the option in (a). In Black Scholes formula, use a volatility of σ = 61.6% and take r = 0.50% as the risk-free rate.

d) Suppose that you bought the at-the-money call for the price in (c). What would your return be
when it expired on April 23rd?

e) Suppose now that you bought a call struck at K = $2500 on March 23, 2020. Repeat the previous part to find the your return 1 month later.