Suppose that you have $10,000 to invest on the market. You are considering two strategies:

 

(i) Invest in the S&P 500 (say by buying an ETF such as SPY or VTI or by putting it into a mutual fund such as FXAIX.).

(ii) Buying at-the-money S&P 500 options (SPX options).

 

You begin your investment on March 31, 2022 when the S&P 500 was at $4,530. At-the-money options (i.e. with strike K = $4530) are available with a Black-Scholes volatility of = 21% with an expiration of T = 3 months.

 

(a) Using the provided data, find the Black-Scholes model price of the at-the-money 3-month option. Take r = 0:50% as the risk-free rate.

 

b) Suppose that in 3-month, the S&P 500 is at $4900. What is the return for strategy (i)? What is the return for strategy (ii)?

c) Repeat (b) assuming that the S&P 500 ends up at $4700.

d) Repeat (b) assuming that the S&P 500 ends up at $4530.

Transcribed Image Text:

Black Scholes Equation SO K (Option Strike) sigma r (interest rate) T (in years) Call price Put Price d1 d2 put+stock (put-call parity) call +bond (put-call parity) Black Scholes Prices N(D1) N(D2) $41 $40 30% 8.0% 0.25 $3.40 $1.61 $42.61 $42.61 Intermediate Calculations in BS formula 0.373 0.223 64.5% 58.8% Black Scholes Equation SO K (Option Strike) sigma r (interest rate) T (in years) Call price Put Price d1 d2 put+stock (put-call parity) call +bond (put-call parity) Black Scholes Prices N(D1) N(D2) $41 $40 30% 8.0% 0.25 $3.40 $1.61 $42.61 $42.61 Intermediate Calculations in BS formula 0.373 0.223 64.5% 58.8%

Answer rating: 100% (QA)

To calculate the returns for each strategy we need to determine the final value of the investment un... View the full answer