4. The opening values of Amazon's shares (AMZN) recorded at the beginning of each month from...

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4. The opening values of Amazon's shares (AMZN) recorded at the beginning of each month from February 2020 to January 2021 are: Feb $2,010.60|| Jun $2,448.00 || Oct $3,208.00 Mar $1,906.49 Jul $2,757.99 Nov $3,061.74 Apr $1,932.97 Aug $3,180.51 Dec $3,188.50 May $2,336.80 Sep $3,489.58 Jan $3,270.00 (i) Compute their monthly logarithmic returns. Hence based on such histor- ical data, estimate the monthly drift and volatility of the AMZN stock. (ii) Using the data computed in (i), model the future AMZN share values with the N-step log-normal random walk model S(t) = S(0) elt+aWN(t) (a) Define what are μ, o, and WN(t) in the above N-step log-normal random walk model. (b) Compute the future values in 2 months of the AMZN stock using first a 2-step model and then a 3-step model. (c) Assume that the risk-free asset's interest rate compounded continu- ously is r = 1% per year. Using the future AMZN stock values ob- tained with the 2-step model, find the premium and the values at each time step of a European put option having strike price X = $3,300, expiry time in 2 months and whose underlying asset is the AMZN stock. (d) Compute the expected value of the AMZN stock values at time t > 0 for the log-normal random walk with N steps and in the limit N → ∞. You may use the resulte-2²/2dz = √2π. 4. The opening values of Amazon's shares (AMZN) recorded at the beginning of each month from February 2020 to January 2021 are: Feb $2,010.60|| Jun $2,448.00 || Oct $3,208.00 Mar $1,906.49 Jul $2,757.99 Nov $3,061.74 Apr $1,932.97 Aug $3,180.51 Dec $3,188.50 May $2,336.80 Sep $3,489.58 Jan $3,270.00 (i) Compute their monthly logarithmic returns. Hence based on such histor- ical data, estimate the monthly drift and volatility of the AMZN stock. (ii) Using the data computed in (i), model the future AMZN share values with the N-step log-normal random walk model S(t) = S(0) elt+aWN(t) (a) Define what are μ, o, and WN(t) in the above N-step log-normal random walk model. (b) Compute the future values in 2 months of the AMZN stock using first a 2-step model and then a 3-step model. (c) Assume that the risk-free asset's interest rate compounded continu- ously is r = 1% per year. Using the future AMZN stock values ob- tained with the 2-step model, find the premium and the values at each time step of a European put option having strike price X = $3,300, expiry time in 2 months and whose underlying asset is the AMZN stock. (d) Compute the expected value of the AMZN stock values at time t > 0 for the log-normal random walk with N steps and in the limit N → ∞. You may use the resulte-2²/2dz = √2π.

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Solution To compute the monthly logarithmic returns we can use the formula Logarithmic Return lnStSt1 where St is the closing price at the end of the ... View the full answer