R Lab (Financial Analytics with R)

I only need help with number #4 (files attach should help)

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1. For the following questions, assume a stock is priced according to a geometric Brownian motion process

Δ = Δ + √Δ

where = 0.10, = 0.15, Δ = 1/52, and is a random draw from a standard normal pdf. Additionally, assume that the initial stock value, 0 = 80 and that the derivatives that follow expire in 26 weeks ( = 1/2). Also, assume that implied volatility every week is 0.30 and the risk-free interest rate is equal to 0.02.

  1. Simulate a single price trajectory using the information above.Note: Make sure to set a seed in simulations to ensure consistency through repeatedsimulations.
  2. Compute the implied call option premium for every week, with K=85, using theinformation from above and simulated values from part (a).
  3. Assume you are looking to sell 10 European call options at t=0 and opt to use a deltahedge throughout the life of the option.
    1. Derive the delta associated with each week.
    2. Derive the number of long futures contract you would use to hedge each week.
  4. Simulate 1,000 price trajectories using the information above. Note: Make sure to set a seed in simulations to ensure consistency through repeated simulations.
    1. Using the simulated outcomes from above, compute the associated returns from a type of Asian put and an Asian call option where the payment is equal to the difference between the average price of the last 5 weeks of the stock and the strike, K.
    2. Using the simulated outcomes from above, compute the associated returns from an all-or-nothing option that pays $100 when ≥ 95 at any time during the life of the contract.
    3. Using the simulated outcomes from above, compute the associated returns from a floating call price, where K=85 and the price is the maximum of all prices within the life of the contract.
    4. Using the simulated outcomes from above, compute the associated returns from a floating put price, where K=85 and the price is the minimum of all prices within the life of the contract.