BF2209: In Week 0, you sell 10,000 European put options

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1. In Week 0, you sell 10,000 European put options on the stock with a strike price of $55 and time to expiration of 20 weeks for $33,000. The price of the underlying stock is $54 in week 0.

(a) Suppose that r = 2%, σ = 20%, δ = 1%, and µ = 8%. They will be constant over the next 20 weeks. Assume that continuously compounded return of the stock for one week is normally distributed with a mean of (µ − δ −σ22) ×152 and standard deviation of σ q 1 52. Generate a path of weekly stock prices for the next 20 weeks.

(b) Compute the BSM price of the put option. How much can you make an arbitrage profit?

(c) Compute Delta of the put option in each week.

(d) You want to delta hedge your portfolio. How many shares of the stock do you need to purchase each week? How much does it cost?
(e) In week 0, you purchase a zero-coupon bond with a face value of $10,000 and a time to maturity of 20 weeks to realize the arbitrage profit in (b) upfront. That is, the initial cost of the delta hedging portfolio should be the negative of the arbitrage profit in (b). How many units of ZCB should be purchased in week 0?

(f) In the following weeks, the purchase of shares is funded by the proceeds from selling the ZCB. If shares are sold, the proceeds will be used to buy more units of the ZCB. That is, the self-financing delta-hedging portfolio is constructed. Compute the total number of units of the ZCB in the portfolio each week.

(g) Compute the value of ZCB holdings, stock holdings, and written put options. For the value of written put options, use the BSM formula. Finally, compute the net portfolio value and the weekly change in the portfolio value. What is the theoretical portfolio value in week 20? Why is the actual portfolio value in week 20 different from the theoretical value?.

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