A financial call option provides the holder the right to purchase an asset (e.g. a stock)...

 

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A financial call option provides the holder the right to purchase an asset (e.g. a stock) at a fixed price called the strike price, w, over a fixed number of days, N. Each day, if the stock price, p, is above w then the holder can choose to exercise the option (where exercise means that they purchase and immediately sell the stock) and get a return of p-w or keep the option. The option can only be exercised once and must be exercised within the N days. If the stock price is below w then there is no value to exercising the option. Each day the price increases by u% with probability q and decreases by d% with probability 1-q. Assume the objective is to maximize expected return from the option. a) Formulate a dynamic program to determine the optimal exercise policy, stating any assumptions you make in your model. b) Prove there are non-increasing numbers p P PN such that at day n the option should be exercised if and only if p Pn. c) Suppose that today you have an option to purchase IBM stock at $50 per share. The stock price is currently $50, and each day increases by 5% with probability 0.55 and decreases by 10% with probability 0.45. You may exercise on any of the next 5 days. i. ii. What is the optimal exercise policy? If someone offers to purchase the option from you today, what is the minimum purchase price you would sell it for if your goal is to maximize expected return? A financial call option provides the holder the right to purchase an asset (e.g. a stock) at a fixed price called the strike price, w, over a fixed number of days, N. Each day, if the stock price, p, is above w then the holder can choose to exercise the option (where exercise means that they purchase and immediately sell the stock) and get a return of p-w or keep the option. The option can only be exercised once and must be exercised within the N days. If the stock price is below w then there is no value to exercising the option. Each day the price increases by u% with probability q and decreases by d% with probability 1-q. Assume the objective is to maximize expected return from the option. a) Formulate a dynamic program to determine the optimal exercise policy, stating any assumptions you make in your model. b) Prove there are non-increasing numbers p P PN such that at day n the option should be exercised if and only if p Pn. c) Suppose that today you have an option to purchase IBM stock at $50 per share. The stock price is currently $50, and each day increases by 5% with probability 0.55 and decreases by 10% with probability 0.45. You may exercise on any of the next 5 days. i. ii. What is the optimal exercise policy? If someone offers to purchase the option from you today, what is the minimum purchase price you would sell it for if your goal is to maximize expected return?