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1. Black-Sholes-Merton Option Model: Let s be the initial stock price, K the strike price, T the maturity time (in years) , r a continuously compounded risk free rate, and σ the volatility of the underlying security. Let c be the value of the European call option and p the value of the put option: where c = sN(d1) − Ke−rT N(d2), p = Ke−rT N(−d2) − sN(−d1) ln(s/K) + (r + 1σ2)T √ ln(s/K) + (r − 1σ2)T d1= √2 ,d2=d1−σT= √2 σT σT (a) Using R, find codes for the equations of the European call and put given above. (b) Assume the current stock price is $40, the strike price for the call is $42, the time to maturity is 6 months (i.e. T=0.5), the risk-free rate is 10% compounded continuously, and the volatility of the underlying stock is 20%. Find the value of the European call. (c) Assume the current price of stock is $38.5, the strike prices for call and put are both $37, the time to maturity is 3 months, the risk-free rate is 3.2% compounded continuously, and the volatility of the underlying stock is .25. What are the prices for the European call and put. (d) Use the put-call parity to verify the above solutions. 2. Simulating stock prices: Assume that the current stock price is $10.25, the mean value in the past 5 years was $9.35 and the standard deviation 4.24. Write a program to generate 1000 future prices. 3. Consider an asymmetric random walk, defined as Sn = ni=1 Xi, where {Xi, i ≥ 1} are independent and identically distributed random variables, with distribution −1 1 1 Xi: q p , p>2. Let σ2 = V ar(Xi) and define Mn = [Sn −(p−q)n]2 −σ2n. Show that {Mn, n ≥ 1} is a martingale with respect to the natural filtration Fn = σ(X1, …, Xn). 4. Consider a stochastic process (similar to Brownian motion) satisfying the following properties: X(0) = 0, it has independent increments, and for s < t, X(t) − X(s) ∼ N(a(t − s), σ2(t − s)). Show that the process {X(t), t ≥ 0} is a Markov process.

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