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3. Assume a stock price follows a lognormal Brownian motion, i.e. its price satisfies So eo)...
Assume an asset price St follows the geometric Brownian motion, dSt = u Stdt+oS+dZt, So =s > 0 where u and o are constants, r is the risk-free rate, and Zt is the Brownian motion. 1. Using the Ito's Lemma find to the stochastic differential equation satisfied by the process X+ = St. 2. Compute E[Xt] and Var[Xt]. 3. Using the Ito's Lemma find the stochastic differential equation satisfied by the process Y1 = Sert'. 4. Compute E[Y] and Var[Y].
PROBLEM 2. [5 points] Let X(t) be a Brownian motion. Assume that the stock price follows the stochastic differential equation dS ơSdX+1Sdt. what stochastic differential equation does the stochastic process (a) Y 25, (b) Y = S (c) Y-es, (d) YeT-/S follow? In each cases express the coefficients of dX and dt in terms of Y rather than S. Use Ito's lemma
PROBLEM 2. [5 points] Let X(t) be a Brownian motion. Assume that the stock price follows the stochastic...
2. Let Bt denote a Brownian motion. Consider the Black-Scholes model for the price of stock St, 2 So-1 and the savings account is given by β,-ea (a) Solve the equation for the price of the stock St and show that it is not a (b) Explain what is meant by an Equivalent Martingale Measure (EMM) martingale. State the Girsanov theorem. Give the expression for Bt under the EMM Q, hence derive the expression for St under the EMM, and...
You own one share of a stock. The price is 10. The stock price changes according to a geometric Brownian motion process, with time measured in months and coefficients µ=.1 σ= 2. Calculate the probability that: a. The stock price is at least 20 in one year. b. The stock reaches the price of 20 before it drops to 5. c. You will be able to sell the stock at 20 in a year if the price at the end...