0.2. The time-t price of a stock is s(t). You are given: (1) A stock's price...
suppose T= 30 to begin
7. Suppose you are high frequency trading of stocks, if the current price of the stock is 5% higher than the purchase price then you will buy. Suppose that a stock follows a Standard Geometric Brownian Motion lifted by the current market price of $20? What is the probability that the stock price is ready to buy after T units of time later? Evaluate for for multiple values of T and the plot the distribution...
A market-maker sells a straddle on a stock. You are given: (i) The stock's price follows the Black-Scholes framework. (ii) S(O)= 45. (iii) The continuously compounded risk-free rate is 0.10. (iv) The stock pays no dividends. (v) The annual volatility of the stock is 0.2. (vi) The straddle consists of European options and expires in one year. The market-maker delta-hedges the sale by buying shares of the underlying stock. Calculate the amount of money the market-maker spends on the stock.
10. 2 marks Suppose that S, is the price of a nondividend paying stock at time t. Sy follows a lognormal model. You are given So = $40, the stock's volatility o = 0.3 and the stock's continuously compounded expected growth rate a = 0.15 (a) (1 marks) What is the average price of the stock after 6 years? (b) 1 marks What is the expected value of the log-returns on the stock in a time interval of 4 years?...
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...
Let S(t), t >=0 be a Geometric Brownian motion process with drift mu = 0.1 and volatility sigma = 0.2. Find P(S(2) >S(1) > S(0))
2. Solving the generalized geometric Brownian motion equation. Let S(t) be a positive stochastic process that satisfies the generalized geometric Brownian motion SDE dS(t) = u(t)S(t) dt + o(t)S(t) dW(t), where u(t) and o(t) are processes adapted to the filtration Ft, t > 0. (a) Use Itô’s lemma to compute d log S(t). Simplify so that you have a formula for d log S(t) that does not involve S(t). (b) Integrate the formula you obtained in (a), and then exponentiate...
1. Consider a stock and assume it follows a geometric Brownian motion dS = µdt+σdz. Consider now a function G = G(S, t). i) Use Itˆo’s lemma to find the stochastic process dG followed by G. ii) Use Itˆo’s lemma to find the stochastic process followed by G(S) = ln S.
3. Assume a stock price follows a lognormal Brownian motion, i.e. its price satisfies So eo) where W(t) is standard Brownian motion, So-$265, ơ 18%. Calculate the probability that in 20 days 22), or less, the stock price reaches the value H 275, i.e. calculate Prob. max St > 275
Problem #1 Imagine that the price of a given stock at time t is given with + (1 - wHp - No100), where R = 1,w = 0.5, Hp = 100,000 = 1. i) Let D, = 200 and N, = 100. What is the price of the stock in this case? ii) Now imagine that an investor purchases a call option that allows her to acquire this stock at time t + 1 for K = 150. Will this...
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...