6. Let S(t) denote the price of a security at time t. A popular model for the process S(t): t 0} supposes that the...
Let S = {S(t), t > 0) denote the price of a continuous dividend-paying stock. The prepaid forward price for delivery of one share of this stock in one year equals $98.02. Assume that the Black-Scholes model is used for the evolution of the stock price. Consider a European call and European put option both with exercise date in one year. They have the same strike price and the same Black-Scholes price equal to $9.37. What is the implied volatility...
The Black-Scholes-Merton model for stock pricing in discrete time Let So be the initial stock price at time t = 0. At time t = 1,2,-. ., the stock price is S,ett+σ Σ. 2. the drift where a 0 is known as the volatility and the independently and identically distributed standard Normal N(0,1) random 0 is known as Zi variables are (a) Show that S, = S¢_1e#+oZ¢ _ St St-1 (b) What is the distribution of ln (c) What is...
6. Extra-credit problem: 6 pts] Let θ denote the unknown proportion of time that an airline operator is busy answering customers, where 0 <8 <1. To estimate 0, a supervisor observes the operator at times selected randomly and independently from other observed times. Let xi1 if the operator is busy at the ith observation and let x, 0 otherwise, i -1,.,n. If the supervisor uses #2X1+--+- to estimate θ, determine the value of n for which the error of eatimation...
Let P(t) represent the number of people who, at time t, are infected with a certain disease. Let N denote the total number of people in the population. Assume that the spread of the disease can be modeled by the initial value problem: dP/dt = k(N − P)P, P(0) = P0. At time t = 0, when 100,000 member of a population of 500,000 are known to be infected, medical authorities intervene with medical treatment. As a consequence of this...
The flow system shown in the figure is activated at time t = 0. Let Qi(t) denote the amount of solute present in the ith tank at time t. Assume that all the flow rates are a constant 10 L/min. It follows that the volume of solution in each tank remains constant; assume this volume to be 1000 L. The concentration of solute in the inflow to Tank 1 (from a source other than Tank 2) is 0.2 kg/L, and...
Let aſt),o(t), and r(t) be non-random adapted processes. For a stock process that follows: dS(t) = a(t) S(t)dt + o(t)S(t)dW(t) and a discount factor given by: D(t) = e-Soudu derive the formula at any time 0 <t< T for a forward agreement to purchase the stock at time T for price K.
Question 1 Consider the derivation of the Black-Scholes model of option pricing. Let S=S(t) be the underlying stock price at time t and let f=f(S, t) be the option price at time t. a) Write down the value P of the portfolio defined in the Black-Scholes model. [2 marks] b) Use Itô’s lemma to find an expression for the change Δf in the discrete time Δt. [5 marks] c) Use the expression you have found in point b) to find...
(2 points) The flow system shown in the figure is activated at time t 0. Let Qi(t) denote the amount of solute present in the ith tank at time t Assume that all the flow rates are a constant 10 L./min. It follows that the volume of solution in each tank remains constant; assume this volume to be 1000 L. The concentration of solute in the inflow to Tank 1 (from a source other than Tank 2) is 0.8 kg/L,...
Homogeneous Poisson process N(t) counts events occurring in a time interval and is characterized by Ņ(0)-0 and (t + τ)-N(k) ~ Poisson(λτ), where τ is the length of the interval (a) Show that the interarrival times to next event are independent and exponentially distributed random variables (b) A random variable X is said to be memoryless if P(X 〉 s+ t | X 〉 t) = P(X 〉 s) y s,t〉0. that this property applies for the interarrival times if...
(3) Suppose that the intensity of the Poisson process describing the crystallization nuclei is time dependent and given by 1 + g(t), where g(0) = 0 and g is continuous and monotonically increasing (take g(t) = et as an example). Follow the method from Exer- cise 1 to derive a reasonable K-A model for this scenario. 1) (The raindrop problem) At time t = 0, rain starts to fall at an even and steady rate of I* droplets per unit...