1. Consider the one period binomial model and assume 0 < So< 00, S1(H) -- uSo and Si (Τ)-dSo for some 0 〈 1 + r 〈 d 〈 u. P is an arbitrage oportunity. rove or disprove There
1. Consider the one period binomial model and assume 0 < So< 00, S1(H) -- uSo and Si (Τ)-dSo for some 0 〈 1 + r 〈 d 〈 u. P is an arbitrage oportunity. rove or disprove There
Consider the following one-period binomial model for stock price. At t = 0 the stock price is $80 and at t = 1 (t is in years) it could be $70 with probability p > 0 and $y with probability 1 − p. The interest rate is assumed to be 8%. (1) Determine the range of values for y that precludes arbitrage in this model. (2) Assume that y = $83. Construct an arbitrage strategy for this model.1
1. (Put-call parity) A stock currently costs So per share. In each time period, the value of the stock will either increase or decrease by u and d respectively, and the risk-free interest rate is r. Let Sn be the price of the stock at t-n, for O < n < N, and consider three derivatives which expire at t - V, a cal option Voll-(SN-K)+, a put option VNut-(X-Sy)+, and a forward option VN(SN contract FN SN N) ,...
2. Consider a two-period (T = 2) binomial model with initial stock price So = $8, u= 2, d=1/2, and “real world” up probability p=1/3. (a) Draw the binary tree illustrating the possible paths followed by the stock price process. (b) The sample space for this problem can be listed as N = {dd, jdu, ud, uu}. List the probabilities associated with the individual elements of the sample space 12. (c) List the events (i.e., the subsets of N2) making...
1.1. Suppose that you have a stock in the one-period binomial model with fixed u, d, and r such that 0< d< 1 +r < u. Suppose that there are positive numbers pi and such that pi, qi < 1, pi + q-1, and (1 + r)So = PiSi (H) + qi Si (T). Show that pi = p ad qi = q. Hint: You know that the risk-neutral probabilities satisfy these equations as well.
1. (Put-call parity) A stock currently costs So per share. In each time period, the value of the stock will either increase or decrease by u and d respectively, and the risk-free interest rate is r. Let Sn be the price of the stock at t n, for O < n < V, and consider three derivatives which expire at t- N, a call option Vall-(SN-K)+, a put option Vpul-(K-Sy)+, ad a forward contract Fv -SN -K (a) The forward...
1.2. You have a stock in the one-period binomial model such that So and r= 4, S1(H) = 8, S, (T) = 2, 1.5. (a) Show that this setup violates the no-arbitrage assumption. (b) Show that there is a portfolio in the one-period model such that Xo X1 > 0. Such a portfolio is an example of arbitrage extraction. 0, Ao #0, and %3D 1.3. With the same stock as in problem 1.2 but with r = 0.25, suppose that...
5. Consider the single period binomial model as in Section 1.2.2. Suppose that d <1+r <u. Show that if there exists an arbitrage opportunity (as in Definition 1.5), then one can find an arbitrage opportunity with V = 0. This means that there is no net cash flow at time 0. (Note: This is a step in the proof of Proposition 1.7 which you should go through carefully.) 1.2.2 Formal logical content The theory we build will be a mathematical...
2. Heat equation Let ult, 2) satisfy the equation 4472(t, 2) +1, 0<r <1, t>0 with initial condition u(0,2) = 0, 0<x<1, and boundary conditions u(t,0) = 0, u(t,1)= 0, t> 0. This equation describes the temperature in a rod. The rod initially has a temperature of 0 (zero degree Celsius), and is then heated at a uniform rate 1. However, its two endpoints are kept at the temperature of 0 at all times. The unknown function u(t, x) describes...