(6 marks) Consider a filtered probability space (2,F,P, Ftte.). a. (2 marks) Let the stochastic p...
4. [20 points] Let {B(t):t0 be a standard Brownian motion. Define a stochastic process (X (t):t20) by the formulas X (t) = tB(1 + t-1)-tB(1), x(0) = 0, t > 0, You may take for granted the fact that imt-«HX(t) = 0, with probability 1 (b) Explain why [X():t20 is a standard Brownian motion 4. [20 points] Let {B(t):t0 be a standard Brownian motion. Define a stochastic process (X (t):t20) by the formulas X (t) = tB(1 + t-1)-tB(1), x(0)...
3. Consider a filtration (F) and an (F)-adapted stochastic process (X) such that Xo 0 and E X] oo for all n 2 0. Also, let () be a sequence of constants. Define Mo 0 and rt 7t j-1)j-1, for n 1 Prove that (Ms n (F)-martingale.
[8 marks] Consider a discrete time stochastic process {Xn,n 2 0j defined by the equation with Xo1 and Rn,n21 are random variables taking their values in (-1,00). Denote Sn-Li-1 Rk for n 〉 1 and So-0 i) [3 marks] Briefly explain why the filtration {F,:n 〉 0} gener- 0 generated by ated by Xo, X1,.. . , Xn and the filtration , n So, S1, , Sn should be identical ії) [5 marks] Show that {X,,n 〉 0} is a...
5. Let (S2,F,P) be a probability space and let {W(t),t 2 0) be Brownian mo- tion with respect to the filtration Ft, t 2 0. By considering the geometric Brownian motion where Q R, σ > 0, S(0) > 0. Show that for any Borel-measurable function f(y), and for any 0 〈 8くthe function 2 2 g(x) =| f(y) da 0 satisfies Ef(S(t))F (s)-g(S(s)), and hence S(t) has a Markov property. We may write qlx as q We may write...
find the probability that a european put option with underlying s finishes in the money a) Let Y e*t, Find the stochastic differential equation satisfied by t. b) Let Zt -eatX. Find the stochastic differential equation satisfied by Zt c) Find XtWdWs, where W, is a Brownian motion. 0 Hint: Set XtaW2 + bWt + ct. Find a, b, and c. 6) Find the probability that a European put option with underlying S a) Let Y e*t, Find the stochastic...
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...
Q6 (4+3+3+ 6=16 marks) Let Xo, X1, X2 be three distinct real numbers. For polynomials p(x) and q(x), define < p(x),q(x) >= p(xo)q(x0) + p(x1)q(x1) + p(x2)q(22). Let p(n) denote the vector space of all polynomials with degree more no than n. (i) Show that < .. > is an inner product in P(2). (ii) Is < ... > an inner product in P(3)? Explain why. (iii) Is <,:> an inner product in P(1)? Explain why. (iv) Consider Xo =...
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...
4. Let Z1, Z2,... be a sequence of independent standard normal random variables. De- fine Xo 0 and n=0, 1 , 2, . . . . TL: n+1 , The stochastic process Xn,n 0, 1,2,3 is a Markov chain, but with a continuous state space. (a) Find EXn and Var(X). (b) Give probability distribution of Xn (c) Find limn oo P(X, > є) for any e> 0. (d) Simulate two realisations of the Markov process from n = 0 until...