![PROBLEM 2. [5 points] Let X(t) be a Brownian motion. Assume that the stock price follows the stochastic differential equation](http://img.homeworklib.com/images/36c46473-88b2-414e-bf52-eee519c66691.png?x-oss-process=image/resize,w_560)
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PROBLEM 2. [5 points] Let X(t) be a Brownian motion. Assume that the stock price follows the stoc...
Assume an asset price St follows the geometric Brownian motion, dSt = u Stdt+oS+dZt, So =s...
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].
1. Consider a stock and assume it follows a geometric Brownian motion dS = µdt+σdz. Consider...
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.
2. Solving the generalized geometric Brownian motion equation. Let S(t) be a positive stochastic process that...
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 point) Let a and b be constants, and let Let f(t,) . Then f is...
1 point) Let a and b be constants, and let Let f(t,) . Then f is a smooth function of variables t and z, and frz Let z-W be a Wiener process. The goal is apply Ito's lemma in the form, to find a stochastic differential equation that is satisfied by Y = f(t, z) After applying Ito's lemma, dt t dz Type z Wt as z. Since Y e is a common factor on the right side, after dividing...
If s follows the geometric Brownian motion process ds - S dt+oS dz what is the...
If s follows the geometric Brownian motion process ds - S dt+oS dz what is the process followed by (a) y = 2S, (b) y S 5.
If s follows the geometric Brownian motion process ds - S dt+oS dz what is the process followed by (a) y = 2S, (b) y S 5.
If s follows the geometric Brownian motion process ds - S dt+oS dz what is the...
If s follows the geometric Brownian motion process ds - S dt+oS dz what is the process followed by (a) y = 2S, (b) y S 5.
4. [20 points] Let {B(t):t0 be a standard Brownian motion. Define a stochastic process (X (t):t20) by the formulas X (...
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)...
Let {X(t), 1 2 0} denote a Brownian motion 8.1. Let Y(t) = tx(1/t). (a) What is the distributio...
let {X(t), 1 2 0} denote a Brownian motion 8.1. Let Y(t) = tx(1/t). (a) What is the distribution of Y(t)? (b) Compute Cov(Y(s), Y()) (c) Argue that {Y(t), t 2 0] is also Brownian motion (d) Let Using (c) present an argument that
let {X(t), 1 2 0} denote a Brownian motion
8.1. Let Y(t) = tx(1/t). (a) What is the distribution of Y(t)? (b) Compute Cov(Y(s), Y()) (c) Argue that {Y(t), t 2 0] is also Brownian motion...
Let aſt),o(t), and r(t) be non-random adapted processes. For a stock process that follows: dS(t) =...
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.
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 a...
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...
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].
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 point) Let a and b be constants, and let Let f(t,) . Then f is a smooth function of variables t and z, and frz Let z-W be a Wiener process. The goal is apply Ito's lemma in the form, to find a stochastic differential equation that is satisfied by Y = f(t, z) After applying Ito's lemma, dt t dz Type z Wt as z. Since Y e is a common factor on the right side, after dividing...
If s follows the geometric Brownian motion process ds - S dt+oS dz what is the process followed by (a) y = 2S, (b) y S 5.
If s follows the geometric Brownian motion process ds - S dt+oS dz what is the process followed by (a) y = 2S, (b) y S 5.
If s follows the geometric Brownian motion process ds - S dt+oS dz what is the process followed by (a) y = 2S, (b) y S 5.
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)...
let {X(t), 1 2 0} denote a Brownian motion 8.1. Let Y(t) = tx(1/t). (a) What is the distribution of Y(t)? (b) Compute Cov(Y(s), Y()) (c) Argue that {Y(t), t 2 0] is also Brownian motion (d) Let Using (c) present an argument that
let {X(t), 1 2 0} denote a Brownian motion
8.1. Let Y(t) = tx(1/t). (a) What is the distribution of Y(t)? (b) Compute Cov(Y(s), Y()) (c) Argue that {Y(t), t 2 0] is also Brownian motion...
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.
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...
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