10. Read the first 8 pages of the paper "Valuing Am...

10. Read the first 8 pages of the paper "Valuing American Options by Simulation: A Simple Least-Squares Approach" by Longstaff and Schwartz 80 and play with the Matlab code Sect_9p12. m on luminus. Answer the following two questions: (a) Let S0 = 100, K = 100, T = 0.03, T = 1. C...

1. (Random number generator) A linear congruential (pseudo-)...

1. (Random number generator) A linear congruential (pseudo-)random number generator generate a sequence of integers by In+1 = f (In) with f (x) = (ax + c) mod m where In is an integer and a > 1, C > 0, m >> 1 are fixed integers. Once xi is obtained, Ui = xi/m will be ...

8. Consider the CIR model dYt = (r - BYt dt + o Y1dW1, Y0=a...

8. Consider the CIR model dYt = (r - BYt dt + o Y1dW1, Y0=a, (6.30) where Y, B, o, a are positive constants. a) Write down equation (6.30) in integral form. b) Let u(t) = E[Yt]. Use the integral form of (6.30) to show that u(t) satisfies the differential equation d = Y - Bu(t), u(0)=a. c) B...

8. Suppose you want to approximate H = E[h(X)] with X ~ U(0,...

8. Suppose you want to approximate H = E[h(X)] with X ~ U(0,1) and = 10 x by Monte Carlo method. (a) If you use simple sampling to approximate u by in = 1/2i-1 h(Xi), where X1, are i.i.d. U(0,1) random variables, what is the variance of in? How is the variance of in related to liin - u/? [Hint: ...

Statistics 1. Let X have PDF: .c -2 if x>1, f(x)...

Statistics 1. Let X have PDF: .c -2 if x>1, f(x) = otherwise. Prove that the m.g.f. (()()) is finite for all t < 0 but not for t> 0. "We only need to show that it is finite for t<= 0 and infinite for t>0. I ended up relying on the general form of ...

6. (Continuation of Questions 4 and 5) Suppose that X1 (t) a...

6. (Continuation of Questions 4 and 5) Suppose that X1 (t) and X2(t) are the Ornstein- Uhlenbeck processes with dXi (t) = -aXi (t) dt + odWi (t) for i = 1,2, where (W1(t), W2 (t)) is a 2-dimensional Brownian motion. Let X = (X1, X2. Recall the 2-dimensional Itô formula (Theorem 5.2) 2 2 ...

6. (Black-Scholes-Mertor equation) Given constant T, functio...

6. (Black-Scholes-Mertor equation) Given constant T, functions b, o, and f, consider the stochastic differential equation dXt= b(t,Xt)dt+o(t,Xt)dW and the deterministic partial differential equation = (6.23) with terminal condition = for all x. (6.24) Fix T > 0, show that a martingal...

6. We now present the derivation of Black-Scholes-Merton equ...

6. We now present the derivation of Black-Scholes-Merton equation in Lishang Jiang Section 5.2. Let c(t,x) denote the value of an option at time t if the stock price at that time is S(t) = x. We want to derive an equation for c(t,x) and hence obtain its formula. Like in (2.2), construct a portfol...

1. Miscellaneous: (a) Find the column space, row space, nul...

1. Miscellaneous: (a) Find the column space, row space, null space and left nullspace of the matrix 2 6 4 A = . (b) Consider the three vectors x¹ = x² = (0,1,0)T, x3 = Does the set A = {x¹,x²,x³} form an orthonormal basis of R3? 2. Trading model: As in class, consider...

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