1. Brownian Motion and Martingales:

Let (W_t)_t≥0, and (Z_t)_t≥0 be two independent Brownian motions. Use the definition of Brownian motion and the definition of a martingale to show whether or not the following stochastic processes are standard Brownian motions and/or martingales, respectively (both for all three).

(a) (S_t⁽¹⁾)_t≥0 where S_t⁽¹⁾ = Ρ W_t + (1 - Ρ) Z_t and 0 ≤ Ρ ≤ 1 is a constant.

(b) (S_t⁽²⁾)_t≥0 where S_t⁽²⁾ = W_t+t0 =  W_t0.

(c) (S_t⁽³⁾)_t≥0 where ,S_t⁽³⁾ = W_t Z_t.

2. Path of Bachelier Model:

Consider the Bachelier model for the stock (S_t)_t≥0:

S_t = S₀ ± a t + b W_t,

where (W_t)_t≥0 is a Brownian motion and a, b > 0.

(a) Download daily data for the S&P 500 index for the twenty year period beginning in January 1994 until the end of 2013 (e.g., from yahoo finance). Use the data to estimate a and b for this model.

(b) Use Excel or another spreadsheet software to simulate a (discretized) sample path of a the Bachelier model over the year 2014 using 250 equidistant time steps, and compare it to the realized path. Just hand in the resulting plot.

3. Quadratic Variation 1:

Let X_t = X₀ (μ - 0.5 σ²) t + σW_t, where (W_t)_t≥0 is a Brownian motion. You are given the following two statements concerning X_t.

(a) Var[X_t+h - X_t] = σ² h, 1 ≥ 0, h ≥ 0.

(b) lim_n→∞j=1Σⁿ [X_jT/n - X_(j-1)T/n]² = σ² T ,T ≥ 0.

Which of them is true? Provide an explanation for your answer.

4. Quadratic Variation 2:

Define:

(a) S_t⁽¹⁾ = [t] , where [t] is the greatest integer part of t; for example, [3.14] = 3, [9.99] = 9, and [4] = 4.

(b) S_t⁽²⁾ = 2t + 0.9 W_t, where (W_t)_t≥0 is a standard Brownian motion.

(C) S_t⁽³⁾ = t².

Let h = T/n and let

V_T⁽²⁾(i) = lim_n→∞j=1Σⁿ [S_jh⁽ⁱ⁾ - S_(j-1)h⁽ⁱ⁾]²

denote the quadratic variation of the process S⁽ⁱ⁾ over the time interval [0, T]. Rank the quadratic variations  V_T⁽²⁾(1) , V_T⁽²⁾(2), and V_T⁽²⁾(3) over the time interval [0, 2.4]. Provide an explanation for your answer.