1. Path of Bachelier Model:

Consider the Bachelier model for the stock (St) t≥0 :

St = S0 + a t + b Wt,

where (Wt) 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.

2. Quadratic Variation 1:

Let Xt = X0 + (µ − 0.5 σ 2 )t + σWt, where (Wt)t≥0 is a Brownian motion. You are given the following two statements concerning Xt.

(a) V ar[Xt+h − Xt] = σ 2 h, t ≥ 0, h ≥ 0.

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