Financial Engineering MRM 8610, Spring 2016 Homework 3
Question # 00188661
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Updated on: 02/05/2016 12:12 AM Due on: 02/05/2016
Homework 3
Due Thursday, 02/04/2016 before class.
1. Brownian Motion and Martingales: (1+1+1 points)
Let (Wt )t?0 and (Zt )t?0 be two independent Brownian motions. Use the de?nition of Brownian motion and
the de?nition of a martingale to show whether or not the following stochastic processes are standard Brownian
motions and/or martingales, respectively (both for all three).
(1)
(a) St
(1)
= ? Wt + (1 ? ?) Zt and 0 ? ? ? 1 is a constant.
(2)
= Wt+t0 ? Wt0 .
(3)
= W t Zt .
t?0
where St
t?0
where St
t?0
where St
(2)
(b) St
(3)
(c) St
2. Path of Bachelier Model: (1+1 point)
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 ?nance). 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: (2 points)
Let Xt = X0 + (ยต ? 0.5 ? 2 ) t + ?Wt , where (Wt )t?0 is a Brownian motion. You are given the following two
statements concerning Xt .
(a) V ar[Xt+h ? Xt ] = ? 2 h, t ? 0, h ? 0.
(b) limn??
n
j=1
2
X jT ? X (j?1)T
n
= ? 2 T , T ? 0.
n
Which of them is true? Provide an explanation for your answer.
4. Quadratic Variation 2: (3 points)
De?ne:
(1)
(a) St
(b)
(c)
(2)
St
(3)
St
Let h =
T
n
= [t], where [t] is the greatest integer part of t; for example, [3.14] = 3, [9.99] = 9, and [4] = 4.
= 2t + 0.9 Wt , where (Wt )t?0 is a standard Brownian motion.
= t2 .
and let
n
(2)
(i)
n??
(i)
(i)
2
Sjh ? S(j?1)h
VT (i) = lim
j=1
denote the quadratic variation of the process S over the time interval [0, T ]. Rank the quadratic variations
(2)
(2)
(2)
VT (1), VT (2), and VT (3) over the time interval [0, 2.4]. Provide an explanation for your answer.
Due Thursday, 02/04/2016 before class.
1. Brownian Motion and Martingales: (1+1+1 points)
Let (Wt )t?0 and (Zt )t?0 be two independent Brownian motions. Use the de?nition of Brownian motion and
the de?nition of a martingale to show whether or not the following stochastic processes are standard Brownian
motions and/or martingales, respectively (both for all three).
(1)
(a) St
(1)
= ? Wt + (1 ? ?) Zt and 0 ? ? ? 1 is a constant.
(2)
= Wt+t0 ? Wt0 .
(3)
= W t Zt .
t?0
where St
t?0
where St
t?0
where St
(2)
(b) St
(3)
(c) St
2. Path of Bachelier Model: (1+1 point)
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 ?nance). 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: (2 points)
Let Xt = X0 + (ยต ? 0.5 ? 2 ) t + ?Wt , where (Wt )t?0 is a Brownian motion. You are given the following two
statements concerning Xt .
(a) V ar[Xt+h ? Xt ] = ? 2 h, t ? 0, h ? 0.
(b) limn??
n
j=1
2
X jT ? X (j?1)T
n
= ? 2 T , T ? 0.
n
Which of them is true? Provide an explanation for your answer.
4. Quadratic Variation 2: (3 points)
De?ne:
(1)
(a) St
(b)
(c)
(2)
St
(3)
St
Let h =
T
n
= [t], where [t] is the greatest integer part of t; for example, [3.14] = 3, [9.99] = 9, and [4] = 4.
= 2t + 0.9 Wt , where (Wt )t?0 is a standard Brownian motion.
= t2 .
and let
n
(2)
(i)
n??
(i)
(i)
2
Sjh ? S(j?1)h
VT (i) = lim
j=1
denote the quadratic variation of the process S over the time interval [0, T ]. Rank the quadratic variations
(2)
(2)
(2)
VT (1), VT (2), and VT (3) over the time interval [0, 2.4]. Provide an explanation for your answer.

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Solution: Financial Engineering MRM 8610, Spring 2016 Homework 3