Continuous Time Finance Homework 2
Deadline: one day before class 7.
Submit a Word or PDF file and supporting programs on Canvas.
Write ALL group member names on the Word or PDF file.
Problem 1 (16 points)
In this problem, you are asked to price an Asian option using the simulation method we
talked about in class. Suppose that the stock price follows Geometric Brownian Motion
dSt = ô dt + ?St dBt .
The initial stock price S0 is $10. The drift = 15% and the volatility ? = 40%. The
continuously-compounded riskfree rate r is 5%. These are exactly the same parameters you
have for Problem 5 of Homework 1.
Consider an Asian put option on the stock with maturity T = 1 and strike price
K = $10. The average stock price is calculated using the closing stock price of each trading
day. That is, letting N = 252 and ?t = T /N , the average stock price is defined as
Savg =
S?t + S2?t + 7 + SN ?t
.
N
To ensure the reproducibility of your results, you can set random seed in Python using
np.random.seed(0).
(a) (8 points) Simulate 100, 000 independent paths of the pseudo-price process
S?0 , S??t , S?2?t , . . . , S?N ?t
under
dS?t = rS?t dt + ? S?t dBt ,
1
S = S0 .
Compute the pseudo-price average as
S?avg =
S??t + S?2?t + 7 + S?N ?t
.
N
(b) (8 points) Calculate the current price of this Asian option using the risk-neutral pricing
formula
h
i
e?rT E max(K ? S?avg , 0) .
Problem 2 (18 points)
This problem asks you to numerically calculate Delta and Gamma for the Asian put option
in Problem 1. All parametric settings are the same as that of Problem 1, so you will need
to build on the program you write for Problem 1.
Choose step size = 0.1. Instead of fixing the initial stock price S0 at $10 as in Problem
1, we now consider the initial stock price S0 on grid points , 2, 3, . . . , 20 ? , 20.
(a) (8 points) For each possible initial stock price S0 on the grid points, calculate the
current price of the Asian put option P (S0 ). Plot the option price P (S0 ) against the
initial stock price S0 . Make sure that you see a downward sloping line. Hint:
s we discussed in class, each time your iteration changes to a different starting S0 ,
make sure you reset to the same random seed in Python using np.random.seed(0).
Can you think of a smarter programming way to implement the same thing?
our codes could take up to 10 minutes to run depending on your computer and
how efficient your codes are. To save time, first do the calculation with step size
= 1. Once you see the results and are sure your codes are correct, change back
to = 0.1 to produce the required figure.
(b) (5 points) The option Delta can be approximated by the following finite-difference
equation,
P (S0 + ) ? P (S0 )
P 0 (S0 ) ?
.

Use the option price P (S0 ) in part (a) to calculate the Delta P 0 (S0 ) of the Asian put
option. Plot the option Delta P 0 (S0 ) against the initial stock price S0 . Is this curve
2
upward or downward slopping? Why?
(c) (5 points) The option Gamma can be approximated by the following finite-difference
equation,
P (S0 + ) ? 2P (S0 ) + P (S0 ? )
P 00 (S0 ) ?
.
2
Use the option price P (S0 ) in part (a) to calculate the Gamma P 00 (S0 ) of the Asian
put option. Plot the option Gamma P 00 (S0 ) against the initial stock price S0 . What is
the shape of this curve? Why?
Problem 3 (16 points)
For this problem, your job is to simulate the stock price St and volatility process Vt under
the Heston model
p
Vt dBt ,
p
dVt = ?(? ? Vt )dt + ? Vt dWt .
dSt = ô dt + St
Simulate the Heston model using the following baseline parameters: S0 = 10, V0 = 0.06,
= 0.1, ? = 5, ? = 0.04, ? = 0.5, and the correlation between two Brownian Motions ? =
?0.6. Simulate from time 0 to T = 10, and use N = 10, 000 number of steps. To ensure the
reproducibility of your results, you can set random seed in Python using np.random.seed(1).
(a) (5 points) Simulate two Brownian Motions Bt and Wt with correlation ? = ?0.6. Plot
both Bt and Wt on the same figure. Make sure you see a negative relationship.
(b) (5 points) Simulate the stock price St and volatility process Vt using the correlated
Brownian Motions Bt and Wt you construct in part (a). Once you get Vt , define
?t =
p
Vt à100,
as the implied volatility in percentage points. Plot the stock price St and the implied
volatility ?t on the same figure. Make sure you see a negative relationship.
(c) (3 points) From the baseline parameters, change ? to 20. Plot the same figure as in
part (b). What differences do you see? Why?
3
(d) (3 points) From the baseline parameters, change ? to 0.6. Plot the same figure as in
part (b). What differences do you see? Why?
4

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