Programming for Finance HW 1
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Finance
Date
Jan 9, 2024
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1. Historical Analysis of Sector ETFs:
(A)
Please see the historical data in the attached Excel file.
(B)
Annualized Return:
SPY
0.125937
XLB
0.089259
XLE
0.067284
XLF
0.100938
XLI
0.121781
XLK
0.172643
XLP
0.104094
XLU
0.092621
XLV
0.129068
XLY
0.150678
Standard Deviation:
SPY
0.174190
XLB
0.213877
XLE
0.281345
XLF
0.227483
XLI
0.199047
XLK
0.213326
XLP
0.139759
XLU
0.176553
XLV
0.164915
XLY
0.201777
(C)
Daily Covariance Matrix: Please see attachment.
Monthly Covariance Matrix: Please see attachment.
Covariance of daily returns tend to be smaller. The reason might be
that the covariance of monthly return sums up the daily returns across
a month.
(D)
The rolling 90-day correlation with S&P 500 don’t tend to be stable
over time.
Possible reason
: There are specific factors that are
affecting certain industries which is not apparently reflected on the
S&P 500 index.
(E)
Betas for the entire historical period:
XLB
: 1.073536034273137
XLE
: 1.1339397351303069
XLF
: 1.150400130213644
XLI
: 1.0440889304051424
XLK
: 1.137283708089871
XLP
: 0.6279913768699096
XLU
: 0.6251513669152604
XLV
: 0.8064018723825596
XLY
: 1.0598427421068164
Please see the chart of Rolling 90-days Betas below:
The rolling betas don’t seem to be consistent over the period. The
rolling betas and correlation are not moving in the same direction.
(F)
Alphas for each ETFs:
SPY
: -0.10855819707198816
XLB
: -0.04953909491095662
XLE
: -0.03470814556153503
XLF
: -0.12065508623373887
XLI
: -0.0489454640984493
XLK
: -0.12038766075932804
XLP
: -0.10176335070414656
XLU
: -0.08193789712358865
XLV
: -0.09052119383830146
XLY
: -0.05548669527327981
These alphas tend to close to 0, which indicates that there’s little to no
autocorrelation. The stock price in the past gives us little information
about the price change in the future.
2. Option Pricing
(A)
Mean
of Terminal Values: 99.141
Variance
of Terminal Values: 603.625
Yes, the mean and variance appear to be consistent with the Black-
Scholes model which claims that the stocks follow a random walk with
variable potential ending price.
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(B)
Mean
of Payoff: 10.0742531050227
Standard Deviation
of Payoff: 12.68903155743441
(C)
Put Option
Price
: 10.0742531050227
(D)
Black-Scholes price:
9.94764496602258
Monte Carlo Simulation price:
10.0742531050227
Two numbers are close enough while due to the limits of the Monte
Carlo simulation, the result is impacted by the number of the stock
samples we take. We took 1000 sample stock paths into the calculation
of the Monte Carlo price, and we could expect the difference between
two prices calculated by two different ways could get smaller if we took
more sample stock paths into the calculation of the Monte Carlo price.
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