Pset 2, ECON 6070
A. Factor models.
The data contains all firms traded from 1989-2015. This problem set is due on May
18th.
1. Read data from cleaned data.csv and FF-5 factors. match the dates from 1989
on monthly basis. Drop those firms if there is any missing data.
2. For each stock i and time t, consider the following factor regression:
Rit = αi + βift + it. Estimate the model using the first 80% of data (aka training
sample )and leave the rest 20% as the testing data.
Report the mean and standard deviation of ˆα across i = 1, 2, ..., N, where N is the
total number of stocks. Report the average in sample R2 and out of sample R2 over i.
3. Consider the Markowitz optimal portfolio w =1γΣ−1
(µ−re), where e = (1, 1, ..., 1)0
is a N × 1 vector of ones. Let γ = 2 and risk free rate r = 1%.
Compute ˆµ and Σ from the training sample based on the factor model: ˆ ˆ µi:= ˆαi +βˆiEn[fit] for i = 1, 2, ..., N, and V ar(Rt) = BV ar d (ft)B0+diag(ˆσ21, ..., σˆ2N ), where V ar d (ft)
is the sample covariance matrix of ft, and ˆσ2i:= 1T −p
PTt=1 ˆ2it, ˆit := Rit − αˆi − βˆift, and
p is the number of factors.
Report: (a) the return and volatility of the plug in optimal portfolio ˆw := 1γΣˆ −1(ˆµ −re) in the training sample.
(b) the return and volatility of the plug in optimal portfolio ˆw := 1γΣˆ −1(ˆµ − re) in
the testing sample. Do you see any differences between (a) and (b)?
B. Simulations and Black-Scholes
Consider a law of motion of returns: rt+1 := α + βrt + t, where α = 0.04, β = 0.8,and t ∼ N(0, σ2), σ = 20%. Initial price P0 = 100. Set r0 = 0.
There is an call option with a strike price K and a maturity T = 10.
Simulate M = 500 price paths from t = 0 to t = T and obtain 500 ”simulated price
at maturity” P1T, ..., P MT.
Compute the “NPV” valuation of the option Cˆ(K) := 1M1(1+r)
TPM
i=1 max(0, PiT −K).
Report a table of Cˆ(K) with K = 100, 120, 140, 160, 180, 200.