Financial Statistics: Time Series, Forecasting,
Mean Reversion, and High Frequency Data
FINM 33170 and STAT 33910 Winter 2019
HW 4, due Friday 8 February, 2019, at 1:30 pm
Homework can be submitted in paper form in class, or you can scan it and then e-mail to youngseok@galton.uchicago.edu and mykland@pascal.uchicago.edu
1. Make sure that you have read Chapter 2.2 of the posted notes (The Econometrics of High Frequency Data, M & Z (2012)).
2. Returns and log returns.
(a) Suppose that a stock (or other financial instrument) has price
dSt = μtStdt + σtStdWt (1) where Wt is a Wiener process. Define the instantaneous return by
Set Xt = log(St). Explain why
dRt = 1 dSt (2) St
dXt = dRt − 1σt2dt. (3) 2
(b) Determine whether Rt and Xt have the same volatility.
(c) Define daily return and daily volatility on day # i as
ri+1= i+1 STi
Ti
ST − ST
Ti+1
σt2dt (4)
i ands2i+1=
where Ti is the closing time on day # i. Also set ∆XTi+1 = XTi+1 − XTi . Explain the
rationale for the following approximation between return and log return:
∆XTi+1 = ri+1 − 1s2i+1. (5) 2
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(d) Consider two trading strategies with daily updates:
(i) θi∆XTi+1 and
i
(ii) θiri+1, i
(6)
where θi is determined at time Ti. Which of these strategies can be implemented in practice? [It does not have to be self financing.]
3. Potentially stationary processes.
Let Xt = log(St) be the log price of a certain stock, which we assume (for simplicity) to be given by
dXt = σtdWt, (7) where (Wt) is a Wiener process. For the moment, we assume that σt is a positive process (and
semi-martingale).
(a) Consider the process satisfying
dY = σ−1dX . (8) ttt
Determine whether Yt or YTi+1 − YTi are stationary, where Ti is the end of day # i. (We assume trading for 24 hours on all days.)
(b) Give an example of a case where ∆XTi+1 = XTi+1 − YTi is non-stationary. (c) Background. Assume that
dσt2 = κ(α − σt2)dt + γσtdBt, (9)
with σt > 0, where (Bt) is a Wiener process independent of (Wt), and where κ > 0, and 2κα ≥ γ2. (Cf. the discussion of the Heston model on p. 122 in the posted notes. Process (9) is also known as the Feller square root process, and the Cox-Ingersoll-Ross (CIR) process.) Also assume that σ02 has distribution with density
ων 2κ 2κα f(ζ)=Γ(ν)ζν−1e−ωζ forζ≥0,whereω=γ2 andν= γ2 (10)
(and f(ζ) = 0 for ζ < 0), where Γ(ν) is the Gamma function (recall that the integral of a probability density must be one). This process is stationary in the strong sense, and also Markov.
Question itself. Determine whether ∆XTi+1 = XTi+1 − XTi is stationary in the weak (second order) sense, and provide the autocovariance function for this process.
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(d) If you would like to implement a trading strategy along the lines of HW 3, problem 2(b) on ∆XTi+1 or ∆YTi+1 , what are the pros and cons of using one process versus the other?
(e) If you are unsure whether your process is on the form ∆XTi+1 or ∆YTi+1, suggest a statistical procedure to determine which of the two it is.
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