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Volatility Forecasting Homework

波动率预测代写 1.You will estimate the parameters of a few GARCH-type models using about ten years of data (from Jan 1, 2013 to Nov 30, 2022) for SPY

1.

You will estimate the parameters of a few GARCH-type models using about ten years of data (from Jan 1, 2013 to Nov 30, 2022) for SPY (the S&P 500 ETF). I would like you to build this from scratch, rather than use a package. As I demonstrated in class, it’s easy to do in Excel, which has a built-in solver, but you can code it in Python and use its optimization library as well. I used the scipy.optimize.minimize function with method=’SLSQP’, which allows you to specify bounds on thevariables.

(a)

Estimate the parameters of a GARCH(1,1) model.

(b)

Estimate a GARCH(1,1) model with conditional returns having a Generalized Error Distribution.

(c)

Estimate a TGARCH(1,1) model with conditional returns having a normal distribution.

(d) 波动率预测代写

Compare the maximum value of the likelihood function for the models in parts (a), (b), and (c) to rank the three models. The maximum value of the likelihood function for parts (b) and (c) can be compared directly since they have the same number of parameters. Use the Schwartz Bayesian Criteria to compare likelihood functions when the number of parameters is different.

One way to download historical prices for SPY in Python is to pip install the yfinance library, import the library using

import yfinance as yf

and download the data with

df = yf.download(“SPY”, start=”2013-01-01″, end=”2022-11-30″)

Use the ”Adj Close” column, which adjusts for dividends.

2. 波动率预测代写

The figure below lists prices and implied volatilities for puts and calls on SPY options as of the close on Wednesday, November 30, 2022. The middle column in the figure are the strikes, and the data on the left side are for the call options, and the data on the right side are for the put options. The column ”IVM” is the midmarket implied volatility. Thecolumn ”DM” is the midmarket delta of the options. The closing price for SPY on November 30, 2022 was 407.68. Use the TGARCH(1,1) model from part (c) above to price two options: the 416 strike call options (the delta is approximately 0.25), and the 400 put options (the delta is approximately -0.25). You should price these using a Monte Carlo simulation. These options expire on Friday, December 9, 2022, which is 7 trading days from Wednesday, November 30. Assume that the annual risk-free rate, rf , is 4.71% and there are no dividends.

Figure 1: Option Prices at the Close on Nov 30, 2022 for expiration Dec 9, 2022

(a)

What are the prices, and the corresponding Black-Scholes implied volatilities, for the two options?

(b)

There have been several stories in the news recently that the skewness in the volatility smile, as measured by the difference in volatilities between out-of-money put options and out-of-the-money call options, is the lowest it’s been since 2009.

The figure below shows a graph of the CBOE SKEW Index. Is that reduction in the skew justified, according to the results above? In other words, given the current difference in implied vols for 0.25 delta puts and calls of 18.88 − 17.01 = 1.87 Would your TGARCH model prefer to buy the puts and sell the calls, or sell the puts and buy the calls? A one-sentence explanation is sufficient.

Figure 2: The CBOE SKEW Index over the past 20 years

3. 波动率预测代写

Consider the stochastic volatility (SV) model we discussed in class,given by:

r_t= µ + σ_tϵ_t

ln(σ_t) = α + ϕ(ln(σ_t−1) − α) + η_t

You can use the class notes to find the expectations above. Just like with the first exercise, I would like you to estimate the parameters from scratch, rather than using a package. Also, to make things easier, you can choose to do two-step GMM rather than iterated GMM, and you can compute the weighting matrix assuming the data are uncorrelated so you don’t have to compute the Newey-West adjustment if you would like.

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