volatility as an asset class

Project 2: Volatility as an Asset Class—improving the mean-variance efficient frontier using volatility as an asset class

MS&E 445 Projects in Wealth Management Professor Peter Woehrmann

Ian Schultz, Linda He Yi, Hai Wei, J.R. Riggs, Andrew Tsai, Vicky Wang, Henry Chen, Erica Jiang

Project 2: Volatility as an Asset Class

Introduction

Part I: Portfolio Optimization & Role of Volatility● Construct the efficient frontier for universe of 30 (DJIA), 100 stocks (S&P100), 500 stocks (S&P500) & 2600+ stocks (NASDAQ)● Include volatility using ETFs tracking the VIX (VXX, VXZ)

Part II: Volatility Forecasting & Estimation● Volatility forecasting using AR(1) Model● Minimize estimation errors by using the Shrinkage Approach to estimate the covariance matrix

Part III: Trading & Implementation● Comparing selective long/short volatility trading strategies

Part I: Portfolio Optimization & Role of Volatility


Efficient Frontier with Market Indices

S&P 500

NASDAQ

S&P 100

DJIA

Efficient Frontier - Markowitz Formula to find two efficient portfolio; i.e. minimum variance for a given return

- Two-Fund Theorem to construct the efficient frontier


Volatility: Negative Correlation

S&P 500

Volatility S&P 500 (VIX)


Price

Today Maturity

Expected Future Spot Price

Forward Price in Normal Backwardation

Forward Price in Contango

Volatility: Contango Effect of VIX Futures

- Each subsequent expiration month of VIX futures prices are traded higher than the closer month's VIX futures prices and the spot VIX overall

How to take advantage of the decay effect that is very consistent and significant over time?


Choice of Volatility Vehicles

iPath S&P 500 VIX Short Term Futures TM ETN (VXX) iPath S&P 500 VIX Mid-Term Futures ETN (VXZ)


Comparing the Effects


Strategy: Short VXX to Speculate, Long VXZ to Hedge

Part II: Volatility Forecasting & Estimation


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Using Calculated CovS&P500S&P100DJIANASDAQVXXVXZAfter Cov Shrinkage

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Traditional covariance estimation methods based on historical data incur lots of error and therefore degrade the results through mean-variance optimizationCovariance Matrix Shrinkage gives better estimation of covariance coefficients [Ledoit & Wolf 2003]

Covariance Matrix Shrinkage


AR(1): Auto-regressive model of order 1

Key assumptions:1) Only t-1 information is used to predict the result at t2) Error term ɛt is independent of time and X

AR(1) Model


For the whole data set of size n, half of the data points are used as training data.

Parameters c, ɛt, ϕ are estimated from data points 1, 2, … n/2

AR(1) Model Training


We use the model to simulate the response from the 1st data point and compare the simulated value with the original data

For each method we run the simulation 100 times and calculate the RMS value from the simulated data for each run

X difference between real and simulated data

AR(1) Model Testing


C = 3.3802 ϕ = 0.8424

Residue Histogram

Mean RMS: 10.5836

AR(1) Model Verification

Accurate in the short term


AR(1) Model Verification – Prediction

Part III: Trading and Implementation


Active Volatility Trading

It is clear positions on volatility products can enhance overall portfolio performanceBut…

• Does active volatility trading outperform static volatility positions?• How can we actually compare performance of different trading strategies?• What is the best way for a long/short volatility hedge fund to operate?

Here we will try to address these issues:1. Examine simple momentum strategies2. Incorporate AR(1) volatility predictions to increase returns3. See how the best returns in active volatility trading can increase

efficient portfolios


2010

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Price, $

VIX Future Closing Price10-Period MA+/- 5% of MA

Moving Average Momentum

Trading Rules:1. Enter long VXX position when VIX futures cross n% above N-day moving average2. Close long VXX position when VIX futures cross N-day moving average3. Enter short VXX position when VIX futures cross n% below N-day moving average4. Close short VXX position when VIX futures cross N-day moving average

First attempt of a simple momentum strategy

Short entry

Short close

Long entry

Long close


Sharpe Ratios to Compare Strategies

Trading Strategy Sharpe RatioStatic short VXX 0.181

Static VXX+VXZ 0.012

PPO & RSI -0.14

Momentum trading 0.524

ARCH Model 0.403

Sharpe Ratio is used to compare excess return and variance against a benchmark

Expected Excess Return

Variance of Excess Return

Here we use the static short VXX as our benchmark portfolio, except for VXX itself


Exploration of Strategy Variations

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ν = 73.8%σ = 44.5%SR = 0.501

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ν = 78.1%σ = 45.5%SR = 0.516

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ν = 79.8%σ = 45.4%SR = 0.524

ν = 63.3%σ = 44.4%SR = 0.453


Trading on the AR(1) Model• Recall AR(1) Model of the form:

• Fit the AR(1) over the first half of historical VIX (1993-2003)• Test the returns of this strategy on the second half of data (2003-2013)

• These results look promising, but perhaps we can simulate many times to eliminate εt noise

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Actual VIX Price MovementARCH Prediction


Tomorrows Volatility Today

• At each time step we use the today’s VIX in the AR(1) Model to predict tomorrow’s VIX level• Repeat this at each step 30 times and take the average to get tomorrows volatility prediction• When tomorrow’s volatility is higher than today’s, buy the VIX• Tomorrow lower, vice versa

Annualized Return = 167%Standard Deviation = 59.9%

Sharpe Ratio = 0.403

Qualitative performance of this method looks exceptional.

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Model Fitting

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no volw/ VXXw/ ARCH-based active trading

Excess Returns Expand Efficiency

Massive expansion in efficiency frontier due to excess returns from trading on AR(1) predictions of VIX

What’s the catch?• The VIX itself is not a tradable product• Can generate VIX sensitivity, however

through:- Volatility Swaps- Options Positions- Volatility Futures


Using The VIX for VXX TimingThree criteria using the VIX to generate a short signal in VXX (and sell long position in VXX):

1.The monthly low is above its 10-month moving average2.The monthly close is at least 10% above its 10-month moving average (PPO more than 10)

- Use the PPO (Percent Price Oscillator) - PPO = (1-day EMA – 10-day EMA)/10-day EMA

3.The monthly close is above the monthly open (Filled Candle Stick)

Three criteria using the VIX to generate a cover signal in VXX (and enter long position in VXX):

1. The high of the VIX is below the 10-day moving average (candlestick must be below the 10-day moving average)2. The monthly close is at least 10% below the 10-month moving average3. The close is below the open (Hollow Candlestick)


Returns Using the PPO Indicator

ν = 18.6%σ = 51.3%SR = -0.1418


Another Method Using RSI Indicator

• When RSI is above 70, VIX is overbought Cover VXX position

• When RSI is below 30, VIX is oversold Initiate short VXX position

• RSI = 100 – 100/(1+RS) RS = (Average Gain / Average Loss ) in a 5 period (5 month) setting


Results Using the RSI indicator

ν = 15.8%σ = 35.6%SR = -0.1406


Conclusions Use of volatility ETFs significantly expands efficient frontier

• Short position on VXX to speculate• Long position on VXZ to hedge

Covariance Matrix Shrinkage technique gives us more reliable covariance estimations and a more accurate efficient frontier

AR(1) model helps us predict future movement in VIX Simple momentum trading strategies and trading using AR(1)

predictions exhibit promising excess returns above benchmark returns

RSI and PPO trading strategies are less viable, returning negative Sharpe ratios

Project 2: Volatility as an Asset Class—improving the mean-variance efficient frontier using volatility as an asset class

Q & A

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volatility products

active volatility trading

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