# Strategy Library

## Exploiting Term Structure of VIX Futures

### Introduction

The Chicago Board Options Exchange (CBOE) introduced the Volatility Index (VIX) in 1993 to provide a measure of the implied volatility of 30-day, at the money S&P 100 index options. Volatility has become a widely accepted asset class since the introduction of the VIX futures contracts in 2004. VIX futures are often used for hedging purpose because of its negative correlation with the equity market return. When we talk about the term structure of futures, we often refer to the forward curve. The VIX forward curve consists of VIX futures prices at various delivery times in the future. Academic research states that volatility follows a mean reverting process. When the VIX futures curve is upward sloped (in contango), the VIX is expected to rise because it is low relative to long-term average levels and vice versa for the downward sloped VIX future curve. In this study, we will create a strategy with the term structure effect of VIX futures and hedge the term structure risk with the S&P500 futures.

### Method

The trading strategy uses VIX futures as a trading vehicle and S&P E-mini for hedging purposes. The spot VIX price data and the continuous front contract price of VIX and E-mini S&P500 futures in the daily resolution are from Quandl.

def Initialize(self): self.SetStartDate(2011, 1, 1) # Set Start Date self.SetEndDate(2018, 9, 1) # Set End Date self.SetCash(10000000) # Set Strategy Cash self.vix = self.AddData(QuandlVix, "CBOE/VIX", Resolution.Daily).Symbol # Add Quandl VIX price (daily) self.vx1 = self.AddData(QuandlFutures, "CHRIS/CBOE_VX1", Resolution.Daily).Symbol # Add Quandl VIX front month futures data (daily) self.es1 = self.AddData(QuandlFutures, "CHRIS/CME_ES1", Resolution.Daily).Symbol # Add Quandl E-mini S&P500 front month futures data (daily) # Add VIX futures contract data self.AddFuture(Futures.Indices.VIX).SetFilter(timedelta(0), timedelta(days=180)) # Add E-mini S&P500 futures contract data self.AddFuture(Futures.Indices.SP500EMini).SetFilter(timedelta(0), timedelta(days=180))

Futures basis is defined as the difference between the underlying product cash price and futures contract price at a given time.

\[Basis_t=S_t-F_t\]Where \(S_t\) is the spot VIX price, \(F_t\) is the VIX futures price. The basis is in contango means the futures price is higher than the spot price. The opposite of Contango is Backwardation. It refers to the market condition in which the futures price is less than the spot price.

\[Contango=S_tWe select the nearest VIX and E-mini futures with at least ten trading days to maturity in the futures chains.

def OnData(self, data): # select the nearest VIX and E-mini S&P500 futures with at least 10 trading days to maturity # if the front contract expires, roll forward to the next nearest contract for chain in data.FutureChains: if chain.Key.Value == Futures.Indices.VIX: if self.front_VX is None or ((self.front_VX.Expiry-self.Time).days <= 1): contracts = list(filter(lambda x: x.Expiry >= self.Time + timedelta(days = 10), chain.Value)) self.front_VX = sorted(contracts, key = lambda x: x.Expiry)[0] if chain.Key.Value == Futures.Indices.SP500EMini: if self.front_ES is None or ((self.front_ES.Expiry-self.Time).days <= 1): contracts = list(filter(lambda x: x.Expiry >= self.Time + timedelta(days = 10), chain.Value)) self.front_ES = sorted(contracts, key = lambda x: x.Expiry)[0]

While this trading strategy takes advantage of the roll by selling VIX futures at a premium to the VIX and by buying VIX futures at a discount to the VIX, it is exposed to the potentially substantial risks associated with adverse moves in the VIX futures curve. However, as the tendency of VIX futures prices to move inversely to equity returns, much of this risk can be hedged by open the E-mini S&P 500 futures position in the same direction.

The number of mini-S&P futures contracts to buy or sell per VIX futures contract is based on the hedge ratio estimates. The hedge ratios are constructed from regressions of VIX futures price changes on a constant and on contemporaneous percentage changes of the front mini-S&P 500 futures contract both alone and multiplied by the number of business days that the VIX futures contract is from the settlement, as shown below.

\[\Delta P^{VX}_t=\beta_0+\beta_1* Return^{ES}_t+\beta_2*(Return^{ES}_t*TimeToSettlement^{VX}_t)+\mu_t\]After we get the parameters \(beta_0\), \(beta_1\) and \(beta_2\), the formula for the hedge ratio is

\[HR_t=\frac{\beta_1*1000+\beta_2*TimeToSettlement_{t-1}*1000}{0.01*P^{ES}_{t-1}*50}\]def CalculateHedgeRatio(self): price_VX = np.array(self.price_VX) price_ES = np.array(self.price_ES) delta_VX = np.diff(price_VX) res_ES = np.diff(price_ES)/price_ES[:-1]*100 tts = np.array(self.days_to_maturity)[1:] df = pd.DataFrame({"delta_VX":delta_VX, "SPRET":res_ES, "product":res_ES*tts}).dropna() # remove rows with zero value df = df[(df != 0).all(1)] y = df['delta_VX'].astype(float) X = df[['SPRET', "product"]].astype(float) X = sm.add_constant(X) model = sm.OLS(y, X).fit() beta_1 = model.params[1] beta_2 = model.params[2] hedge_ratio = abs((1000*beta_1 + beta_2*((self.front_VX.Expiry-self.Time).days)*1000)/(0.01*50*float(self.Securities[self.es1].Price))) self.Plot("Trade", "Hedge Ratio", hedge_ratio) return hedge_ratio

Our daily futures price data is from the continuous front contract. To measure the time to settlement, we import the custom data of the VIX futures expiration dates from 2011 to 2018. We populate the date index with the backward fill method to make it easy to calculate the time to settlement.

# import the futures expiry calendar url = "https://www.dropbox.com/s/5k4rbuzfsfn3w0h/expiry.csv?dl=1" df_date = pd.read_csv(url, index_col = 'date') # convert the index and expiry column to datetime format df_date.index = pd.to_datetime(df_date.index) df_date['expiry']=pd.to_datetime(df_date['expiry']) idx = pd.date_range('03-16-2011', '04-19-2019') # populate the date index expiry_date = df_date.reindex(idx, method='bfill') df = pd.concat([settle, expiry_date], axis=1, join='inner')

To perform the regression, we save the history price in deque list and update the list every day.

# the rolling window to save the front month VX future price self.price_VX = deque(maxlen=252) # the rolling window to save the front month ES future price self.price_ES = deque(maxlen=252) # the rolling window to save the time-to-maturity of the contract self.days_to_maturity = deque(maxlen=252) # initialize the deque list for index, row in df.iterrows(): self.price_VX.append(row[self.vx1]) self.price_ES.append(row[self.es1]) self.days_to_maturity.append((row['expiry']-index).days)

The daily roll is defined as the difference between the front VIX futures price and the VIX, divided by the number of business days until the VIX futures contract settles. Short VIX futures positions are entered when the VIX futures basis is in contango and the daily roll exceeds 0.10 and long VIX futures positions are entered when the VIX futures basis is in backwardation and the daily roll is less than -0.10.

# calculate the daily roll daily_roll = (self.Securities[self.vx1].Price - self.Securities[self.vix].Price)/(self.front_VX.Expiry-self.Time).days self.Plot("Trade", "VIX", self.Securities[self.vix].Price) self.Plot("Trade", "VIX Futures", self.Securities[self.vx1].Price) self.Plot("Trade", "Daily Roll", daily_roll) if not self.Portfolio[self.front_VX.Symbol].Invested: # Short if the contract is in contango with adaily roll greater than 0.10 if daily_roll > 0.1: hedge_ratio = self.CalculateHedgeRatio() self.Plot("Trade", "Sell", self.Securities[self.vx1].Price) self.SetHoldings(self.front_VX.Symbol, -0.5) self.SetHoldings(self.front_ES.Symbol, -0.5*hedge_ratio) # Long if the contract is in backwardation with adaily roll less than -0.10 elif daily_roll < -0.1: hedge_ratio = self.CalculateHedgeRatio() self.Plot("Trade", "Buy", self.Securities[self.vx1].Price) self.SetHoldings(self.front_VX.Symbol, 0.5) self.SetHoldings(self.front_ES.Symbol, 0.5*hedge_ratio)

Trades are exited when the motivating conditions no longer exist. The exit condition is defined as the daily roll being less than 0.05 for short trades and higher than -0.05 VIX futures points for long trades. If these exit conditions are not triggered, trades are exited two days before the contract expires.

# exit if the daily roll being less than 0.05 if holding short positions if self.Portfolio[self.front_VX.Symbol].IsShort and daily_roll < 0.05: self.Liquidate() self.front_VX = None self.front_ES = None return # exit if the daily roll being greater than -0.05 if holding long positions if self.Portfolio[self.front_VX.Symbol].IsLong and daily_roll > -0.05: self.Liquidate() self.front_VX = None self.front_ES = None return if self.front_VX and self.front_ES: # if these exit conditions are not triggered, trades are exited two days before it expires if self.Portfolio[self.front_VX.Symbol].Invested and self.Portfolio[self.front_ES.Symbol].Invested: if (self.front_VX.Expiry-self.Time).days <=2 or (self.front_ES.Expiry-self.Time).days <=2: self.Liquidate() self.front_VX = None self.front_ES = None return

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