Vix Term Structure Lows and Vix returns N Days Later

As of yesterday (25’th january 2017) Vix Term Structure (1mo/3mo implied vol) closed at 0.794. Taking a quick look at what low Vix Ts values have meant for Vix returns going forward and run a bare bones backtest to gauge the initial validity of low Ts as a signal

import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
from quantopian.interactive.data.quandl import cboe_vix, cboe_vxv
from odo import odo
import datetime
import pyfolio as pf

xiv = get_pricing(symbols("XIV"), fields="close_price",
    start_date=datetime.date(2011,1,1),
    end_date=datetime.date(2017,1,25))

dfvix = odo(cboe_vix, pd.DataFrame)
dfvix = dfvix.set_index(dfvix["asof_date"])
dfvxv = odo(cboe_vxv, pd.DataFrame)
dfvxv = dfvxv.set_index(dfvxv["asof_date"])

df = pd.concat([dfvix, dfvxv], axis=1, join_axes=[dfvix.index])
df.sort(inplace=True)
df = df.rename(columns={"close":"vxv_close"})
df = df[["vix_close", "vxv_close"]]
df["vix_ts"] = df["vix_close"] / df["vxv_close"]

First looking at just the overall distribution of absolute Vix Ts values, readings below .8 are rare. Data starts from 2008

ecd = np.arange(1, len(df)+1, dtype=float) / len(df)

plt.figure(figsize=(10, 10))
plt.plot(np.sort(df["vix_ts"]), ecd, linewidth=1, color="#555555")
plt.axvline(0.794, linestyle="--", color="crimson", linewidth=1)
plt.xlabel("Vix Ts values")
plt.ylabel("Percent of Vix Ts values that are smaller than corresponding x")
plt.grid(alpha=0.21)

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Rather than using an absolute Vix Ts threshold as a signal or gauge to determine how low is low, i tried to use a rolling min value, meaning making the Vix Ts value relative. The threshold is simply defined as Ts – Rolling21 Min of Ts. Took two months as the trail amount, since it seems a significant enough time for Ts to make new lows

df["min_ratio"] = df["vix_ts"] - df["vix_ts"].rolling(42).min()
df.dropna(inplace=True)

def formatPlot(x, where):
    x.grid(alpha=0.21)
    x.legend(loc=where)

fig, (ax1, ax2) = plt.subplots(2, sharex=True, figsize=(16, 10))
ax1.plot(df["vix_close"], label="Vix", linewidth=1, color="#555555")
mins = df.index[df["min_ratio"] == 0]
ax1.vlines(mins, ymin=df["vix_close"].min(), ymax=90, color="crimson",
 linewidth=0.55, alpha=0.55, label="VixTs hits rolling 2 month min")

formatPlot(ax1, "center left")

ax2.plot(df["min_ratio"], color="#555555", linewidth=1, label="VixTs - rolling 2 month min")
formatPlot(ax2, "center left")

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Now that the threshold is defined, one can look at the Vix returns going forward from the 2 month low being hit. Added two projected periods of 21 and 10 days to see if theres any significant difference. Average Vix returns going forward from Vix Ts hitting new 2 month lows are positive only enough so on occasions where Vix Ts itself is below 0.8 when hitting a new 2 month low

def retsShift(values, amount):
ret = (values.shift(-amount) / values) -1
return ret

df["ret21"] = retsShift(df["vix_close"], 21)
df["ret10"] = retsShift(df["vix_close"], 10)
df2 = df.copy() # Making a copy, otherwise masking will go crazy, probably doing masking wrong?
df2 = df2[df2["min_ratio"] == 0]

slope, intercept, r_val, p_val, std_err = scipy.stats.linregress(
np.array(df2["vix_ts"]), np.array(df2["ret21"]))
predict = intercept + slope * df2["vix_ts"]

plt.figure(figsize=(10, 10))
plt.plot(df2["vix_ts"], predict, "-", label="r2 = {}".format(format(r_val**2, ".2f")))
plt.scatter(df2["vix_ts"].where(df2["vix_ts"]<=0.8), df2["ret21"].where(df2["vix_ts"]<=0.8),
c="crimson", linewidth=0, s=28, alpha=0.89, label="Vix return 21 days later")
plt.scatter(df2["vix_ts"].where(df2["vix_ts"]<=0.8), df2["ret10"].where(df2["vix_ts"]<=0.8),
c="royalblue", linewidth=0, s=28, alpha=0.89, label="Vix return 10 days later")
plt.scatter(df2["vix_ts"].where(df2["vix_ts"]>0.8), df2["ret21"].where(df2["vix_ts"]>0.8),
c="#666666", linewidth=0, alpha=0.89, label=None)

plt.axhline(0, linestyle="--", color="#666666", linewidth=1)
plt.axvline(0.794, linestyle="--", color="crimson", linewidth=1, label="Vix Ts as of 25 Jan 2017")
plt.legend(loc="upper right")
plt.xlabel("Vix Ts level at which it hits a new 2 month low")
plt.ylabel("Vix return N days later")
plt.yticks(np.arange(-0.5, 1.5, 0.25))
formatPlot(plt, "upper right")

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Next, looking at Vix behaviour, while filtering out only instances where Ts was below 0.8 when hitting its 2 month low.  Which is what we have in Vix Ts as of this writing

But also added mean of all instances where Ts was above 0.8

for index, row in df3.iterrows():
if row["min_ratio"] == 0 and row["vix_ts"] <= 0.8:
ret = df3["vix_close"].iloc[index:index+22]
ret = np.log(ret).diff().fillna(0)
ret = pd.Series(ret).reset_index(drop=True)
df_instances[index] = ret.cumsum()

for index, row in df3.iterrows():
if row["min_ratio"] == 0 and row["vix_ts"] > 0.8:
ret = df3["vix_close"].iloc[index:index+22]
ret = np.log(ret).diff().fillna(0)
ret = pd.Series(ret).reset_index(drop=True)
df_instances2[index] = ret.cumsum()

plt.plot(df_instances, color="#333333", alpha=0.05, linewidth=1, label=None)
plt.plot(df_instances2, color="#333333", alpha=0.05, linewidth=1, label=None)
plt.plot(df_instances.mean(axis=1), color="crimson", label="Mean Vix returns where Ts <= 0.8")
plt.plot(df_instances2.mean(axis=1), color="royalblue", label="Mean Vix returns where Ts > 0.8")
plt.axhline(0, linewidth=1, linestyle="--", color="#333333")
plt.xlabel("# Of days from Vix Ts rolling 2 month low")
plt.ylabel("Vix % return")
plt.xticks(np.arange(0, 22, 1))
plt.ylim(-0.2, 0.4)
plt.xlim(0, 21)
formatPlot(plt, "upper right")

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However, since one can not directly trade Vix, the results are spurious, meaning when applying the signal to buy Vxx when Vix Ts hits a new 2 month low and holding it for N days (21 in the case below), the results are horrible. The reason being that Vxx (or any long volatility etf) experiences siginificant decay when Vix futures are in contango (which is most of the time in a bull market), so that holding it (just it alone) is not viable in any way. It can be useful however as a hedge in a more reasonable position sizing and allocation context

Just to make the contango effect on Vxx plain, also added backtest version which does the opposite, meaning it goes into Xiv when Vix Ts is hitting new 2 month lows and holds for 21 days

However adding the Xiv version illustrates the Vix futures contango benefits on Xiv (at least while the underlying trend in Spx is up). The backtest’s are no way anything viable, i just added them in order to gauge validity of the raw signal itself.

One could run an optimizer in order to determine the best possible Ts min trailing threshold, but i wont do that. In my limited experience, if a signal is not immediatley evident, it wont help massaging it to fit the curve

bt_vxx = get_backtest("588a07ff3d6aed61f57ca491").daily_performance.returns
bt_vxx_rets = pf.timeseries.cum_returns(bt_vxx, starting_value=1.0)

bt_xiv = get_backtest("588a069f66ba2e6145ca1aca").daily_performance.returns
bt_xiv_rets = pf.timeseries.cum_returns(bt_xiv, starting_value=1.0)

plt.figure(figsize=(16, 8))
plt.plot(bt_vxx_rets, color="royalblue", label="Strategy trading Vxx")
plt.plot(bt_xiv_rets, color="crimson", label="Strategy trading Xiv")
plt.plot((xiv.pct_change().cumsum()+1), label="Xiv b&amp;h", color="#555555")
formatPlot(plt, "center left")

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I will look further into the underlying signal though, with a more reasonable position sizing and trade logic and will post a follow up once im somewhere with it

If anyone can pitch in regarding any mistakes, misconceptions or any other comments, please do – Thanks for your time

Discretionary trades update

Scaling out of Vix short (via long Xiv) position that i have held since election day
Also starting to scale out of Tesla long that i acquired below 200, however will just reduce position and looking to add again once it takes a breather

Have smaller open positions in
Short miners
Short biotech

Recent closed positions
Short natgas

Vix single day spikes & their historical returns

Taking a look at Vix single day spikes, since there have been two rather significant and rare single day spikes of 39 and 49% in 2016. Note, im not talking about Vix daily swings, but single day spikes from close to close. The intention here is to gauge how Vix has historically behaved after significant single day spikes

First, importing modules, vix data etc. Im running this on Quantopian so importing data is straightforward


from quantopian.interactive.data.quandl import yahoo_index_vix
from odo import odo
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns
import statsmodels.api as sm
import scipy as sp

Setting up and cleaning the vix dataframe that came from quandl and adding the single day % change values

data = odo(yahoo_index_vix, pd.DataFrame)
data = data.drop(["open_", "high", "low", "adjusted_close", "volume", "timestamp"], axis = 1)
data["close"].loc[18] = 14.04 #Fixing error data
data = data.set_index(["asof_date"])

data["pct_change"] = data["close"].pct_change()
data = data.dropna()

This years second biggest spike was 39%. There have been only 12 single day spikes of 39% or greater since 1990 (again, from close to close)

len(data[data["pct_change"] &gt; 0.39])

12

When they occurred and what their actual spike percentages were

data[data["pct_change"] &gt; 0.39].sort()

screen-shot-2016-12-09-at-18-26-13

First, looking at all Vix daily percent changes as an empirical cumulative distribution, so one gets better idea how the daily percent changes are distributed. In this post, we are interested in the positive daily percent changes. As one can observe, spikes above 20% are rather rare

ecd = np.arange(1, len(data)+1, dtype=float) / len(data)
ticks = np.arange(-0.3, 1.1, 0.1)

plt.plot(np.sort(data["pct_change"]), ecd)
plt.xlabel("Vix daily percent changes (from close to close)")
plt.ylabel("Fraction of daily pct changes that are smaller than corresponding x")
plt.axvline(linestyle="--", color="#333333", linewidth=1)
plt.xticks(ticks)
plt.yticks(ticks)
plt.grid(alpha=0.21)
plt.margins(0.05)

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We can now plot the returns N days later from a spike, iterations are for 10% and 20% spike returns 5 days later in order to have a large enough sample set. I wanted to see weither or not the spike % was correlated to the forward returns, it seems to be the case. The larger the single day spike, the more likely a negative Vix return down the road. Though it must be noted that the sample set for spikes larger than 20% is low

def pct_ret(close, amount):
rets = (close.shift(-amount) / close) - 1
return rets

ret10 = pct_ret(data["close"], 5).where(data["pct_change"] > 0.1).dropna()
pct10 = data["pct_change"][data["pct_change"] > 0.1]

ret20 = pct_ret(data["close"], 5).where(data["pct_change"] > 0.2).dropna()
pct20 = data["pct_change"][data["pct_change"] > 0.2]


slope, intercept, r_val, p_val, std_err = sp.stats.linregress(pct10, ret10)
ret10_predict = intercept + slope * pct10


plt.plot(pct10, ret10_predict, "-", label="Linreg")
plt.scatter(data["pct_change"][data["pct_change"]>0.1], ret10,
color="#333333", alpha=0.55, label="Vix spike >= 10%, return 5 days later")
plt.scatter(data["pct_change"][data["pct_change"]>0.2], ret20,
color="crimson", s=34, label="Vix spike >= 20%, return 5 days later")

plt.ylabel("Vix % return 5 days later")
plt.xlabel("Vix single day spike % (from close to close)")
plt.axhline(linestyle="--", linewidth=1, color="#333333")
plt.legend(loc="upper right")
plt.ylim(-0.5, 1)
plt.grid(alpha=0.21)
plt.title("R2={}".format(r_val**2))

 

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For example, mean Vix return after a 20% single day spike or greater, 5 days later is about -16%

np.mean(ret_20[ret_20 &lt; 0])

-0.16629674607687678

For a clearer picture on how Vix actually looks like after significant spikes, we can also plot the N day returns of all instances where a significant spike occurred (in the chart below, its 64 trading days). There are notable rebound tendencies at the 10th, 20-23rd and 40th trading days after a spike. The higher spike means are more pronounced since the sample size is rather smaller on those instances

data2 = data.copy().reset_index()

def rets(df, days, pct, pct_to):
ret_df = pd.DataFrame()
for index, row in df.iterrows():
if row["pct_change"] &gt; pct and row["pct_change"] &lt; pct_to and df["pct_change"].iloc[index-1] &lt; pct:
ret = df["close"].iloc[index:index+days]
ret = np.log(ret).diff().fillna(0)
ret = pd.Series(ret).reset_index(drop=True)
ret_df[index] = ret
return ret_df

twenty = rets(data2, 65, 0.2, 0.3).mean(axis=1).cumsum()
thirty = rets(data2, 65, 0.3, 0.4).mean(axis=1).cumsum()
forty = rets(data2, 65, 0.4, 1).mean(axis=1).cumsum()

plt.plot(twenty, color="royalblue", label="Vix mean return after a spike &gt; 20% and &lt; 30%")
plt.plot(thirty, color="crimson", label="Vix mean return after a spike &gt; 30% and &lt; 40%")
plt.plot(forty, color="cadetblue", label="Vix mean return after a spike of &gt; 40%")

plt.xlabel("# Of days from spike")
plt.ylabel("Vix % return")
plt.grid(alpha=0.21)
plt.ylim(-0.3, 0.1)
plt.xlim(0, 64)
plt.legend(loc="upper right")

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The mean returns chart is desceptive, since there are of course plenty of instances where Vix just keeps going up, so one can get a better picture by looking at all the instances. In the case below, i plotted all spike instances of 20% or greater, 64 days forward

plt.plot(rets(data2, 65, 0.2, 1).cumsum(axis=0), linewidth=1, alpha=0.21, color="#333333")
plt.plot(forty, color="crimson", label="Mean")

plt.title("All instances of Vix singe day spikes &gt; 20%, returns 64 days forward")
plt.xlabel("# Of days from spike")
plt.ylabel("Vix % return")
plt.grid(alpha=0.21)
plt.axhline(0, linewidth=1, linestyle="--", color="#333333")
plt.xlim(0, 64)
plt.legend(loc="upper right")

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One additional way of looking at the dataset is to make a heatmap of all single day spike instances, meaning we plot out all Vix single day spikes and their mean returns, regardless of the size of the spike or the direction of the spike. First converted the pct changes to integers and then grouped all the data by those. From that a heatmap of mean returns for all Vix spike % instances can be summoned

Nothing meaningful happens in the middle of the % change range, but the edges are more pronounced, however again its worth noting that the sample size on the edges is also smaller

df3 = data.copy()

for i in range(1, 35):
df3[str(i)] = pct_ret(data3["close"], i)

df3["pct_change"] = df3["pct_change"].apply(lambda x: int(round(x*100)))
df3.reset_index(inplace=True)
df3.drop(["asof_date","close"], axis=1, inplace=True)

grouped = df3.loc["1":].groupby(df3["pct_change"], as_index=True, squeeze=True).mean()
grouped.drop("pct_change", axis=1, inplace=True)

plt.figure(figsize=(16, 13))
sns.heatmap(grouped, annot=False, cmap="RdBu")
plt.ylabel("Vix single day spike %")
plt.xlabel("Number of days after spike")
plt.title("Vix single day % spikes vs. mean returns N days later")

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If anyone can pitch in regarding any mistakes, misconceptions or any other comments, please do – Thanks for your time

100 Years of dow jones returns

A quick look at annual returns over the 100+ years of daily percent change (close to close) data that we have on dow jones

import matplotlib.pyplot as plt
import pandas as pd
import numpy as np
import datetime

dj = local_csv("DjiaHist.csv", date_column = "Date", use_date_column_as_index = True)
dia = get_pricing("DIA", start_date = "2016-01-01", end_date = datetime.date.today(), frequency = "daily")

First cleaning up the data, especially the dates. Also adding day of the year into the df in order to sort all returns based on the day of the year and plot em all at once later

dj.sort_index(ascending=True, inplace=True)
dj.index = pd.to_datetime(dj.index)
dj.rename(columns={"Value" : "value"}, inplace=True)
dj["pct"] = np.log(dj["value"]).diff()
dj["year"] = dj.index.year
dj["day"] = dj.index.dayofyear

dia["day"] = dia.index.dayofyear
dia["pct"] = np.log(dia["price"]).diff()
dia = dia.drop(["open_price", "high", "low", "volume", "close_price"], axis=1)
dia.set_index(dia["day"], inplace = True)
dia.fillna(0, inplace=True)

Pivoting the returns table, so that we get returs for all years and all days of the year in separate columns

daily_rets = pd.pivot_table(dj, index=["day"], columns=["year"], values=["pct"])
daily_rets.convert_objects(convert_numeric = True)
daily_rets.fillna(0, inplace = True)
daily_rets.columns = daily_rets.columns.droplevel()
daily_rets.drop(2016, axis =1, inplace = True)
daily_rets.rename(columns = lambda x: str(x), inplace=True)

daily_rets.head(8)

screen-shot-2016-11-07-at-12-28-50

Heres how it looks with all years plotted along with 2016

f, ax = plt.subplots(figsize=(18, 12))
ax.plot(daily_rets.cumsum(), color="#333333", linewidth=1, alpha=0.1, label=None)
ax.plot(dia["pct"].cumsum(), linewidth=2, color="crimson", label="2016 returns")
plt.grid(False)
plt.ylabel("Annual return")
plt.xlabel("Day of the year")
plt.ylim(-0.7, 0.7)
plt.xlim(0, 365)
plt.axhline(0, linewidth= 1, color="#333333", linestyle="--")
plt.legend(loc="upper left")

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Adding the mean returns of all years so one can compare with 2016.
Also added a daily returns histogram so the historical day to day fluctuatios are more clear and positive or negative periods are painted out clearly

daily_rets["mean"] = daily_rets.mean(axis=1)
daily_rets["2016"] = dia["pct"]

plt.figure(figsize=(18, 12))

ax1 = plt.subplot2grid((4,1), (0,0), rowspan=3)
ax1.plot(daily_rets.index, daily_rets.cumsum(), color="#333333", linewidth=1, alpha=0.06, label=None)
ax1.plot(daily_rets["mean"].cumsum(), color="#333333", linewidth=2, alpha=0.8, label="Mean returns since 1896")
ax1.plot(daily_rets["2016"].dropna().cumsum(), linewidth=2, color="crimson", label ="2016 returns")
plt.title("Cumulative 2016 Returns Vs Mean Historical Returns Since 1896")
plt.axhline(0, linewidth= 1, color="#333333", linestyle="--")
plt.ylim(-0.15, 0.15)
plt.grid(False)
plt.legend(loc="upper left")

ax2 = plt.subplot2grid((4,1), (3,0), rowspan=3, sharex=ax1)
ax2.fill_between(daily_rets.index, 0, daily_rets["mean"], where= daily_rets["mean"]<0, color="crimson")
ax2.fill_between(daily_rets.index, daily_rets["mean"], 0, where= daily_rets["mean"]>0, color="forestgreen")
plt.title("Mean Daily Returns")
ax2.grid(False)
plt.xlim(1, 365)

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Now that the show is over, its time to look at returns around elections. Up to 1936 the votes were cast in early january, from there on the vote has been in early november, so i used returns from 1936 onward in the calc. Also plotted the mean return of post-election year

daily_rets["el_year"] = daily_rets.loc[:, "1936"::4].mean(axis=1)
daily_rets["post_el"] = daily_rets.loc[:, "1937"::4].mean(axis=1)

f, ax = plt.subplots(figsize=(18, 12))

ax.plot(daily_rets.index, daily_rets.cumsum(), color="#333333", linewidth=1, alpha=0.06, label=None)
ax.plot(daily_rets["mean"].cumsum(), color="#333333", linewidth=2, alpha=0.8, label="Mean returns since 1896")
ax.plot(daily_rets["el_year"].cumsum(), color="darksage", linewidth=2, alpha=0.8, label="Election year mean returns since 1936")
ax.plot(daily_rets["post_el"].cumsum(), color="steelblue", linewidth=2, alpha=0.8, label="Post election year mean returns since 1936")
ax.plot(rets_df["pct"].dropna().cumsum(), linewidth=2, color="crimson", label ="2016 returns")
plt.grid(False)
plt.ylabel("Annual return")
plt.xlabel("Day of the year")
plt.ylim(-0.15, 0.15)
plt.xlim(1, 365)
plt.axhline(0, linewidth= 1, color="#333333", linestyle="--")
plt.legend(loc="upper left")

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We can also pull up decade returns. 80’s and 90’s were good times indeed. Applied a 21 day mean to the returns to the trends would be more clear

def decadeMean(start, end):
return daily_rets.loc[:, start : end].cumsum().mean(axis=1)

decade_rets = pd.DataFrame({#"1900’s" : decadeMean("1900", "1909"),
#"1910’s" : decadeMean("1910", "1919"),
#"1920’s" : decadeMean("1920", "1929"),
#"1930’s" : decadeMean("1930", "1939"),
#"1940’s" : decadeMean("1940", "1949"),
#"1950’s" : decadeMean("1950", "1959"),
#"1960’s" : decadeMean("1960", "1969"),
"1970’s" : decadeMean("1970", "1979"),
"1980’s" : decadeMean("1980", "1989"),
"1990’s" : decadeMean("1990", "1999"),
"2000’s" : decadeMean("2000", "2009"),
"2010’s" : decadeMean("2010", "2015")
}, index= daily_rets.index)

mean_rets = decade_rets.rolling(21).mean()

plt.figure(figsize=(18, 12))
mean_rets.plot(linewidth=1)
rets_df["pct"].dropna().cumsum().rolling(21).mean().plot(color="crimson", linewidth=2, label="2016")
plt.legend(loc="upper left")
plt.ylabel("Annual return")
plt.xlabel("Day of the year")
plt.grid(False)

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The most revealing thing about this to me, is that the day to day fluctuations havent really changed over 100+ years – market still behaves the same.

For example, if we randomly reshuffle the order of daily returns of 1910 and compare it to 2015 reshuffled daily returns, its impossible to say which one is which. The nature and behaviour of day to day fluctuations is still the same.

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If anyone can pitch in regarding any mistakes, misconceptions or any other comments, please do – Thanks for your time

Regarding election

What i think is happening is that the Brexit vote is fresh on speculators minds, the polls were dead even before the vote but few actually believed the exits would win. Perhaps we are seeing something similar with us elections right now, meaning (almost) all predictions and polls point to Clinton (Stay), but a surprise like Trump (Exit) weighs on peoples minds.

Here are some of the good research and analysis done
FiveThirtyEight Elections
NY Times 2016 Election