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by CMarch 4th, 2020

Specifically, this article will look at various options strategies that can capitalize on BTC’s volatility clustering (i.e. heteroscedasticity) and compare these strategies to how well a simple buy & hold portfolio performed. At the end, I will cover portfolio optimization and diversification in order to show that, overall, the best performing portfolio comes from a combination of the option strategies and buy & hold.

This article should not be looked to for investment advice. Instead, this article is interested in studying the dynamics between daily and weekly volatility. From these dynamics, we can gain insight into how to quantify risk and possible ways to mitigate such risk. However, before that can be looked at, it's necessary to dive deeper into how we can think about volatility and volatility clustering.

“Heteroscedasticity” sounds intimidating, but it’s not overly complicated. Simply put, heteroscedasticity refers to non-constant conditional variance of error terms in a model (e.g. a linear regression model). “Error terms” in this case refers to the distance between the projected line of best fit for a stock’s returns and the observed returns when making a regression model of stock returns and time.

In this sense, when a data set shows heteroscedasticity, it means that there is a time varying condition for the value of each error term so that the change between returns is not constant.

To get a little more specific, in an effort to show what heteroscedasticity looks like, let *r* denote the daily returns for a time series (stock). “Variance” refers to the measure of dispersion (variability), in the data. Variance is denoted as sigma (*σ*) squared. The variance of returns can be given as:

This depiction of variance however is unconditional. The variance given above does not change. This is a simplistic depiction of variance because daily returns are rarely static. Rather, daily returns for any asset tend to vary a lot. Thus, a conditional variance must be given in order to capture the changes between daily returns. This can be given by:

In the above representation of conditional variance, 𝝮 denotes the last observation of daily returns for the given set. As can be parsed from the equation above, the conditional variance is now given for period “*t*.” Yet, even though the variance is conditional upon the which return at period *t *is being looked at, it still could be the case that the variance is constant.

This could happen when the changes between daily returns are constant. This phenomenon is homoscedasticity, where the variance of *r* is the same regardless of which period of time *r* is observed.

**Figure 1**. BTC daily returns of 2019

For BTC, the variance of *r* is definitely not the same for any given period of time. As can be seen from Figure 1, a scatter plot of BTC daily returns for 2019 reveals that there’s plenty of periods with low returns followed by dramatic spikes.

Besides just a high degree of variability in the returns, the scatter plot also shows how many observations are clustered together. This is indicative of heteroscedasticity. From Figure 1, there appears to be some correlation between *r *at time *t *and *r* at *t*-1.

This correlation is important because it suggests that today’s returns are not completely random, but rather are influenced by yesterday’s, the day before yesterday, or even the previous week’s returns. Thus, from BTC’s returns, there is some evidence that volatility clusters depending on time varying conditions.

In order to see volatility clustering more clearly, a correlogram can be plotted to identify which lags (previous observations) are significantly correlated to *r* at time *t*.

In Figure 2, such a correlogram is given whereby the autocorrelation of absolute daily (log) returns of BTC for 2019. The reason why absolute returns are used, instead of just plotting the autocorrelation of log returns, is because we’re interested in measuring variance, not just the returns. Thus, negative values are treated as positive so as to see how correlated deviations are from some line.

So, when imagining that a line is given in Figure 1 right in the middle of the plot through the 0% percent mark, an autocorrelation of absolute log returns is treating points of 10% and -10% as simply .01 because that’s the variance (deviation) of those points.

The logic for this comes from the idea of volatility clustering, in that, high volatility days follow previous high volatility days and, likewise, low volatility days follow previous low volatility days. Such volatility though may result in positive or negative returns.

**Figure 2.**ACF plot of absolute returns

Ideally, for a correlogram, the bars should stay below the dotted blue line. This indicates that no lag is significantly correlated to *r* at time *t*. In Figure 2, many lags cross the dotted blue line, which represents a 95% confidence interval in order to account for white noise. Thus, from Figure 2, it appears that BTC returns are not random.

However, to confirm the suspicion that BTC returns are serially correlated and that BTC’s conditional variance is time-varying, an autocorrelation plot can be given for permuted (randomly ordered) BTC 2019 returns. In Figure 3, BTC’s returns for 2019 are randomly sorted in order to confirm that the serial correlation in Figure 2 is time sensitive.

As can be seen from Figure 3, no lag crosses the dotted blue line and any evidence of volatility clustering has disappeared. Figure 3 clearly shows that how BTC’s returns are ordered matters. In this sense, it should be safe to conclude that heteroscedasticity is present and therefore BTC’s returns are influenced by previous returns.

**Figure 3**. ACF plot of permuted absolute returns

Knowing that heteroscedasticity is present in BTC’s returns is important because specific options strategies can be tailored to take advantage of it. For simplicity, we will focus on option strangles and straddles. Briefly, an option strangle is when an out-the-money (OTM) put option is bought in addition to an OTM call option.

An option straddle is when at-the-money (ATM) put and call options are bought. Importantly, both strangles and straddles can be short positions. This means that instead of buying the options, an investor is on the other side of the trade and sells the options. Figure 4 shows the payoff diagrams from long strangles and straddles as well as short strangles and straddles.

**Figure 4**. Payoff diagram for strangles and straddles

As for implementing the strategies, the autocorrelation of BTC returns comes in handy. This is because we know that lags of 1 to 5 days are most strongly correlated with *r* at time *t*. Thus, BTC’s daily volatility clustering follows, roughly, a weekly time frame.

However, as shown in Figure 5, BTC’s weekly returns are not serially correlated. From this, it can be assumed that weekly returns are more independent than daily returns. So, if the previous week showed a significant spike in volatility, the correlogram suggests that the current week isn’t influenced by such spike.

Since BTC’s weekly volatility shows no clustering, let’s assume that it is mean reverting. Mean reversion is the idea that, over time, an asset’s volatility will return to an average volatility for a given time-frame. This assumption is not baseless, but it also can’t be proven.

The reason why volatility is mean reverting is because if it wasn’t (i.e. if it could be truly random and not influenced by agent’s expectations), then for an arbitrarily large time scale, volatility could likely be some order of magnitude larger or smaller than the initial observation. Since that prospect seems unlikely, it’s easier to believe that volatility reverts to its mean.

In Figure 6, BTC’s weekly volatility for 2018 is displayed as a histogram. The mean, denoted by the dotted blue line, is 10.7%. For the option strategies, it will be assumed that the volatility taken at the beginning of each week will revert to the mean.

So, if the weekly volatility is 5% at the beginning of the week, long strangles and straddles will be used in the hope that volatility will increase. Conversely, if the volatility is 15%, short strangles and straddles will be used in the hope that volatility will decrease.

**Figure 5.**ACF plot of absolute weekly returns

It should be noted that there are ways to model volatility more accurately than using the simple assumption above. Namely, because heteroscedasticity is present, an autoregressive model like ARCH or GARCH may be preferred.

However, such models are beyond the scope of this article. Rather than actually modeling volatility, the option strategies presented herein are only concerned with how we can delineate between when it’s best to buy or sell options given what we know about BTC’s volatility at the beginning of the week.

Now that it’s been established when to buy or sell a straddle or strangle, it’s necessary to determine which strike prices will be used for the strangles.

Since long and short straddles concern buying or selling options at the strike price nearest to the current underlying price, choosing the strike price will be straightforward. For strangles, the strike prices are OTM. For this strategy, the spot price at the beginning of the week will be multiplied by the weekly volatility in order to give the estimated upper and lower strike range.

Thus, if the spot price is $5000 at the beginning of the week and the historical volatility is 10%, the upper strike price will be $5500 and the lower will be $4500.

**Figure 6**. 2018 BTC weekly volatility

In addition to assuming weekly volatility mean reversion and daily volatility clustering (indicating that when volatility spikes, it will be sustained), there remains some less significant details to flesh out. First, since access to historical option prices was limited, I will assume that OTM options cost $25 each and ATM options cost $200 each.

This assumption is rooted in observations of BTC's option chain on Deribit. Figure 7 shows the option chain for the week of December 27th. Second, I will assume that option payouts will be denoted in USD. Third, at least 4 contracts will be bought or sold every week. Fourth, the option contracts will be held for the entire week until expiration. And fifth, the differences between strike prices will be increments of $250.

The data for the option strategies is collected in a google sheet that can be found here. Observations of the spot price at the beginning of the week and at expiration were taken from BitMex, as well as the 7 day historical volatility.

To begin, Figure 8 shows the performance for the long and short strangles.

**Figure 8**. Strangles performance

Notably, the short strangles, while not all that profitable, were successful. As can be seen by Figure 4, short strangles and straddles can be risky. In fact, their risk is infinite. However, by placing the OTM strikes at the end of the range calculated by multiplying the historical volatility by the spot price, the estimated upper and lower prices ranges were correct 100% of the time.

The reason for this is probably because the weekly volatility, when over the 2018 mean, was often very high. The average weekly volatility for weeks over 10.7% was 15.4%. The highest weekly volatility was 22.8%. By assuming mean reversion, the strategies expect that these spikes in volatility will not be sustained in the following week.

Figure 9 depicts the yearly performance for long and short straddles. The straddles were significantly riskier than the strangles. Contrary to the short strangles, the short straddles were not successful. This is because the payoff for short straddles only provides a very limited range of profitability.

As opposed to short strangles, short straddles require that the price remains virtually unchanged at the end of the week. The largest loss (-$1126) occurred during the week with the highest volatility (22.8%). Whereas the short strangle provided a comfortable range (+/- >$2500), the straddle was only profitable if the price stayed within a range of a couple hundred dollars.

**During that week, the price changed by $1433. **

The long straddles were also much more risky. In the end however, the long straddles outperformed the long strangles ($8388 and $7724, respectively). The most successful week for the long straddles was right before the highest volatility week. At the beginning of the week, the historical volatility was 10.6% - representing a 12.2% difference. Such dramatic changes in volatility were not uncommon.

For a handful of weeks, the change in weekly volatility was almost double. This can be attributed to volatility clustering. Even though it's assumed that weekly volatility is mean reverting, the daily volatility still shows signs of heteroscedasticity.

This means that changes in volatility are not smooth and gradual; rather, BTC's volatility 'jumps' and volatility clustering sustains such jumps. Because of this, long straddles (and strangles) worked well in this study.

To get a perspective on how well the strangles and straddles did, Figures 10 and 11 compare the option strategies with a 'buy and hold' strategy. The buy and hold (B&H strategy assumes that you buy 1 bitcoin at the beginning of 2019 and sell on the December 27th (at the end of the last full week in 2019). The total profit for the B&H strategy was $3374.

**Figure 10**. Strangle portfolio relative to B&H

While the B&H portfolio benefited greatly from the spike in BTC's price during the summer of 2019, Both the strangle and straddle portfolios outpaced B&H as the year wore on. This is because the B&H portfolio is only able to take advantage of positive returns. Naturally, if the asset you're holding drops in value, so does your portfolio.

The option strategies presented here are neutral on the direction of BTC's price. As long as BTC's volatility expands (for long options) or contracts (for short options) the strategies will benefit, regardless of whether that means BTC will go up or down.

However, this does not mean that they're without risk. Rather, as Figure 11 shows, the strategies may experience extensive losses. Mitigating this risk will be looked at in the next section.

**Figure 11**. Straddle portfolio relative to B&H

Imagine that you're faced with the decision to invest in three assets, *A*, *B*, and *C*. These investments can either represent single assets (like shares of Google) or entire portfolios (like shares in multiple companies). In the case of the three assets before you, you know that *A* has an expected return of 7% and expected risk of 10%, *B *** **has an expected return of 5% and expected risk of 10%, and *C *has an expected return of 8% and expected risk of 11%.

In order to maximize returns and minimize risk, you should prefer *A *over *B *and *C * over *B. *However, from this information alone, it's not clear that you should prefer *C* over *A. *Even though *C* is expected to perform better, the risk is also higher.

Given the dilemma of *A *and *C*, it may be beneficial to split your investment between them. In the event that both *A* and *C *aren't perfectly correlated, diversification can reduce overall risk - thereby making a portfolio with higher expected returns than *A *and lower expected risk than *C*. Assuming that borrowing money isn't an option, the new portfolio (combining *A* and *C*) must apply weights to each asset that sums to 1.

Thus, weight ω will be applied to *A* and ω*C *will equal 1-ω*A*. This can be expressed as 0 ≤ ω*A *≤* *1, 0 ≤ ω*C *≤ 1, and ω*A *+ ω*C* = 1. The respective allocation of *A *and *C* in the new portfolio can take on as many combinations as desired.

The range of combinations for the new portfolio can then be plotted in order to find the most efficient allocation. In a similar manner, a new portfolio made of the strategies presented above can be examined by simulating the risk-reward payoff for many different combinations and plotting them.

*Before a new portfolio can be made, it's necessary to find the risk-return tradeoff of each strategy. This is done by calculating the expected returns and the expected risk (volatility) of each strategy. *

For the expected returns, simply calculate the mean of returns each strategy had for every week. It should be noted that the predictive utility of expected returns should not be taken as dogma. Expected returns are taken from historical performance. Historical performance is not indicative of future performance.

However, expected returns are used for the risk-reward payoff so that each strategy's performance can be measured and compared. Next, to find volatility, calculate the standard deviation (sigma) of the returns. With the expected returns and the volatility of each strategy, it's now possible to plot the risk-return tradeoff.

**Figure 12**. Risk-return payoff

In Figure 12, the risk-return tradeoffs for each portfolio is given. As indicated by the red dot, the straddle portfolio provides the highest expected return. However, it also has a higher volatility than B&H. Likewise, the strangle portfolio has a higher volatility than B&H while also having much worse expected returns.

*Thus, the strangle portfolio is the least preferred of the three. Though, it's not necessarily clear that the straddle portfolio is the most preferred. Whereas B&H's expected returns are much lower, the lower risk is attractive. *

In any event, B&H (in regards to this study) should not be ignored. In this respect, a combination of both B&H and straddles would appear to give the most efficient balance between risk and reward (a look at all three will also be given).

As can be seen from Figure 11, B&H and the straddle portfolio are slightly correlated. At times, as B&H rises, so does the straddle portfolio. Yet, they are not perfectly correlated. This means that diversification between the two will reduce risk.

Creating a portfolio of both B&H and the straddle strategy can give the preferred balance between expected returns and volatility. In order to see the preferred balance, many different combinations must be simulated.

This can be done by applying weights ω to the two strategies. Letting B&H be *x* and the straddle portfolio be *y*, with the covariance of *x *and *y *be *σ x*,*y, *the possible expected risks can by found by:

Additionally, the possible expected returns can be found by:

Using the *x* and *y* weights, 1000s of different combinations can be simulated in order to give a representation of the "efficient frontier."

The efficient frontier is the upper edge of the line given in Figure 13. The lower edge is the inefficient frontier. This is because it's possible to create portfolio combinations that have the same volatility as others, but give higher expected returns.

In Figure 13 the ω*X* represents the weight applied to B&H. On the inefficient frontier, the lighter blue line starts with a portfolio comprised of only B&H. As was already known, B&H had much lower expected returns compared to straddles. However, a portfolio only made up of straddles had a higher risk.

In Figure 13, it can now be seen that allocating the majority of capital in straddles while also leaving some for B&H makes it possible to achieve the lower risk of a 100% B&H portfolio as well as slightly lower expected returns than a 100% straddles portfolio.

*This means that a portfolio combining B&H and straddles is more "efficient" than just B&H. *

Yet, this characterization of efficient prioritizes expected returns. What if you just wanted to reduce overall risk? For that, adding a third asset, the strangles, will be useful.

Adding a third asset requires the same process as above, but in addition to *x* and *y*, let strangles be *z*. Thus, the possible expected risks for a portfolio combining all three can be found by:

Similar to what's presented above, the possible expected returns can be found by:

By creating a portfolio where the expected returns and volatility are calculated by applying different weights to the three assets, the area of possible portfolios becomes three dimensional.

**Figure 14**. Diversification between all three portfolios

Figure 14 shows the possible portfolio combinations of all three strategies. Similar to Figure 13, the blue color represents straddles (with 100% straddles being at the top right).

*In the bottom right corner, red represents 100% strangles. *

For the middle point, yellow represents 100% B&H. By combining all three strategies it can now be seen that the portfolio is able to reduce risk the most. In fact, a mixture of all three strategies can provide the same expected returns as B&H while dramatically reducing risk from >90% to ~60%.

Bitcoin's volatility can be erratic. As has been laid out above, it is far from uniformly dispersed. Rather, BTC experiences frequent volatility clustering that suggests BTC's returns are not independent and random. Even though randomness and normal distributions may be preferred when it comes to financial markets, the presence of volatility clustering can be factored into investment strategies.

This article has presented two strategies that use options in order to take advantage of BTC's volatility clustering. While these two strategies, overall, preformed better than simply buying and holding BTC, that's only half of the story. When trying to balance expected returns and risk, combining all three strategies becomes preferable.

Just using one can be inefficient. In this sense, risk can be better managed by knowing the characteristics of an asset's volatility and diversification of strategies.

This article benefited greatly from the tremendous work by David Zimmermann and his blog.

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