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Original | "Kurtosis and Bitcoin: A Quantitative Analysis"

Here we introduce the concept of a Gaussian random walk, which is the underlying assumption used in the Black-Scholes option pricing model. This option pricing model regards the time interval of asset price changes as an independent variable, and assumes that the price or asset return changes over time to obey a normal distribution, in other words, transactions are uniformly distributed in various time periods, daily, weekly or The monthly trading volume is huge, so according to the Central Limit Theorem (Central Limit Theorem), these prices will follow a normal or Gaussian distribution. When the return distribution of an asset is a normal distribution, the probabilities of different return situations are known. Understanding these probabilities can provide investors with an idea to better quantify the risks that may arise when holding these assets.

On this basis, we can't help thinking, can this model be applied to Bitcoin, a new type of asset? The sharp rise and fall of Bitcoin is a well-known fact, and there is no debate here. This paper aims to explore how to construct a risk framework and examine the application of the assumptions implicit in the pricing of traditional financial derivatives to Bitcoin.

This article will initially introduce the derivatives market, outline the Black-Scholes model, discuss the importance and scope of application of the model, analyze its limitations based on the unrealistic assumptions of the model, and discuss its feasibility in the Bitcoin market sex. Based on historical data such as Bitcoin’s daily returns from January 2016 to August 2019, we compared the results of the Black-Scholes model applied to Bitcoin and the Standard & Poor’s 500 Index (S&P500). Finally, the conclusion that "the Black-Scholes model may not be applicable to the cryptocurrency market" is drawn, and some inspirations for the rapidly growing token derivatives market are drawn from this conclusion.

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Derivatives and Hedging Risk

Suppose you are a corn farmer and you want to harvest 5,000 bushels (about 127 tons) of corn and sell as much of it as possible. However, the price is affected by the supply and demand situation in the market, and the selling price of corn may be lower than the production cost, and the application of financial derivatives can minimize the losses caused by these situations.

The above examples explain the role of financial derivatives. Of course, this derivatives portfolio can become more complex when the full range of futures, options, swaps, etc. are taken into consideration. The basis of all these portfolios is that markets and prices reflect risk and uncertainty, and derivatives will minimize this uncertainty.

Armed with this basic understanding, we have a special consideration for the price of any derivative product. The prerequisite for derivatives to play a role is that they can represent the actual hedging of the uncertainty of the underlying object. How to use options for effective investment is a question worth thinking about.

The actual risk of an option is actually reflected in the actual price of the underlying. In the above example, if the put option was priced at 2 dollars instead of 10 cents, corn would still be at 3.50 cents. Then through the Black-Scholes model, it can be calculated that the volatility of corn prices at this time is higher than 200% (see note), this figure is unusual for the agricultural market, based on this, your expectation of the future price of corn Also changes. Second, even if your expectations remain the same, buying the put at $2 will greatly reduce your profit margin, and if the price of corn falls below $3, then you will lose money because of the option premium. Third, if your expectations change and the implied volatility in corn prices is credible, the risk of losing money on producing corn at $1 per bushel becomes significant. Therefore, the validity of option pricing is crucial, which reflects the market's expectations for the future.

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Black-Scholes model

The pricing process of options contracts is actually quite mechanical. As we all know, the Black-Scholes model plays a very important role in option pricing and hedging. At the same time, investors and exchanges also use this model to determine the Greeks or calculate δ, Vega, θ in options and other portfolios , equal partial derivatives of γ. These partial derivatives are of great help to the risk management of exchanges/brokers, and they are coefficients to measure the price sensitivity of derivatives. For example, when Deribit, a large encrypted derivatives exchange, is liquidating high-risk positions, their risk engine is actually creating a "(delta neutral)" hedging position, allowing the positive and negative deltas to cancel each other out, so that the portfolio value does not change. Affected by changes in the price of the underlying asset.

The news of the listing of these new derivatives exchanges can’t help but trigger our thinking: Can the Black-Scholes model play a role in Bitcoin’s risk management? If so, how useful is this?

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Figure 1: Lognormal distribution (left) vs. normal distribution (right)

Figure 2: Probability Distribution of Option Strike Prices

The higher the volatility, the larger the area of ​​the normal distribution curve and the higher the option price. Therefore, option prices can be thought of as a probability distribution.

If the volatility is very stable and a stock is 100% above the strike price of the call option or 100% below the strike price of the put option at the option expiration, then the option has no value. In fact, from the perspective of hedging, it does not make sense to choose options at this time, because there is no risk to hedge. Or assuming that there is a 50% chance that the stock will be higher than the strike price of the call option or 50% likely lower than the strike price of the put option at the expiration of the option, then this option is valuable because it can Investors are attracted to buy options to hedge the risk of holding the underlying stock.

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The Black-Scholes model is by no means perfect. To some extent, the Black-Scholes model's assumptions about the market do not match the actual situation. Through this model, traders only need to input parameters such as strike price, remaining time to maturity, underlying asset price, underlying asset volatility, and risk-free interest rate to obtain the corresponding option price.

Among the above parameters, the values ​​of four parameters can be obtained accurately from the market, and only the underlying asset price volatility needs to be estimated. Instead, the model assumes that volatility is not only constant, but also known in advance. This assumption is problematic because volatility itself can be erratic. The CBOE created the Victim Index (VIX), which refers to the implied volatility of the S&P 500 index for the next 30 days. In 2018, the panic index (VIX) fell as low as 8.5% and as high as 46%. Volatility is therefore not always consistent throughout the year.

The accuracy of the Black-Scholes model will also be affected by changes in the market. When the financial markets collapsed in 1987, the derivatives markets were also affected. Prior to 1987, there was not much relationship between implied volatility and strike price, with out-of-the-money puts and in-the-money calls having roughly the same volatility. However, in 1987 there was a creepy "volatility smile". As shown in Figure 3, when the current price of the option deviates from the strike price, the implied volatility of the option rises, showing a middle low A smiling mouth with high sides. For different financial options, the shape of implied volatility is also different. Generally speaking, the volatility curve of stock options may be skewed, which is called volatility skew.

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This skew can signal panic in the market. If a put option has a much higher implied volatility than a call option, this can be explained by a disproportionate amount of investors hedging against downside risk.

In the example in Figure 3, the chart shows that the S&P 500 exhibits a negative volatility skew. One reason is that more and more investors prefer to buy in-the-money call options instead of buying stocks directly, because buying options can bring about a leverage effect, that is, paying an option fee equivalent to a part of the stock price can enjoy the same share price. The benefit of the increase, and buying stocks needs to take up funds equivalent to 100% of the stock price. The result of this is that the return rate for investors will be improved, so the market demand for call options in the price of the stock will increase, and the implied volatility level of call options with lower strike prices will rise.

Thus, although the normal distribution curve of the Black-Scholes model gives equal probabilities at both ends, in reality the stock options market tends to behave more pessimistically. Interestingly, the Bitcoin market is far more optimistic in comparison.

Figure 4: Bitcoin options table with expiry date December 27, 2019

Figure 4 shows Bitcoin derivatives expiring on December 27, 2019 on the Bitcoin options platform Deribit. It can be seen that two prices (7,000 and 13,000) that deviate from the current price of Bitcoin (10,000+) by the same degree show different implied volatilities: the implied volatility of the put option bought at 7000 (right side) The rate (IV) is 86.6%, while the 13,000 call option (on the left) has a slightly higher IV of 90.2%. This shows that out-of-the-money puts are far less valuable than out-of-the-money calls, and while this options table is not representative of the entire Bitcoin options market, it also shows that a considerable number of speculators/investors underestimate the downside risk.

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Unpredictability of kurtosis

Kurtosis is the flatness of the data distribution. The data distribution with a large tail has a large kurtosis value, which reflects the risk characteristics of the future tail. The sample excess kurtosis formula is:

The kurtosis is:

Deviation from the mean (difference between each random variable X and the mean of all values) of the daily return is necessary when calculating the kurtosis of an asset's return distribution. This deviation can be expressed as:

In statistics, moments describe the shape of a probability distribution. Generally speaking, the first-order distance and the second-order moment represent the mean and variance of the distribution, respectively, and the third-order moment represents the skewness (Skewness). As mentioned earlier, skewness is a measure of how asymmetric or skewed a distribution is. The fourth distance reflects the sharpness of the distribution and alters the curve of the normal distribution in different ways. It can be expressed as:

Figure 5: The shape of a normal distribution with the same variance and positive kurtosis

Kurtosis can be used to measure risk, and here we temporarily ignore basic assumptions such as "random walk". Finding the kurtosis of returns over any given time frame can give investors an idea of ​​how volatility is distributed. Different risk profiles can be described by whether asset returns are normally distributed or not. Most people in the investment community choose to treat volatility and risk as the same thing, arguing that the more volatile an asset is, the more risky it is. Conversely, the less volatile an asset is, the safer it is. However, this volatility/risk dualism ignores the nature of volatility and even lumps returns with a normal distribution into the "risk" category.

When we restrict the return to obey the normal distribution, then the probability of different return situations is known. For example, assuming that the edges of the normal distribution of an asset's returns reach -50% and 50%, this asset will be considered extremely unstable, but if the return is subject to a normal distribution, then the curve can be drawn The tails and margins of are 2 and 3 standard deviations from the mean, respectively. Once this information is known, investment strategies can be adjusted around this possibility, and even very volatile assets can be traded like less volatile ones. So instead of confusing risk and volatility, let them establish an orthogonal relationship that can serve as a volatility risk compass.

Figure 6: Orthogonal relationship between volatility and risk

In Figure 6, it is assumed that the vertical axis is the predictability of the price, and the horizontal axis is the knowability of the probability distribution of returns. In this figure, the upper quadrant is the unpredictable price, which represents "random walk"; the left quadrant is the known probability distribution of the rate of return, which means that the rate of return obeys a normal distribution.

In Figure 7, the upper left corner represents the assets in the Black-Scholes model based on the "ideal" assumptions. Such assets go through a "random walk", the price is unpredictable, and the rate of return is normally distributed, so the probability distribution is known. The lower right corner represents the opposite of the Black-Scholes model. The price of an asset is predictable, but the probability distribution is unknowable. This asset can be considered to be manipulated. Whether it is through insider trading or technical analysis in general, the price is Fully predictable, but the probability of occurrence is uncertain. The price of this asset can be fully manipulated over time, eliminating the need to predict the probability distribution of returns. The price in the lower left corner can be predicted, and its probability can also be known. Such an asset can be considered "stable", there should be no price deviation, and the future rate of return is also known. Finally, the top right corner represents assets that follow a "random walk" but have an abnormal distribution of return probabilities.

Figure 7: Volatility-Risk Compass

With the Volatility-Risk Compass, it is possible to paint a clearer picture of an asset's risk profile. On this basis, kurtosis can be used to determine which quadrant an asset falls into. The kurtosis of the asset quantifies the risk of the option because excess kurtosis means that the pricing by the Black-Scholes model is not necessarily reliable.

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Bitcoin kurtosis

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Figure 8: Bitcoin daily returns 2016 (left), 2017 (middle), 2018 (right)

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Figure 9: Bitcoin Daily Returns in 2019

So far (September 2019), Bitcoin’s kurtosis has not decreased. On the contrary, compared with 2018, it has slightly increased, and the excess kurtosis is 3.92. While the probability distribution of daily returns within the year is larger near its mean, the probability distribution is relatively uniform at its tail. This exhibits a classic positive kurtosis with thicker tails and a wider range of values ​​on either side of the mean than a normal distribution.

Figure 10: 2016-2019 Bitcoin excess kurtosis

Overall, the excess kurtosis shows that the probability skew of Bitcoin’s daily returns using the Black-Scholes model will be larger than expected with both mean and tails. Pricing options can become very difficult as implied volatilities become less reliable. Excess kurtosis means that most price changes become unpredictable and volatility does not follow a normal distribution.

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Comparison and Conclusion

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Figure 11: Changes in S&P 500 Kurtosis (Source: Financial Times)

The histogram in Figure 12 shows that the daily return distribution of the S&P500 is closer to a normal distribution than Bitcoin. Although some daily returns far exceed the normal distribution curve, there is obviously excess kurtosis. But overall, only 6 of these 250 observations for the S&P500 fall outside the normal distribution curve. Of the 364 observations of Bitcoin’s daily return in 2017, 28 fell outside the normal distribution curve. According to the data comparison, we found that the S&P500 daily yield curve has a very fast decline rate, and the possibility of extreme price fluctuations is low, which makes it difficult for people to predict extreme changes in events, and the fluctuations are unpredictable.

Figure 12: S&P 500 Daily Returns in 2018

So what enlightenment can we get from it? While it’s true that the S&P 500 doesn’t fall into the normal distribution curve, it does fit a more normal distribution curve than Bitcoin. The specific reasons for this result are yet to be determined, but I think there are at least three possibilities. First, Bitcoin represents a different asset class than the broader stock market and obeys different underlying assumptions. Secondly, the current Bitcoin market is still immature and lacks the management and control of specialized investment institutions. Third, in general, the reliability of the Black-Scholes model is doubtful, and the unpredictability of fluctuations has become the "new normal".

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notes

From Brenner and Subrahmanyan's "A Simple Formula for Computing Implied Standard Deviation" and the Black-Scholes model we get:

σ=√(2π/5)*(2/3.5)

σ=202.73%

Using the above formula we get: