8.2 Estimated Shape Invariant Pricing Kernels
8.2.2 Maturity in 18 Days
Figure 9: Pricing Kernels estimated on different observation Dates. Maturities of the individual PKs: 2020-04-24, 2020-06-26,2020-07-31,2020-10-09.
SIM
Figure 10: Convergence of the Shape Invariant Pricing Kernel. Final Iteration.
SIM
Figure 11: Estimated Parameters for horizontal and vertical shift.Theta 1,Theta 2, Theta 3,Theta 4.
SIM
9 Trading on Density Deviations
Knowledge of the SPD unveils investor preferences for all possible Bitcoin price realizations at a pre-specified point in time. The SPD has a forward-looking nature since it contains information about investor preferences and their expectations about the future. Acknowledging that investors may misjudge, over- or underemphasize certain outcomes, the argument can be made that a simulation from a neutral point of view may yield better results. If this holds true, then a profitable trading strategy might be found that may counteract shortcomings of investor valuations.
The estimated PKs indicate that investors are willing to insure themselves against large, negative returns. This willingness is measurable by the substantial premia for FOTM puts that are being offered and traded. Since SPDs and PDs are probability densities, they are formally required to be strictly positive and integrate to one. This is verified for all estimated densities. If investor’s are willing to insure themselves against large, negative returns, then this implies additional probability mass on one end of the tail, which means a comparative lack of probability mass somewhere else on the domain. To continue with the example, if investors put too much weight on left-sided fat tail events, their behavior may be counteracted by shorting FOTM puts and buying ATM puts (and vice versa for calls). As a result, a simple trading strategy may be constructed, consisting of a call spread and a put spread. Two spreads are traded on the evening of each day when the SPD and the PD have been calculated for that day and a fixed time-to-maturity𝜏. Both are held until maturity and standard Deribit transaction costs are applied.
On the left-hand side of Figure 12 the SPD is compared to the PD. The right-hand
call spread and a put spread can be traded, hence if at one instruments exist in each of the designated four regions on the moneyness domain.
Recall that moneyness 𝑚is defined as the ratio of spot to strike price. Usually, a SPD is depicted as a density over a range of strike prices. In order to ensure comparability of the PKs, moneyness is chosen as the domain. Since the end-of-day spot price is used to calculate moneyness, an increase in moneyness is actually associated with a decrease in wealth (negative return). Vice versa, a decrease in moneyness is associated with an increase in wealth (positive return).
For example the densities observed on 2020-03-18, depicted on Figure 14, reveil that investors are willing to pay a substantial premium for puts, as the difference in the right tail shows. Naturally the payoff of the trades committed on each day varies due to degree of density deviations as well as available instruments and direction (long/short). One example of a payoff function is depicted on Figure 13 for the densities observed on 2020-05-18, where time-to-maturity is four days.
Figure 12: State Price Density vs. Physical Density on 2020-05-18 for 4 Days until Maturity.
Strategies
Figure 13: Payoff Function for the Trading Strategy on 2020-05-18 for 4 Days until Maturity. Denoted in USD.
Strategies
Figure 14: State Price Density vs. Physical Density on 2020-03-18 for 2 Days until Maturity.
Strategies
Figure 15: State Price Density vs. Physical Density on 2020-05-25 for 2 Days until Maturity.
Strategies
Across all observations and from the beginning of March until mid December, a set of SPDs, PDs and eventually PKs has been created. For each PK, the discussed strategy is employed. A total of 53 trading opportunities have been identified and simulated. The rather small amount of trading simulations indicates that the SPD and PD do not allow for too large deviations, which leads to the conclusion that market inefficiencies can rarely be found using these methods.
The Profit and Loss distribution (PnL) is estimated by means of a Gaussian kernel density estimator applied to the absolute profits (Virtanen et al. 2020). Deribit (2020d) charges a fee of 0.015% on the delivery of options. As all options in the simulated strategy are held until maturity, the fee is subtracted at maturity. As the PnL distribution shows, the expected value is positive, but comes at the price of a large variance. The Sharpe Ratio, which summarizes the performance of an investment strategy over the risk free rate, is 0.1598. The risk-free rate is set to zero.
Despite the large amount of orderbook observations, few trading opportunities could actually be identified. In conjunction with the low Sharpe Ratio we can conclude that the market is relatively efficient since the SPD and the PD are often too close to create a valid trading strategy in the defined manner. Those opportunities that are identifiable, create a positive payoff at such a high cost (standard deviation) that more profitable and less risky strategies should exist.
While the proposed trading strategy performs relatively well, the risk is too high to actually be preferable over other trading strategies. Nevertheless, a variety of modifications can be implemented in order to counter the shortcomings. Comparing the results with Huynh, Kervella, and Zheng (2002a), a common property of
such an extend that it is very likely to find less risky strategies, which come at a similar payoff.
Figure 16: Profit and Loss Distribution of the Trading Simulation. Denoted in USD.
PnL
10 Conclusion
A novel data set containing high-frequency orderbook snapshots of the major Bitcoin Derivatives exchange Deribit has been analyzed in this thesis. The market is found to price contracts reasonably in terms of implied volatility. State Price Densities are estimated for various instruments in order to allow arbitrage-free pricing for arbitrary options and the provided Quantlets allow practicioners to use the provided results. A Stochastic Volatility with Correlated Jumps framework is fitted and found to adequately describe the Bitcoin asset process. Pricing kernels are calculated and evaluated. Those, which share a common time-to-maturity also share a common shape, which is summarized in the Shape Invariant Pricing Kernel. This allows to study the evolution of pricing kernels over time as well as their common features.
It also provides a link to study investor’s absolute risk aversion. A trading strategy has been designed based on deviations between the physical density and the State Price Density. As the low Sharpe Ratio indicates, the expected profit does not outweigh the risk.
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Declaration of Authorship
I, Julian Winkel, hereby confirm that I have written this thesis independently. All sources, that have contributed to this thesis are marked as such. This work has neither been submitted to any examination nor published before.
19.02.2021