Geometric modeling and computational methods to detect shock events in cryptocurrency markets and to propose alternative portfolio allocations

1. Shock detection and portfolio allocation in
cryptocurrency markets
Apostolos Chalkis 1, 2
1Dept Informatics & Telecommunications
National & Kapodistrian U. Athens, Greece
2ATHENA Research & Innovation Center, Greece
June 2, 2021

2. Shock event detection (Part I)
Copula representation.
Define an indicator.
Detection of shock events according to the value
of the indicator.
[Chalkis, Christoforou, Emiris ’21]

3. Stock markets’ types of behavior
Financial markets exhibit 3 types of behavior e.g. [Billio,
Getmansky,Pelizzon’12]:
1 In normal times, stocks are characterized by slightly positive
returns and a moderate volatility.
2 In up-market times (typically bubbles) by high returns and low
volatility.
3 During financial crises by strongly negative returns and high
volatility.
These observations motivate us to describe the time-varying
dependency between portfolios’ returns and volatility.

4. Geometric representation of portfolios
Set of portfolios
{x ∈ Rn
|
P
i xi = 1, xi ≥ 0}
The set of portfolios can be seen
as the simplex/polytope in Rn
.
Ptf x= (0.3, 0.25, 0.45)

5. Portfolio return and volatility
Ptf x= (0.3, 0.25, 0.45)
– Asset returns R ∈ Rn
– R = (0.09, 0.13, −0.038)
– Return Rx = RT
x = 18%
– Covariance matrix Σ ∈ Rn×n
– Σ =


3.66 0.25 −0.66
0.25 6.36 0.50
−0.66 0.50 0.18


– Volatility Vx = xT
Σx = 73%

6. Consider k consecutive days
Date . . . asset j asset j + 1 . . .
day i . . . Ri,j Ri,j+1 . . .
day i + 1 . . . Ri+1,j Ri+1,j+1 . . .
.
.
. . . . . . . . . . . . .
day i + k − 1 . . . Ri+k−1,j Ri+k−1,j+1 . . .
– Compute compound return
– Cut the simplex with mqqqa df
e hyperplanes
– Equidistant in terms of volume
– Here m = 3
–Estimate the covariance matrix
– Cut the simplex with mqqqa df
e ellipsoids
– Equidistant in terms of volume
– Here m = 3

7. Computing the copula of a given time period
– Cut the simplex with bothqqqa
q hyperplanes and ellipsoids
– Compute the volume of the q
d bodies defined by the qdfxcvxc
d intersections
→
– Obtain a Copula
– Bivariate distribution
– Each marginal is uniform
Apostolos Chalkis Practical Volume Computation & Crises detection

8. Q: How to detect a shock event?
We use the daily returns of the 12 cryptocurrencies with longest
history.
16th December 2017 15th March 2020
Left Copula corresponds to normal and right copula to crises times.
⇒ Normal times: The mass of portfolios on the up diagonal.
⇒ Crisis times: The mass of portfolios on the down diagonal.

9. Answer: Define an indicator I
I :=
mass in red area
mass in blue area
When I > 1 for > 60 days then we report a shock event.

10. Detecting shock events
When I > 1 for > 60days we report a shock event.
When I > 1 for > 30days we report a short shock event.

11. Alternative portfolio allocations (Part II)
Directions:
Use an alternative score to evaluate a portfolio.
Define new performance measures based on the score.
Compute the corresponding optimal portfolios.
Compare them with mainstream portfolios.
[Calès, Chalkis, Emiris, ’21]

12. Score of a portfolio
Ptf x= (0.3, 0.25, 0.45), Asset returns R = (0.09, 0.13, −0.038)
ewrtertsdfhsdgh Ptf return Rx = RT x = 18%
Score of x:=percentage of the Ptfs with worse return than x.
0 ≤ score of a Ptf ≤ 1.

13. The score as a random variable
Given a portfolio x ∈ Rn (use Monte Carlo),
Assume a distribution for the asset returns R ∼ D, and
estimate the distribution of the scores of x (sample from D), or
Use in-sample observations of asset returns R1, . . . , RN

14. New performance measures
f (x) the PDF of the score distribution of a Ptf.
Define 4 performance measures, e.g.,
PerfA =
R 1
0 (x − S∗)f (x)dx
qR 1
0 (x − S∗)2f (x)dx
=
µ − S∗
σ
,
PerfB =
R 1
0 xf (x)dx
qR 1
0 x2f (x)dx
=
µ
σ
,
...

15. Allocation exercise
We consider the daily returns of the 12 cryptocurrencies with
longest history.
We let a sliding window (s.w.) of length W = 100 days.
We shift the s.w. by one day.
For each s.w. we compute the following Ptfs (average
volatility):
The optimal Markowitz Mean-Variance Ptf (OptMV).
The closest single asset Ptf on BTC (OptBTC).
The optimal PerfA, PerfB, PerfC, PerfD (in-sample).
We evaluate the portfolios with the (W + 1)th asset returns.

16. Score and Return distributions
All portfolios are difficult to be distinguished in Markowitz
framework (very similar average return and variance).
Both OptMV and OptBTC the most of the times among the
best or the worst performers.
PerfA, PerfB, PerfC, PerfD are more stable in terms of score
(new concept of risk).

17. Diversified Portfolios
PerfA, PerfB, PerfC, PerfD are much more diversified than
both OptMV and OptBTC.

18. Performance Evaluation

19. Performance Evaluation

20. Performance Evaluation

21. Open Source Code
The computational tools are implemented in package
volesti of Open Source Org. Geomscale.
github.com/GeomScale/volume_approximation.
geomscale.github.io/.