Description

Anchor Labs
1 Problem Setup
Volly, the volatility trader runs proprietary trading algorithms in various crypto-currency markets. Volly’s favorite market is the BTC/USDT pair, which has very high volume offering good liquidity and generally high volatility. We model the daily performance of Volly’s algorithms as an output Ot ∈ {−1,+1}. A simple stochastic model for the performance of Volly’s algorithms is a Bernoulli random variable with two separate regimes: low volatility days and high volatility days. We can treat the daily returns as a suitable proxy for intraday volatility leading to the following model:
w.p θL
(1)
, w.p 1 − θL
w.p θH
. (2)
, w.p 1 − θH
Here Rt is the daily arithmetic returns at time t as
. (3)
Volly’s algorithms generally perform well on on days with large price swings and and perform worse on days of smaller price swings, so we expect θL < θH. In this exercise, we use this model to estimate the performance of a particular market makers Profit and Loss (PNL) defined here as the cumulative algorithm output over time.
2 Questions
You are given a file containing the following columns: (1) Time; (2) BTC/USDT price; (3) Volly’s aggregate PNL.
1. Load the provided file and visualize the following elements:
(a) Price series: BTC/USDT price vs. time
(b) Returns series: use arithmetic returns as opposed to geometric.
(c) PNL series vs time.
(d) Daily output series Ot.
2. Given returns Rt and daily algorithm output Ot, propose a methodology to estimate the following parameters: λ = {α,θL,θH}. Formulate a reasonable objective function.
3. Propose a methodology for arrive at the parameters of interest.
4. (a) How does your methodology scale with the length of the time-series: give an O(T) estimate? (b) Do you believe your methodology is optimal in terms of parameter estimation and time complexity? (c) How would you modify your algorithm if the data was not a single file but distributed between various machines. (This would not be done on daily data but much more granular series).
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