Lotso Insurance Architecture | ChubGame Whitepaper

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3.1 Data collection

We collect historical data for 12 different cryptocurrencies that constitute the portfolio of interest. The data was collected in CSV format and included the daily closing prices of the cryptocurrencies over a defined historical period.

3.2 Calculation of Weights

The weight of each individual cryptocurrency is calculated as the ratio of its market capitalization to the total market capitalization of all cryptocurrencies in the portfolio. The max weight is set to 50% and assigned to the ETH.

(1)

where - the weight of the individual asset, - market capitalization of the individual asset, - total market capitalization of all cryptocurrencies in the portfolio.

3.3 Portfolio Value Modeling

In this step, we will model the portfolio value using a multivariate Jump-Diffusion model, which is a stochastic process that incorporates both continuous and discrete jump components to capture portfolio asset dynamics.

This model allows for the simulation of various scenarios for the price evolution of individual assets in the portfolio, considering factors such as volatility, drift, and correlation, as well as the occurrence of jumps in asset prices.

The multivariate Jump-Diffusion model can be represented by the following equations:

(2)

(3)

Here, represents the change in the price of the asset at the time , is the drift (expected return) of asset , is the volatility (standard deviation of returns) of asset 𝑖, denotes the differential of a standard Wiener process for the asset , is the jump size as a multiple of stock price, while is the number of jump events that have occurred up to time 𝑡. 𝑁(𝑡) is assumed to follow the Poisson process , where is the average number of jumps per unit of time. Where the jump size follows log-normal distribution

(4)

where is the standard normal distribution, is the average jump size, and is the volatility of jump size. The three parameters , , characterize the jump-diffusion model.

3.3.1 Parameters Estimation

Then we will estimate the parameters of the multivariate Jump-Diffusion model using the historical returns of the cryptocurrencies. The returns will be calculated as the natural logarithm of the ratio of successive daily closing prices.

(5)

where is the return at time , is the price at time , and is the price at time − 1.

The drift and volatility σ will be calculated from these historical returns. The drift will be the mean of the returns, mathematically represented as:

(6)

where is the return at time , and is the number of observations.

The volatility will be the covariance matrix of the returns, calculated as follows:

(7)

where is the vector of returns, and Cov denotes the covariance operation.

The jump amplitude () will be estimated as the mean of the returns that exceeds two standard deviations from the mean return.

(8)

where is the return at time , is the mean return, is the standard deviation of returns, and is the number of returns that exceed two standard deviations.

The jump frequency will be set as a fixed parameter, representing the average number of jumps per day.

3.3.2 Estimation Method

We will estimate the parameters using Maximum Likelihood Estimation (MLE).

The MLE method involves finding the parameter values that maximize the likelihood function, which measures the probability of observing the data given the parameters.

(9)

We will maximize the log-likelihood function with respect to and to obtain the MLEs of these parameters.

The estimation of the parameters will be done separately for each cryptocurrency, assuming independence between cryptocurrencies. The correlation between cryptocurrencies will be accounted for in the simulation step by using the covariance matrix of the returns to generate correlated random time series.

3.3.2 Simulation

To forecast potential price trajectories of cryptocurrencies over a predefined futures trading period (30 days), we will employ a Monte Carlo simulation. This simulation will be executed 10,000 times to derive a spectrum of potential portfolio values. Each iteration of the simulation will incorporate random jumps and standard price fluctuations, as modeled by the Jump-Diffusion model.

For each price path generated by the simulation, we will calculate the corresponding portfolio value and construct a distribution of potential future portfolio values. This distribution will shed light on a range of potential outcomes, elucidating the probability of realizing specific portfolio values and the risks involved. Such analysis is crucial for understanding the portfolio’s vulnerability to extreme market movements and for assessing the robustness of futures trading strategies.

(10)

Here, denotes the portfolio's value at the conclusion of the futures trading period (time + 1). The weight of each asset within the portfolio is represented by , and signifies the simulated price of each asset at that future point. The resultant distribution of portfolio values will be instrumental in risk management and strategic decision-making, enabling an evaluation of potential profits or losses over the selected time frame. This approach is vital for investors in navigating the volatile cryptocurrency market, particularly in the context of futures trading.

3.4 Futures Exposure Calculation

Utilizing the expected portfolio value distribution derived from the Monte Carlo simulation, the futures exposure of the portfolio will be calculated. Futures exposure refers to the sensitivity of the portfolio value to changes in the value of an underlying futures contract or portfolio.

To calculate this futures exposure, we will use a multivariate regression approach to estimate the portfolio's elasticity with respect to the ETH price, a key cryptocurrency, and the primary asset for which futures contracts are available.

(11)

where represents the expected daily returns of the portfolio, symbolizes the expected daily returns of cryptocurrencies, are the regression coefficients, and denotes the error term.

The estimated coefficient in the multivariate regression model indicates the portfolio's price elasticity, reflecting the sensitivity of the portfolio returns to changes in the returns of ETH.

This portfolio price elasticity (PE) offers critical insights into how the portfolio value responds to fluctuations in the ETH price. A higher elasticity suggests greater sensitivity of the portfolio to ETH price movements, whereas a lower elasticity indicates reduced sensitivity.

Now, using the estimated portfolio price elasticity in relation to ETH, we can compute the futures exposure (FE) of the portfolio. FE represents the variation in the portfolio value for a 1% change in the ETH price. This can be calculated as follows:

(12)

where is the futures exposure of the portfolio, denotes the current value of the individual assets, is the expected yield, and is the portfolio's estimated price elasticity in relation to the ETH price.

The calculation of futures exposure is critical for understanding the potential impact on the portfolio value due to ETH price changes. It provides invaluable information for formulating effective futures trading strategies, aimed at managing risks associated with price movements in ETH.

3.5 Futures Trading Techniques Simulation

This section simulates the performance of strategies involving futures trading, primarily focusing on short futures contracts, to assess their effectiveness in safeguarding the portfolio against adverse price movements. Due to the complexities in predicting futures prices, we employ the theoretical Jump-Diffusion Model for Futures Pricing. This approach enables us to identify the optimal futures trading strategy that offers an ideal balance between risk mitigation and trading costs.

Figure 1. Short perps and Futures option payoff diagrams

The strategies we design, particularly short futures and futures options, aim to shield the portfolio from negative price shifts and limit potential downside risk.

Short Futures involve a contract where the investor profits from a decline in the price of the underlying asset, in our case, Ethereum (ETH). This strategy offsets potential changes in the total portfolio value due to price movements of underlying assets.

The number of short futures required for full portfolio protection is determined by dividing the futures exposure by the contract size.

We simulate a strategy that involves initiating a short futures position at the start of the trading period and closing it at its end. The cost of this strategy is influenced by the funding rate required to maintain the position throughout the trading period.

The funding rate in a perpetual futures contract relates to the disparity between the futures price and the spot price. In bullish markets, the futures price often exceeds the spot price, leading to a positive funding rate, while in bearish markets, the futures price might be lower, resulting in a negative funding rate. This dynamic suggests a correlation between daily price changes and the funding rate.

A linear regression model is used to quantify this relationship, aiding in the prediction of future funding rates.

(13)

where f denotes the ETH perpetual futures funding rate, represents daily ETH returns, are the regression coefficients, and is the error term.

Futures Options involve acquiring options contracts that grant the right to trade the underlying futures at a predetermined price within a specific timeframe. Purchasing futures options for our portfolio serves as insurance against price declines. As the underlying futures' prices drop, the value of these options increases, counteracting any losses in the portfolio. The cost of this strategy is calculated as the total premium paid for the options contracts.

To protect the portfolio with futures options, we calculate the number of contracts needed by dividing the futures exposure by the contract size.

Strike Selection Process: For simplicity, we select the strike price equal to the current price of the underlying asset at the time of option purchase, known as at-the-money (ATM) trading. ATM options are chosen due to their balance in cost and potential payout, being more favorable in volatile markets like cryptocurrencies.

By selecting an ATM strike, we ensure that our futures options start gaining value immediately if the underlying asset price drops, providing prompt protection for our portfolio.

Futures Options Pricing: We estimate the theoretical prices of futures options using the Jump-Diffusion model.

In this model, the asset price follows a defined random process.

For European futures options, we can derive the price within the jump-diffusion model in terms of a modified Black-Scholes formula:

(14)

where and . The term represents the scenario of jumps occurring during the option's life.

The Black-Scholes Formula for Options Pricing:

(15)

where:

(16) (17)

here:

C - call option price for futures; S - current ETH/USDC price; K - strike price; r - risk-free interest rate; t - time to maturity; N - a normal distribution;

The corresponding put option price, based on put-call parity with a discount factor , is:

(18)

3.6 Optimal Futures Trading Compensation Strategy

The culmination of our analysis is the application of our devised futures trading strategies to the simulated scenarios generated from portfolio modeling. These strategies undergo rigorous evaluation, with a specific focus on balancing the highest expected value against the incurred trading costs. The objective is to pinpoint the strategy that delivers the most effective protection for the portfolio against potential adverse market movements while also optimizing the costs linked to the implementation and maintenance of the futures positions.

For the short futures strategy, a key component of the cost equation is the funding rate necessary to sustain the position over the trading period. The expense for upholding short futures is quantified as follows:

(19)

where represents the funding rate, is the size of the position, and is the number of days the position is held.

For futures options, the trading cost is the total premium paid for the options contracts. This cost is a crucial factor in determining the overall effectiveness of the strategy.

This comprehensive approach to identifying the optimal futures trading compensation strategy ensures a detailed examination of the available options. It encompasses not only the tangible aspects, such as cost and potential returns but also the intangible elements like strategy flexibility and robustness under different market scenarios. The selected strategy, therefore, is designed to effectively balance risk and reward, providing a sophisticated and effective solution for managing portfolio risk amidst the complexities and uncertainties of the digital asset market. This optimal strategy is geared towards safeguarding the portfolio's value while also striving to maximize returns.