Category: Uncategorized

• The Science and Practice of Trend-following Systems: paper and presentation

  Posted at 5:35 pm by artursepp, on June 22, 2025

  I would like to introduce the updated draft of my paper co-authored with Vladimir Lucic and entitled “The Science and Practice of Trend-following Systems”.

  Trend-following systems have been employed by many quantitative and discretionary funds, also known as commodity trading advisors (CTAs), or managed futures, since the early 1980s. Richard Dennis, a commodity trader on the CME, organised and instructed two classes of novice traders in late 1983 and 1984 with the idea that trading skills can be taught. The underlying ideas and methods included strict adherence to rule-based trading and risk-management. A few graduates of these classes created their own quantitatively-driven CTA funds and gave the rise of managed futures industry.

  Lintner in 1983 provided the first evidence that managed futures deliver better risk-adjusted returns and offer strong diversification benefits for long-only portfolios. The following passage is from Lintner 1983, The Potential Role of Managed Commodity-Financial Futures Accounts (and/or Funds) in Portfolios of Stocks and Bonds:
  “The combined portfolios of stocks (or stocks and bonds) after including judicious investments in appropriately selected sub-portfolios of investments in managed futures accounts (or funds) show substantially less risk at every possible level of expected return than portfolios of stock (or stocks and bonds) alone. This is the essence of the ‘potential role’ of managed futures accounts (or funds) as a supplement to stock and bond portfolios suggested in the title of this paper.”

  Subsequent studies reinforce the role of managed futures for the diversification of broad long-only portfolios, so that currently many private and institutional portfolios have some exposure to managed futures.

  The purpose of this paper is to provide both theoretical and practical insights about trend-following (TF) systems. Let me note that practitioners refer to implemented systematic futures-based strategies as systems or programs.

  Theoretical insights

  For theoretical insights, we establish regimes in which TF systems perform well. For any systematic strategy, it is important to understand under which market dynamics it is expected to out-perform or under-perform.  We derive an exact analytical formula linking the performance of the TF system to the autocorrelation of instrument returns under generic processes. We show that the TF system is expected to be profitable when the autocorrelation of returns is positive even if the drift is zero. We also show that the TF system is expected to be profitable for a white noise process with a large positive or negative drift if the filter span is large.

  For the illustration of obtained analytical results, we focus on fractional ARFIMA process which incorporates both short- and long-term mean reversion and / or trend features, which allows for extensive profitability analysis of TF systems.

  In Figure 1, we illustrate that the TF system can be profitable if the fractional order is positive, so the dynamics are trending in the long-term. In this case, the TF system can be profitable even if the short-term dynamics are mean-reverting.

  Figure 1. Panel (A) shows analytical value of expected annual return of TF system and MC confidence intervals using ARFIMA process with positive long-term memory with fractional order $d=0.02$, which implies long-term mean-reversion, with AR-1 feature phi={-0.05, 0.0, 0.05, and with zero drift. Panel (B) shows expected value of volatility-adjusted turnover and corresponding MC 95% confidence interval.

  

  In Figure 2, we illustrate that if the fractional order is negative and dynamics are mean-reverting in the long-term, the TF system can still be profitable if drift is present and span of the filter is large.

  Figure 2. Panel (A) shows analytical value of expected annual return of TF system and MC confidence intervals for ARFIMA process with negative long-term memory with fractional order d=-0.01. with AR-1 feature phi={-0.05, 0.0, 0.05} and with drift mu=0.5 (interpreted as Sharpe ratio) . Panel (B) shows expected value of volatility-adjusted turnover and corresponding MC 95% confidence interval.

  

  Practical Insights

  We have considered three distinct approaches for the construction of trend-following (TF) approaches which we term as European, American, and Time Series Momentum (TSMOM) systems. In Figure 3, we show the simulated performance of three TF systems assuming 2%/20% management/performance fees.

  Figure 3. Simulated performance of of European, American and TSMOM systems along with the historical performance of SG Trend Index. Panels (A1), (B1), and (C1) show the cumulative log-performance, running drawdown, and EWMA correlations with one year span. Panel (A2i) shows risk-adjusted performance table with P.a. returns being annualised return or CAGR, Vol being annualised volatility of daily log returns, Sharpe (rf=0) being Sharpe ratio, Max DD being the maximum drawdown; Skew being the skewness of quarterly log-returns, beta and R^{2} being the the slope and R^{2} of the linear regression of monthly returns relative to 60/40 equity/bond portfolio. Panel (A2ii) shows annual returns. Panels (B2) and (B3) show one year rolling volatility-adjusted turnover and cost, respectively. The background colour is obtained by ordering the quarterly returns of the benchmark 60/40 portfolio from lowest to highest and the splitting the 16% of worst returns into the “bear” regime (pink colour), 16% of best returns into the bull regime (dark green colour), and remaining regimes into “normal” regimes (light green colour). The period of performance measurement is from 31 December 1999 to 1 June 2025.

  

  This illustration emphasises the robustness of TF systems, as different quantitative models can provide first-order exposure to trending features of financial markets. Most CTA managers pursue to deliver outperformance over the benchmark index by second-order proprietary features including exposures to style factors (carry, value, cross-sectional momentum, etc.), risk-management (portfolio volatility targeting, asset class exposure management, etc.), operational capabilities (exposure to smaller or alternative futures markets, enhanced execution, etc.), and other risk premia (e.g. volatility carry) — see Carver 2023, Advanced Futures Trading Strategies, for a detailed overview of additional features and strategies commonly combined with managed futures.

  Smart Diversification of Long-only Portfolios

  We also analyse the diversification benefits of how blending of TF systems long-only portfolios with long-only portfolios. In Figure 4, we generate blended portfolios with (1-x)% weight to 60/40 Equity/Bond portfolio and with x% weight to each of the three TF systems with x varying from 0% to $100%. Blended portfolios are rebalanced quarterly and, for TF systems, we use their net performance. The initial portfolio on the left is 100%/0% blend of 60/40 portfolio and 0% TF system. The final portfolio on the right is 0%/100% blend. Hereby, we measure portfolio risk by the Bear-Sharpe ratio (the performance in 16% worst quarters of 60/40 equity / bond portfolio) and portfolio performance by total Sharpe ratio.

  Figure 4. Bear-Sharpe ratio vs total Sharpe ratio for blended portfolios with (1-x)% weight to 60/40 portfolio and x% weight to each of the three TF systems. The initial portfolio on the left is 100%/0%$ blend of 60/40 portfolio and 0% TF system. The final portfolio on the right is 0%/100% blend. The specification of TF systems is the same as for generation of Figure 3.

  

  We observe that the best combination of European and American TF systems that generates the highest Sharpe ratio is the 40%/60% combination of the 60/40 portfolio / TF system. In this case, the realised Bear-Sharpe ratio is close to zero, while the total Sharpe ratio is about 0.9, which is almost double the Sharpe ratios of its components. As we see in Figure 3, the TSMOM system has a Bear-Sharpe ratio attribution of 50% smaller than that of European and American TFs. Thus, the Bear-Sharpe ratio emphasises the diversification efficiency for long-only portfolios.

  We note that, because implementation of a TF system requires only a limited capital for margin requirements of trading futures, a TF system can implemented as an overlay to 100% exposure to a long-only portfolio. If we take the 50%/50% blend (which is not far from the optimal blend 40%/60% in Figure 4 and leverage it twice, we obtain the portfolio with 100% exposure to the 60/40 portfolio and 100% exposure to a TF system. We note that recent advances in portfolio products termed “stacking alphas” or “portable alphas”  (see Gordillo-Hoffstein, 2024, Return Stacking: Strategies For Overcoming A Low Return Environment) are based on the same concept of blending a fixed 100% exposure to a long-only portfolio and 100% (or similar) exposure to a TF system or a general managed futures program.

  Further Applications

  Our results, allow for prediction of the performance of TF systems conditional on certain dynamics, such as ARFIMA process. This could be applied for instrument selection and signal/weights adjustments.

  Given that we also derive a very good approximate formulas for the expected turnover of European TF system, our results can be applied for quick optimisations of TF systems.

  Finally, our “Smart Diversification” based on regime-conditional Sharpe ratios enables for design of overlays using TF systems for long-only portfolios. In particular, we show that the optimal weight, according to our “Smart Diversification”, of TF system for 60/40 portfolio is 50%. Return stacked portfolios are obtained by 2x leverage of 50%/50% blend of 60/40 portfolio / TF system.

  Links

  Our paper is available on SSRN: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3167787

  I presented our paper at CQF Volatility and Risk conference with slides available here and Youtube video of my presentation is available here

  Disclosure

  This research is a personal opinion and it does not represent an official view of my current and last employers.

  This paper and the post is an investment advice in any possible form.

  Share this:

  • LinkedIn
  • X
  • WhatsApp
  • Email
  • Print
  Posted in Asset Allocation, Quantitative Strategies, Trend-following, Uncategorized | 1 Comment

• Optimal allocation to cryptocurrencies in diversified portfolios – update on research paper

  Posted at 3:08 pm by artursepp, on August 18, 2024

  Cryptocurrencies have been acknowledged as an emerging asset class with a relatively low correlation to traditional asset classes and independent drivers of their long-term performance (see for an example excellent papers by Harvey et al (2022) and Adams at al (2024)).

  A year ago in Summer of 2023, I published research article in Risk Magazine (SSRN draft) on quantitative methods for optimal allocation to cryptocurrencies within alternative and balanced portfolios. The metrics for consideration include metrics for portfolio diversification, expected risk-return relationships and skewness of the returns distribution. Using roll-forward historical simulations, I showed that all four allocation methods produce a persistent positive allocation to Bitcoin and Ether in alternative and balanced portfolios with a median allocation of about 2.7%.

  This time, I would like to present the updated outcomes from my model given that Bitcoin and Ether had a strong performance of 95% and 35%, respectively, since the last update to today (from 30Jun2023 to 16Aug2024).

  Spoiler: the performance of all four methods for balanced and alts portfolios have been in line with what has been reported in the article with optimal allocation weights to Bitcoin and Ether largely unchanged. Python code for this analysis is available in OptimalPortfolios packadge github repo.

  First I start with the analysis of annual rolling performance. In Subplot (A) of Figure 1, I  show Sharpe ratios (through the paper and this post, the Sharpe ratio is computed using monthly log-returns adjusted by 3m UST rate) for trailing holding periods with the period start given in the first column and the period end given in the first row. For an example, Sharpe ratio realized from the investment period from 31Dec2020 to 16Aug2024 is 0.29.

  Clearly, the early periods before 2017 are characterized with higher realized Sharpe ratios. What is remarkable that any investment period that starts at the end of each calendar year from 2010 to today generated positive Sharpe ratio. In Subplot (B) of Figure 1, I  show the realised skeweness of monthly returns. In early periods, the monthly performance exhibits highly positive skewness. Also more recently the skeweness became positive again.

  

  Figure1. Realized Sharpe ratios from the period start (given in the first column) to the period end (given in the first row). Subplot (A) shows Sharpe ratio using average monthly log-returns; Subplot (B) shows skewness of monthly returns.

  Methodology

  The long-term positive performance and positive skeweness of cryptocurrency returns pose well for quantitative allocation methods.

  In the paper I consider four quantitative allocation methods for construction of optimal portfolios:
  1) Two risk-based methods which include portfolios constructed using equal risk contribution and with maximum diversification methods.
  2) Two risk-return based methods which include portfolios constructed using maximum Sharpe ratio and maximum CARA-utility methods.

  For the investment universe, I consider the two mandates:

  1) Alternatives (Alts) or unconstrained mandate that targets absolute returns by investing into alternative assets. This mandate is typical for high net worth private investors and family offices.

  2) Benchmarked (Balanced) mandate which targets excess returns over a benchmark by allocating to a balanced equity/bond portfolio with additional overlay to alternative assets. Such a mandate is typical for institutional investors such as pension funds, insurance companies, and endowments.

  As the balanced benchmark, I use the classic 60/40 equity/bond portfolio. I fix the target weight of the balanced portfolio for this mandate to 75% and assign $25%$ allocation to alternative assets. As a result, I consider the modern 70%/30%$approach for allocation portfolio of institutional mandates (see, for an example, McVey et al (2022)) with 30% allocation to bonds, 45% to public equities and 25% to alternative assets.

  I refer to the paper for the investment universe of this mandates (In this analysis I change the benchmark for macro funds from NEIXMTI Index to HFRIMDT Index). For each allocation method, I evaluate the following portfolios given in Table 1 below. Portfolios 1, 2, 3 provide insights into the marginal contribution of including cryptocurrencies to investable universe alternative portfolios. Portfolios 4, 5 and 6 provide with insights into including cryptocurrencies to alternatives for blending with the 60/40 equity/bond portfolio. The marginal contribution of including cryptocurrencies is estimated using 4 portfolios with either BTC or ETH using 4 allocation methods, with total of 16 different portfolio schemes allocated to cryptocurrencies. I sue spot returns for performances of cryptocurrencies. This provides a sufficient depth for making insights.

  

  Table 1. Simulated mandate portfolios with cryptocurrencies.

  Optimal Portfolios and Their Performances

  I use quarterly rebalancing and roll-forward analysis for generation and backtest of optimal portfolios. I describe the methodology in the paper  and in github package

  Here, I present the result of roll forward simulations from 31Mar2016 t0 16Aug2024. I will present some key figures here, all outputs can be found in pdf report of backtests.

  Maximum Diversification

  Maximum Diversification is my favorite method because it takes into account only the covariance matrix. Also, unlike Equal Risk Contribution method, Maximum Diversification method may produce zero weights to unattractive instruments. In Table 3, I show the risk-adjusted performance of the simulated portfolios without crypto and with inclusion of BTC and ETH cryptocurrencies. The Sharpe ratio is computed using monthly log-returns adjusted by 3m UST rate, beta and (annualised) alpha are computed by regression of monthly returns against 60%/40% equity/bond (Balanced) portfolio.

  The marginal gain of including BTC and ETH is of +0.24 (=0.70-0.46) and +0.29 (=0.75-0.46) in Sharpe ratio for Alternative portfolios and of +0.23 and +0.21 for Balanced portfolios, which is significant.

  In the last 4 rows I show the weight allocated to cryptocurrencies. The median allocation weight is 2.2%/1.9% and 3.13%/3.04% for BTC or ETH in alternatives and balanced portfolios, respectively.

  

  Table 3. Risk-adjusted performance of Maximum Diversification allocation method.

  In Figure 2, I show the time series of cumulative performances and drawdowns of Maximum Diversification portfolios. Adding cryptocurrencies to the portfolio universe did not materially impact realised drawdowns.

  

  Figure2. Cumulative performance of portfolios computed using Maximum Diversification allocation method.

  In Figure 3, I show the stack plot of optimal weights for BTC for alternatives and balanced mandates. We observe that the optimal weight of BTC has been persistent through the backtest period, in contract to other asset classes. It is interesting, that the optimal allocation to alternatives within balanced portfolio includes only Bitcoin and SG Trend instruments for the past two years.

  

  Figure 3. Optimal Allocation weights for alternative and balanced mandates with universe including BTC.

  Equal Risk Contribution

  Equal risk contribution allocates equal buckets for risk (for Balanced mandate, 75% of risk is assigned to the balanced portfolio). We observe that adding cryptocurrencies improves the risk-adjusted performance of alternatives mandate. Interestingly, from the standpoint of the equal risk contribution method, allocations to BTC and ETH are almost same.

  

  Table 3. Risk-adjusted performance of Equal Risk Contribution allocation method.

  Maximum Sharpe Ratio

  I use the rolling window of 5 years to estimate asset return and covariances for the estimation of the Sharpe ratio. For alternatives portfolio, the contribution to the performance (+0.80 and +0.67 in Sharpe) from adding cryptocurrencies is significant with their median weights of 9% and 4% for BTC and ETH. It is clear that using past returns as inputs to the optimiser may not be robust, however increasing the universe may lead to better results because of higher degree of freedom.

  

  Table 4. Risk-adjusted performance of Maximum Sharpe Ratio allocation method.

  Carra Mixture Utility

  To estimate the 3-state mixture of returns distribution for the Carra Mixture utility, I also use the rolling window of 5 years. As I explain in the paper, the Carra Mixture Utility allocation method favors instruments with positive skeweness. Similarly to the Maximum Sharpe ratio, adding cryptocurrencies to the alternatives portfolio improves the realised Sharpe ratio considerably by +0.84 and +0.64 with BTC and ETH, respectively. The median allocated weight is 21% and 8% for alternatives mandate and 19% and 8% for the balanced mandate. The higher weights are the result of overweighting instruments with positive skeweness.

  

  Table 5. Risk-adjusted performance of Carra Mixture Utility allocation method.

  Summary of Weights

  In the summary, I would like to the review the optimal weight to cryptocurrencies. The major goal of my article is to show that cryptocurrencies deserve an allocation for broad portfolios. In my analysis, I did not impose any allocation constraints to make a fair argument.

  In Figure 4 I show the time series of optimal allocations to BTC and ETH by each method and for each mandate. In Table 6, I show summary of weights aggregated from time series.

  Carra Mixture (CARRA-3) allocation method assigns the highest allocation to cryptocurrencies because it favors assets with high positive skewness.

  We observe that the Maximum Sharpe ratio and Carra Mixture, which take into account the rolling performance of assets, have been producing smaller allocation weights in recent years following smaller the risk-adjusted performances of cryptocurrencies.

  However, the risk based methods including Equal Risk Contribution (ERC) and Maximum Diversification (MaxDiv) produce largely stable allocation to cryptocurrencies, which stay largely intact in past couple of years.

  The median of the time series median allocation is 5.7%, 3.8%, 3.0%, 2.4%, which gives a “median”allocation of 3.4% which slightly increased from 2.7% which I reported originally in the paper.

  

  Figure 4. Optimal weights to BTC and ETH by allocation methods.

  

  Table 6. Summary of weights

  Further reading

  Enjoy reading the paper and experiment with Python code

  Disclosure

  This research is a personal opinion and it does not represent an official view of my current and last employers.

  This paper and the post is an investment advice in any possible form.

  Cryptocurrencies are associated with high risk.

  Share this:

  • LinkedIn
  • X
  • WhatsApp
  • Email
  • Print
  Posted in Asset Allocation, Crypto, Decentralized Finance, Python, Quantitative Strategies, Uncategorized | 2 Comments

• Unified Approach for Hedging Impermanent Loss of Liquidity Provision – Research paper

  Posted at 7:16 pm by artursepp, on July 9, 2024

  Let me introduce our research paper co-authored with Alexander Lipton and Vladimir Lucic for hedging of impermanent loss of liquidity provision (LP) staked at Decentralised Exchanges (DEXes) which employ Uniswap V2 and V3 protocols.

  Uniswap V3 protocol allows liquidity providers to concentrate liquidity in specified ranges. As a result, the liquidity of the pool can be increased in certain ranges (typically around the current price) and the potential to generate more trading fees from the LP is increased accordingly. I illustrate the dynamics of staked LP using ETH/USDT pool as an example. A liquidity provider stakes liquidity to a specific range using initial amount of ETH and USDT tokens as specified by Uniswap V3 CFMM. When the price of ETH falls, traders use the pool to swap USDT by depositing ETH, so that the LP accrues more units of ETH. Thus when ETH falls persistently, the liquidity provider ends up holding more units of the depreciating asset, which is similar to being short a put option. In opposite, when ETH price increases, traders will deplete ETH reserves from the pool by depositing
  USDT tokens. Thus, the liquidity provider ends up holding less units of the appreciating asset, which is similar to being short a call option. The combined effect of increasing / decreasing the exposure to depreciating / appreciating asset leads to what is known as the impermanent loss in Decentralised Finance (DeFi) applications.

  In Figure 1, I show ETH units (left y-axis) and USDT units (right y-axis) for LP on Uniswap V3 with 1m USDT notional and p_{0}=2000, p_{a}=1500, p_{b}=2500. The initial LP units of (ETH, USDT) are (220, 559282). The red bar at p=1500 shows LP units of (543, 0) with LP fully in ETH units when price falls below lower threshold p_{a}. The
  green bar at $p=2500$ shows corresponding LP units of (0, 1052020) with LP fully in USDT units when price rises above upper threshold p_{b}. In subplot (B), we show USDT values of 50%/50% ETH/USDT portfolio, Funded LP positions (funded LP involves the purchase of ETH for staking without any delta hedge) and Borrowed LP positions (Borrowed LP is produced by static delta hedge of the initial staked position in ETH).

  The value profile of funded LP resembles the profile of a covered call option (long ETH and short out-of-the-money call). The value of the borrowed LP resembles the payoff of a short straddle (short both at-the-money call and put).

  Figure 1. The impremanent loss of funded and borrowed LP position

  (A) ETH units (left y-axis) and USDT units (right y-axis) for LP on Uniswap V3. (B) USDT value of 50%/50% ETH/USDT portfolio, Funded LP position and Borrowed LP position. Uniswap V3 LP position is constructed using 1m USDT notional with p_{0}=2000, p_{a}=1500, p_{b}=2500.

  

   

  We define the protection claim against the impermanent loss (IL) as a derivative security whose payoff at time T equals to negative value of the IL.

  We develop static model-independent and dynamic model-dependent approaches for hedging of the IL of liquidity provision (LP) staked at Decentralised Exchanges (DEXes) which employ Uniswap V2 and V3 protocols.

  For staking of BTC and ETH with liquid options market, the liquidity provider can apply out static model-independent replication to eliminate the IL completely.

  In Figure 2, I illustrate the replicating of IL for borrowed Uniswap V3 LP. I use strikes with widths of 50 USDT in alignment with ETH options traded on Deribit exchange (for options with maturity of less than 3 days, Deribit introduces new strikes with widths of $25$). In subplot (A), I show the IL of the borrowed LP position, and the payoffs of replicating calls and puts portfolios (with negative signs to align with the P&L). In subplot (B), we show the residual computed as the difference between the IL and the payoff of the replication portfolios. In Subplot (C), I show the number of put and call option contracts for the replication portfolios. It is clear that the approximation error is zero at
  strikes in the grid, which is illustrated in subplot (B). The maximum value of the residual is 0.025% or 2.5 basis points, which is very small. A small approximation error with a similar magnitude will occur in case, p_{0}, p_{a}, p_{b} are not placed exactly at the strike grid.

  Figure 2. Replication of IL of borrowed Uniswap V3 LP for allocation of 1m USDT notional, p_{0}=2000 ETH/USDT with p_{a}=1500 and p_{b}=2500. (A) Impermanent loss in USDT and (negative) values of replicating puts and call portfolios; (B) Residual, which is the spread between IL and options replication portfolios; (C) Number of option contracts for put and calls portfolios.

  

   

  For cryptocurrencies without a liquid options market develop the model-dependent valuation and dynamics hedging of IL protection claims for Uniswap V2 and V3 protocols. Model-based valuation can be employed by a few crypto trading companies that currently sell over-the-counter IL protection claims. When using model-based dynamics delta-hedging for the replication of the payoff of the IL protection claim, the profit-and-loss (P&L) of the dynamic delta-hedging strategy will be primarily driven by the realised variance of the price process. Thus, the total P&L of a trading desk will be the difference between premiums received (from selling IL protection claims) and the variance realised through delta-hedging. Trading desk can employ our results for the analysis of price dynamics and hedging strategies which optimize their total P&L.

  The simplest dynamic model is of course the Black-Scholes-Merton model which allows to analyze the sensitivity of the price for IL protection as a function of a single parameter for log-normal volatility

  In Figure 3, I show the annualised cost (APR) % for the cost of BSM hedge for the borrowed LP as a function of the range multiple m such that p_{a}(m)=e^{-m}p_{0} and p_{b}(m)=e^{m}p_{0}. I use two weeks to maturity T=14/365 and different values of log-normal volatility \sigma. All being the same, it is more expensive to hedge
  narrow ranges.

  Figure 3. BSM premium annualised (U^{borrower}(t, p_{t})/T) for borrowed LP with time to maturity of two weeks and notional of 1 USDT as function of the range multiple m such that p_{a}(m)=e^{-m}p_{0} and p_{b}(m)=e^{m}p_{0}.

  

   

  Further, we consider a wide class of dynamics models with jumps and stochastic volatility for which the moment generating function (MGF) for the log-return  is available in closed-form. The closed-form solution for the MGF is available under a wide class of models including jump-diffusions and diffusions with stochastic volatility. Thus, we can
  develop analytic solution for model-dependent valuation of IL protection under various models with analytic MGF.

  In particular, we apply the log-normal SV model which can handle positive correlation between returns and volatility observed in price-volatility dynamics of digital assets (see my paper with Parviz Rakhmonov for details).

  In Subplot (A) of Figure 4, I show the implied volatilities of the log-normal SV model for a range of volatility of residual volatility with zero volatility beta (which is typical for ETH skews). In Subplot (B), I show the premium APR for IL protection as a function of range multiple for a range of volatility-of-volatility. We see that the model-value of IL protection is is not very sensitive to tails of implied distribution (or, equivalently, to the convexity of the implied volatility). The reason is that the most of the value of IL protection is derived from the center of returns distribution.

  Figure 4. (A) BSM volatilities implied by log-normal SV model as function of volatility-of-volatility parameter ; (B) Premiums APR computed using log-normal SV model for borrowed LP as function of the range multiple m such that p_{a}(m)=e^{-m}p_{0} and p_{b}(m)=e^{m}p_{0}.

  

  For liquidity providers, who buy IL protection claims for their LP position, the total P&L will be driven by the difference between accrued fees from LP positions and costs of IL protection claims. The cost of the IL protection claim can be estimated beforehand using either the cost of static options replicating portfolio or costs of buying IL protection from a trading desk. As a result, liquidity providers can focus on selecting DEX pools and liquidity ranges where expected fees could exceed hedging costs. Thus, liquidity providers can apply our analysis optimal allocation to LP pools and for creating static replication portfolios using either traded options or assessing costs quoted by providers of IL protection.

  We leave the application of our model-free and model-dependent results for an optimal liquidity provision and optimal design of LP pools for future research.

  Enjoy reading the paper available on SSRN https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4887298

  Share this:

  • LinkedIn
  • X
  • WhatsApp
  • Email
  • Print
  Posted in Crypto, Decentralized Finance, Uncategorized, Volatility Modeling | 1 Comment

• Stochastic Volatility for Factor Heath-Jarrow-Morton Framework – research paper

  Posted at 9:29 am by artursepp, on March 2, 2024

  Let me present our recent research paper with Parviz Rakhmonov on the stochastic volatility model for Factor Heath-Jarrow-Morton (HJM) interest rate framework (available on SSRN: Stochastic Volatility for Factor Heath-Jarrow-Morton Framework).

  Factor Heath-Jarrow-Morton (HJM) model

  Under the risk-neutral measure, the interest rate curve can be conveniently modeled using the forward curve f_t(tau) where tau is rate tenor. It is well known that the bond prices can be reconstructed from f_t(tau). When we develop a model for the evolution of forward curve f_t(tau), the HJM framework imposes conditions on the drift of the forward curve f_t(tau) so that the future forward curve reprices (in expectation) the term structure seen today. In practice, this requires using multi-dimensional models and handling path-dependency. See a brief intro to HJM and references in wikipedia.

  When we apply a finite dimensional basis for modelling of the forward rate under statistical measure P, the core difficulty is to derive the corresponding factor dynamics under the risk-neutral measure Q. The paper by Lyashenko and Goncharov provides a straightforward way for augmentation of P-dynamics so that the Q-dynamics are arbitrage-free and consistent with the initial term structure of forward rates by construction.

  Nelson-Siegel Term Structure model

  Nelson-Siegel Term Structure model provides a convenient way to model the forward curve f_t(tau) under the statistical P-measure using just 3 factors for the level, slope, and convexity of the terms structure of forward rates. This model is widely used by central banks due to its intuitiveness and due to its good consistency with time series of rates data.

  In our paper, we develop a generic Factor HJM model extended this model with the stochastic volatility using our previous paper for modelling stochastic volatility of one-factor interest rate model. As a base case, we apply the dynamics of Nelson-Siegel factors under P-measure with stochastic log-normal volatility of this factors.

  We find that this approach is well aligned with the popularity of Nelson-Siegel model and extends this model for realistic P-modeling of factors with stochastic volatility. It is well-established that stochastic volatility models can model empirical features such as volatility clustering, auto-correlations, and heavy-tails, while log-normality of rates volatility for one-factor models is well documented. We apply our developed log-normal stochastic volatility with quadratic drift as a driver for volatility of Nelson-Siegel factors.

  For valuation purposes, we derive the model dynamics under risk-neutral Q-measure. The advantage of our framework is that is fully analytic, and it allows for consistent valuation and risk management of interest rate derivatives including swaps, swaptions, futures rates and options on futures rates.

  Simulations of Nelson-Siegel Term Structure model with Log-normal Stochastic Volatility
  

  In this post, I will illustrate some possible outputs from our model using Monte Carlo simulations. First, I apply the inference of  Nelson-Siegel factors using Diebold-Li approach. Then I use the term structure of US rates observed at the end of February 2024 and I apply model parameters calibrated to swaptions data. In Figure 1, I show the term structure of US Treasury yields and fitted Nelson-Siegel curve. The model fit is very good. For pricing purposes under risk-neutral measure Q, we introduce a small deterministic curve so that the given yield curve is fitted exactly.

  

  Figure 1. Initial US Treasury yields and fitted Nelson-Siegel curve

  Next I simulate the factors of Nelson-Siegel model under P-measure as shown in Figure 2 for the simulation horizon of one year using the initial Nelson-Siegel curve in Figure 1. For brevity, in Figure 2, I show only 10 paths. Factors 1, 2, 3 are the level, slope, and convexity drivers of the forward curve. Realisations of factor X1 model possible evolution for overall level of rates: we observe a range of outcomes from 1.5% to 5.0% in 1 year. Paths of factor X2 model the (negative) slope: all paths indicate mean reversion back to positive slope indicating upward looking forward curves. Paths of factor X3 show the evolution of the convexity of the forward curve.

  

  Figure 2. Simulated paths of Nelson-Siegel factors for 1y horizon starting from initial values (X1, X2, X3) = (4.36%, 1.3%, -1.0%) with mean-reversion lambda=0.55. See Eq (1) along with Eq (23) in the paper.

  Along with the factors I simulate the log-normal stochastic volatility of these factors using the calibrated model. I apply the extension of Karasinki-Sepp stochastic volatility model augmented with the quadratic drift as developed in our paper with Parviz. I show the paths of volatility in Figure 3. The starting value of the stochastic volatility is 100%. For each factors, we apply deterministic volatility scale which is a part of model calibration (see sections 2.1 and 7.5 in the paper). Interestingly, the correlation between different factors and the volatility driver has different signs: the volatility is positively correlated with level factor X1, while the volatility is negatively correlated with  slope factor X2 and convexity factor X3. We see that we can obtain a rich set of realization for both the forward curve and the volatility of factors. We can compare paths of the volatility sigma_t with the move index for implied rates volatility.

  

  Figure 3. Simulated paths of the Log-normal stochastic volatility of Nelson-Siegel factors. See Eq (13) in the paper.

  The realisations of Nelson-Siegel factors and their volatilities in 1y allows us to construct the forward curve f_t(tau) under measure Q as seen in 1 year. We observe different shapes of forward curve in 1y compared to the today curve as function of tenor, as shown in Figure 4. Overall, the level of the yield curve is expected to decline following the initial curve shown in Figure 1, yet we have scenarios with higher curve (path 2), inverted U-shape curve (path 5), upward sloping curve (path 1), and flattish curve with different rate levels. Thus, Nelson-Siegel model can generate a rich set of scenarios of the yield curve evolution which can be applied either for valuation of interest rate derivatives or for risk and stress management of fixed-income portfolios.

  

  Figure 4. Realisations of the forward rates in 1 year using simulated paths of Nelson-Siegel factors. See Eq (22) in the paper for factor loadings.

  Simulations of the yield curve allow us to construct realisations of interest rate derivatives as swap curves, as shown in Figure 5 for swap rate starting in 1y as function of tenor, and rates futures, which require a convexity adjustment (see Section 3.3 in the paper).

  

  Figure 5. Realisations of swap rate starting in 1Y as function of tenor computed using simulated forward rates. See Eq (29) in the paper.

  Furthermore, we can value call and put options on swap futures rates. In Figure 6, I show model implied volatilities for 1y5y swaptions computed using simulated paths. Option strikes are set as fix moneyness in basis points (bps) relative to 1Y5Y swap rate in each paths. We see that the model can generate a variety of implied volatility curves from convex curves (as today), curves with strong positive skeweness (as seen in middle of year 2022), to curves with negative skeweness (as seen during 2010s).

  

  Figure 6. Model implied volatilities of swaptions on 1Y5Y swap rate seen in 1 year as functions of moneyness in bps relative to 1Y5Y swap rate in each path. See Eq (88) in the paper for valuation of swaptions.

  Summary

  Given the popularity of Nelson-Siegel term structure model, our model can provide a valuable toolkit for building scenarios for the shape of both the yield curve and the implied volatilities and for risk-management of fixed-income derivatives. I emphasize that the model is arbitrage-free and consistent with the initial forward curve by construction and the model can value different interest rate derivatives (swaptions and options on rate futures) consistently.

  Enjoy the reading of our paper in full and feel free to provide comments.

  Share this:

  • LinkedIn
  • X
  • WhatsApp
  • Email
  • Print
  Posted in Uncategorized | 2 Comments

• AD Derivatives podcast on volatility modeling and DeFi

  Posted at 7:28 pm by artursepp, on December 7, 2023

  I had a pleasure talking with Greg Magadini from Amberdata Derivatives. Greg is a seasoned options trader and he co-founded of GVol which provides awesome analytics for crypto options: check it out!

  We discussed many interesting topics including my background in becoming a quant, volatility modelling and trading, and my latest work in crypto options and DeFi.

  Greg put a nice summary to get you engaged

  https://blog.amberdata.io/ad-derivatives-podcast-feat-artur-sepp-head-quant-at-clearstar-labs

  and to watch the podcast on Youtube

  https://www.youtube.com/watch?v=3Km02FDIpxM

  Enjoy!

  Share this:

  • LinkedIn
  • X
  • WhatsApp
  • Email
  • Print
  Posted in Crypto, Uncategorized, Volatility Modeling, Volatility Trading | 0 Comments

• Paper on Automated Market Making for DeFi: arbitrage-fee exchange between on-chain and traditional markets

  Posted at 2:37 pm by artursepp, on September 29, 2021

  I have been delighted to collaborate with Alexander Lipton on a paper where we develop a quantitative approach for making arbitrage-free pricing between decentralized exchanges (DEX), relying on Automated Market Making (AMM), and traditional exchanges, relying on the order book. As a very relevant case for developing central bank digital coins (CBDC) on interoperable blockchains, we simulated our model using high-frequency FX data from a traditional exchange to validate our approach.

  This post is a small communication of the background and key results from our paper that can be downloaded from SSRN https://ssrn.com/abstract=3939695

  Automated Market Making

  Automated market making (AMM) for crypto asset has become one of the most interesting developments in the Decentralized Finance (DeFi) space.

  Vitalik Buterin, the founder of Ethereum protocol, originally proposed AMM in 2016 as a concept to exchange on-chain assets on decentralized exchanges which operate entirely on-chain . The purpose was to reduce the spreads and gas fees, that had been excess of 10% at the time. The solution was suggested to create two-sided pools of different coins (for an example, ETH vs BTC) and to fix the exchange rate relative to the pool depth (liquidity).

  This concept was formalized by the Uniswap protocol that introduced the so-called constant function market maker (CFMM) using product rule as for marginal pricing of one token vs the other by mean of smart contracts (SC).

  The AMM is an interesting concept like a dark pool (in a good sense) where investors can place a large orders and get immediate executions without revealing their intentions prior to their trades.

  In Figure 1, I show the relative pricing of a representative USDC-EUDC (US Dollar – Euro) pool (the initial parameters are EUR/USD rate of 1.25) using the three CFMM rules:

  1. Sum rule that allows to swap full balances of one token into another so that the change in the relative rate is a constant.
  2. Product rule that fixes the relative exchange rate inversely proportional to pool balances. Outside of the equilibrium rate of 0.8 EUDC per 1.0 USDC, the relative rate of EUDC will decline or increase faster than the constant exchange rate
  3. Mixed rule with a parameter alpha which is a blended rule between the sum and the product rule.

  

  Bid/Ask marginal rates

  Using the CFMM we can derive the marginal exchange rates as functions of the ratio of the order size to the pool liquidity. This is a very convenient feature that enables to explicitly assign the exchange rate to each order size.

  In Figure 2, I show the marginal AMM rates as functions of the CFMM specification. I use the EUR-USD FX spot of 1.25 and equivalent USD-EUR spot of 0.8. Then we can present a representative bid/ask book for trading in both EUDC and USDC from the same USDC-EUDC pool.

  It follows that the sum rule enforces no feedback from pool liquidity for the marginal exchange (zero slippage costs) while the product rule produces strong feedback from the pool liquidity (slippage costs proportional to the ratio of traded order to the pool liquidity). By introducing the mixed rule with a parameter alpha between 0 (product rule) and infinity (sum rule), we can design flexible CFMM.

  

   

  Pool arbitrage

  One of the most interesting challenges for on-chain exchanging of different CBDCs is how to avoid arbitrage opportunities between on-chain exchanges and traditional markets. We solve this problem by introducing a pool arbitrageur (either a pool operator or designated market-maker) who follows an optimization problem to arbitrage opportunities between the on-chain pool and traditional markets. Because of the pool arbitrageur, the pool bid/ask spreads for small orders are consistent with a traditional exchange.

  We apply our model for simulation of hypothetical CBDC pools using actual high-frequency data FX data. In Figure 3, I show the simulation of USDC-EUDC pool using intraday EUR-USD FX spot rate on 3rd June 2021. For convenience, I normalize the sport FX rate to 1.0 at the start of the trading session. I apply the constant product CFMM.

  In the first panel I show the optimal pool balances that are determined by the pool arbitrageur to exclude arbitrage between the pool and the FX spot rate. In the second panel I show the bid/ask spreads for trading 1bp of the pool liquidity. We see that the actual FX spot rate is sandwiched between the AMM bid/ask rates. The final figure is the arbitrage profits.

  

   

  Application to G-10 currencies

  As as a final validation, we also included the volumes for simulations of CBDC pools using the actual FX buy and sell orders. Intraday volumes are normalized so that the pool daily turnover is 100% for each day in our sample of last 3 years of FX data.

  In the Figure 4, I show the boxplot of key variables from the simulation of the CBDC pools for G-10 currencies including the Chinese Yuan. I apply the mixed rule CFMM with alpha equal to 5 and the transaction fees of 1bp.

  In the first panel, I show the volume-weighted average bid-ask spread. The average spread is about 1.3 across all FX pair, which is competitive to traditional FX markets. The second panel shows the annual P&L (daily P&L multiplied by 260). The last panel shows the Hedged P&L which is produced by hedging the spot exposure or equivalent by allocation to the pool using borrowed CBDCs. It is clear that liquidity providers benefit from both pool fees and the convexity generated by the trading volumes

  

   

  Summary

  Automated market making is one of the core elements for on-chain exchange of digital assets. Of course, one of the most important questions is the arbitrage between on-chain and off-chain exchanges. Alexander Lipton and myself have developed a quantitative approach in this direction.

   

  References

  Lipton, A. and Sepp, A., Automated Market-Making for Fiat Currencies (2021). Working Paper, available at SSRN: https://ssrn.com/abstract=3939695

  Share this:

  • LinkedIn
  • X
  • WhatsApp
  • Email
  • Print

  Like this:

  Like Loading...
  Posted in Crypto, Decentralized Finance, Uncategorized | 2 Comments

• Tail risk of systematic investment strategies and risk-premia alpha

  Posted at 2:55 pm by artursepp, on April 9, 2019

  

  Everyone knows that the risk profile of systematic strategies can change considerably when equity markets turn down and volatilities spike. For an example, a smooth profile of a short volatility delta-hedged strategy in normal regimes becomes highly volatile and correlated to equity markets in stressed regimes.

  Is there a way to systematically measure the tail risk of investment products including hedge funds and alternative risk premia strategies? Further, how do we measure the risk-premia compensation after attribution for tail risks? Finally, would we discover patterns in cross-sectional analysis of different hedge fund strategies?

  I have been working through years on a quantitative framework to analyse the above raised questions and recently I wrote two articles on the topic:

  1. The regime-conditional regression model is introduced in The Hedge Fund Journal (online paper).
  2. A short review of the methodology and results is presented for QuantMinds

  I would like to highlight the key results of the methodology so that interested readers can further follow-up with the original sources.

  Regime conditional index betas

  In the top Figure, I show the regime conditional betas for a selection of hedge fund style from HFR indices data using the S&P 500 index as the equity benchmark.

  We can classify the strategies into defensive and risk-seeking based on their return profile in bear market regimes:

  1. Defensive strategies (long volatility, short bias, trend-following CTAs) have negative equity betas in bear regime so that these strategies serve as diversifiers of the equity downside risk.
  2. Risk-seeking strategies (short volatility, risk-parity) have positive and significant equity betas in bear regime. Equity betas of most of risk-seeking strategies are relatively small in normal and bull periods but equity betas increase significantly in bear regimes. I term these strategies as Risk-seeking risk-premia strategies.
  3. I term strategies with insignificant betas in normal bear regimes as Diversifying strategies. Examples include equity market neutral and discretionary macro strategies because, even though these strategies have positive betas to the downside, the beta profile does not change significantly between normal and bear regimes. As a result, the marginal increase in beta exposure between normal and bear periods is insignificant.

  Risk-premia alpha vs marginal bear beta

  I define the risk-premia alpha as the intercept of the regime-conditional regression model for strategy returns regressed by returns on the benchmark index. To show a strong relationship between the risk-premia alpha and marginal bear beta (the marginal bear betas are computed as the difference between betas in normal and bear regimes), I apply the cross-sectional analysis of risk premia for the following sample of hedge fund indices and alternative risk premia (ARP) products, using quarterly returns from 2000 to 2018 against the S&P 500 total return index:

  1. HF: Hedge fund indices from major index providers including HFR, SG, BarclayHedge, Eurekahedge with the total of 73 composite hedge fund indices excluding CTA indices;
  2. CTA: 7 CTA indices from the above providers and 15 CTA funds specialized on the trend-following;
  3. Vol: 28 CBOE benchmark indices for option and volatility based strategies;
  4. ARP: ARP indices using HFR Bank Systematic Risk-premia Indices with a total of 38 indices.

  In figure below, I plot risk-premia alphas against marginal bear betas grouped by strategy styles. For defensive strategies, their marginal bear betas are negative; for risk-seeking strategies, the marginal bear betas are positive and statistically significant.

  

  We see the following interesting conclusions.

  1. For volatility strategies, the cross-sectional regression has the strongest explanatory power of 90%. Because a rational investor should require a higher compensation to take the equity tail risk, we observe such a clear linear relationship between the marginal tail risk and the risk-premia alpha. Defensive volatility strategies that buy downside protection have negative marginal betas at the expense of negative risk-premia alpha.
  2. For alternative risk premia products, the dispersion is higher (most of these indices originate from 2007), yet we still observe the pattern between the defensive short and risk-seeking risk-premia strategies with negative and positive risk-premia alpha, respectively.
  3. For hedge fund indices, the dispersion of their marginal bear beta is smaller. As a result, most hedge funds serve as diversifiers of the equity risk in normal and bear periods; typical hedge fund strategies are not designed to diversify the equity tail risk.
  4. All CTA funds and indices have negative bear betas with insignificant risk-premia alpha. Even though their risk-premia alpha is negative and somewhat proportional to marginal bear beta is proportional, the risk-premia alpha is not statistically significant. In this sense, CTAs represent defensive active strategies. The contributors to slightly negative risk-premia alpha may include transaction costs and management fees.

   

  References

  Sepp A., Dezeraud L., (2019), “Trend-Following CTAs vs Alternative Risk-Premia: Crisis beta vs risk-premia alpha”, The Hedge Fund Journal, Issue 138, page 20-31, https://thehedgefundjournal.com/trend-following-ctas-vs-alternative-risk-premia/

  Sepp, A. The convexity profile of systematic strategies and diversification benefits of trend-following strategies, QuantMinds, April 2019

  Share this:

  • LinkedIn
  • X
  • WhatsApp
  • Email
  • Print
  Posted in Asset Allocation, Quantitative Strategies, Trend-following, Uncategorized, Volatility Modeling | 1 Comment

• Trend-Following CTAs vs Alternative Risk-Premia (ARP) products: crisis beta vs risk-premia alpha

  Posted at 3:00 pm by artursepp, on February 5, 2019

  

  Year 2018 was eye-opener for investors in alternative risk-premia products. A lot of these products have been sold as market-neutral but they did not live up to expectations… I think, the reason is simple: most of ARP products have been driven by marketing with nice looking back-tested results obtained by over-fitted models. I made a presentation on this topic back in early November 2018. Yet, traditional alternatives had a bad year too.

  We published an article in the Hedge Fund Journal to explain the difference between traditional trend-following CTAs and Alternative Risk Premia. Here I will post the introduction and the key insight from our model to define the risk-premia alpha.

   

  Introduction

  The turbulence of 2018 made it a difficult year for most systematic investment products. To the surprise of several investors, many of these quant products had been sold as market neutral. In particular, the new breed of alternative risk-premia (ARP) products – that had flooded the market a few years prior to 2018 – performed exceptionally badly. For example, the composite HFR Bank Systematic Risk-premia Multi-Asset Index lost -18%, in comparison with a loss of -4% on the S&P 500 total return index. However, traditional alternative asset classes also underperformed, with the flagship HFRX Global Hedge Fund Index losing -7% and the SG Trend Index losing -8%.

  In the face of such losses, both investors and managers are asking how and why so many quant strategies underperformed? Still more importantly, what are the implications for the diversification of traditional equity-bond portfolios and alternative investments? In particular, since trend-following CTAs belong to a handful of tried-and-tested diversifiers, why did trend-followers not diversify in 2018?

  To address such questions, we first intend to look at how trend-following programs are expected to perform when crises last for extended periods of at least two months, because trend-followers need to adjust to profit from sustained crises in equity markets. Second, we shall focus on the way in which the risk profile of ARP products, hedge funds, and trend-following CTAs can change in bear and bull market regimes because of their potential exposures to tail-risk. We analyse the risk-premia alpha in these products by taking into account regime-conditional risk.

  For this analysis, we are proposing a new quantitative model to explain the risk of investment strategies by accounting for extreme market conditions and for their exposure to tail risk, such as selling volatility and credit protection. We apply this model to the cross-sectional risk attribution of about 200 composite indices of hedge funds and ARP products. We show that there is a strong linear relationship between risk-premia alpha and the tail risk of systematic ARP strategies. We can demonstrate that our model explains nearly 90% of the risk-premia for volatility strategies and about 35% of the risk-premia for hedge fund and ARP products. In this way, most ARP and hedge fund type products can be seen as risk-seeking strategies. Importantly, our model predicts that ARP products offer smaller risk-premia compensation compared to hedge funds.

  We are able to illustrate that, interestingly, trend-following CTAs are exceptions since they belong to defensive strategies with negative market betas in bear regimes, yet risk-premia alphas for CTAs are insignificant. CTAs cannot be seen either as ARP products with positive risk-premia alpha from exposures to tail risk, or as defensive products with negative risk-premia designed to reduce tail risk, such as long volatility strategies. Instead, trend-following CTAs should be viewed as an actively managed defensive strategy with the goal to deliver protective negative market betas in strongly downside markets along with risk-seeking positive market betas in strongly upside markets. Overall, after adjusting for the downside and upside betas, the risk-premia alpha of CTAs is insignificant. Yet, because of the negative protective betas in bear markets, trend-followers well deserve their place as diversifiers in alternative portfolios to improve risk-adjusted performance and capture risk-premia alpha on a portfolio level, as we will show in the last section.

  Finally, since our risk-attribution model assumes conditional equity betas in specific market regimes, we are able to illustrate the misunderstanding behind strategies claiming to be “zero-correlated” and “market-neutral”. Given a specific market regime, most typically in the bear regime, many risk-premia strategies tend to produce a strong exposure to equity markets because of their hidden tail exposures. For example, a strategy selling delta-hedged put options would have a small market beta during normal regime; yet the strategy would exhibit a significant market beta during crisis periods because of its negative gamma and vega exposures. When we analyse systematic strategies unconditional to market regimes, the performance may appear to be smooth and uncorrelated because of the aggregation across different regimes.

  We will conclude the introductory section and our article by answering the above questions in the following way. Firstly, ARP strategies are expected to perform well during normal regimes. However, since the excess performance of these strategies is derived from a hidden tail risk, these strategies are expected to underperform during turbulent markets, as in 2018. To earn risk-adjusted alpha from these products, investors need to look at long time horizons that include both bull and bear markets. Second, while the performance of trend-following CTAs is not derived from risk-premia alpha as compensation for hidden tail risks, the performance of trend-followers is conditional on trends lasting for sustained periods. Since trends reversed rapidly multiple times during 2018, trend-followers underperformed. As a result, in what proved to be an extraordinary year, both ARP products and trend-followers underperformed, but for different reasons.

  Going forward, investors and allocators need to understand how different strategies are expected to perform during bear and normal markets and how to diversify their portfolios accordingly. Our results provide a valuable aid in quantifying the hidden tail behaviour of systematic strategies as well as suggesting an approach for the risk attribution and diversification of alternative portfolios.

  Risk-premia Alpha

  Risk-premia alpha measures the excess return on a strategy after adjusting for conditional beta exposures. According to the regime conditional CAPM, a strategy should produce higher risk-premia alpha if it assumes higher equity risk in a bear market measured by marginal bear market betas.

  The figure in the top illustrates different risk profiles of hedge fund and ARP products. We apply the regime conditional model to a large universe of indices grouped into three categories:

  1. Hedge fund indices from major index providers including HFR, SG, BarclayHedge, Eurekahedge with the total of 73 composite hedge fund indices excluding CTA indices;
  2. 7 CTA indices from the above providers; and
  3. ARP indices using HFR Bank Systematic Risk-premia Indices with a total of 38 indices.

  According to our model we see a clrear differentiation among risk-seeking strategies, defensive strategies and trend-following CTAs.

  Risk-seeking strategies: the marginal bear beta is positive (increased risk in bear regime) compensated by positive risk-premia alpha. Most hedge fund and ARP products are risk-seeking strategies with tail risk. We observe almost a linear relationship between risk-premia alpha and marginal bear betas for the cross-section of hedge funds and ARP indices. ARP products deliver less risk-premia alpha for the same level of tail risk compared to hedge funds.

  Defensive strategies: the marginal bear beta is negative (reduced risk in bear regime) compensated by negative risk-premia alpha. Defensive strategies diversify equity risk in bear regimes but deliver negative risk-premia alpha.

  Trend-following CTAs: produce negative marginal bear market betas and hence strongly diversify equity risk in bear market regimes. Because their risk-premia alpha is flat, trend-followers can be considered as an anomaly, or as a differentiation from ARP and traditional hedge fund products.

  Share this:

  • LinkedIn
  • X
  • WhatsApp
  • Email
  • Print
  Posted in Uncategorized | 1 Comment

• My talk on Machine Learning in Finance: why Alternative Risk Premia (ARP) products failed

  Posted at 2:56 pm by artursepp, on November 27, 2018

  

  I have recently attended and presented at Swissquote Conference on Machine Learning in Finance. With over 250 participants, the event was a great success to hear from the industry leaders and to see the recent developments in the field.

  The conference featured very interesting talks ranging from an application of natural language processing (NLP) for industry classifications to a systematic trading in structured products using deep learning. For the interested, the slides and videos are available on the conference page.

  I would like to share and introduce my talk presented at the conference on applications of machine learning for quantitative strategies (the video of my talk available here).

  In my talk, I address the limitations of applying machine learning (ML) methods for quantitative trading given limited sample sizes of financial data. I illustrate the concept of probably approximately correct (PAC) learning that serves as a foundation to the complexity analysis of machine learning.

  In particular, the PAC learning establishes model-free bounds on the sample size to estimate a parametric function from the sample data for a specified level of approximation and estimation error. I recommend very nice textbooks An Elementary Introduction to Statistical Learning Theory and The Nature Of Statistical Learning Theory to study more about the PAC learning.

  I also present an example of using supervised learning for the selection of volatility models for systematic trading from my earlier presentation.

  Finally, I touch on the important topic of the risk-profile of quantitative investment strategies and, in particular, Alternative Risk Premia (ARP) products. For the past few years, since about 2015, the sell-side have been marketing a plethora of ARP products as “cheap” substitutes for hedge fund strategies. However, ARP products fared miserably throughout year 2018 despite the fact that most of these products were marketed as market-neutral. I wanted to share my view why ARP products failed…

  The typical creation process of ARP products is as follows. First, a research team runs multiple back-tests of “academic” risk factors (value, carry, momentum, etc) across many markets until a specific parametrization of their strategy produces a satisfactory Sharpe ratio (around 1.0 or so). Once the necessary performance target is achieved in the back-test, the research team along with a marketing team would write a research paper with economic justification of the strategy. Then the marketing team would pitch the strategy to institutional clients. If the marketing team is successful, they would raise money for the strategy. Finally, the successful strategy (out of dozens of attempted) would reach to the execution team who would implement the strategy in a trading system and execute on behalf of clients.

  The creation of ARP products serve as a prime example why we need to understand the limitations of statistical learning given limited sample sizes of financial data. Also, there is the incentive to fit a rich model to the limited sample to optimize the in-sample performance. For an example, using PAC learning, to estimate a model with 10 parameters at an approximation error within 10% we need to apply 2,500 daily observations!

  It is no coincidence that ARP product suffered a major blow once market conditions changed. As we speak, post October 2018, quants are facing a crisis of confidence.

  In the hindsight, year 2018 brought to the failure the two very popular strategies:

  1) The short volatility ETNs: the figure at the top of the post illustrates how would a naive 5-parameter regression fit the in-sample data of past two years with the accuracy of 98%, but the fitted model fails miserably in February 2018 (I posted a detailed statistical analysis of the crash).

  2) The alternative risk-premia products: the figure below shows the risk-profile of Bank Systematic Risk Premia Multi-Asset Index compiled by the Hedge Fund Research.

  In the figure below, as the predictor, I use the quarterly returns on the S&P 500 index which I condition into the three regimes: bear (16% of the sample), normal (68%), and bull (16%). Then I consider the quarterly returns on the HFR index conditional on these regimes and illustrate the corresponding regression of returns on the HFR index predicted by returns on the S&P 500 index.

  It is clear that the HFR index sells 3 puts to buy 5 calls to obtain the leveraged exposure to the S&P 500 index. Well, over the past decade these models learned to leverage the upside at the cost of selling the downside.

  

  The key message from my talk is that, we may be able to avoid the traps of applying machine and statistical learning methods for systematic trading strategies by understanding the theoretical grounds of the ML methods and the potential limitations of using only limited sample sizes for the estimation of these models.

   

  Disclaimer

  All statements in this presentation are the author personal views. The information and opinions contained herein have been compiled or arrived at in good faith based upon information obtained from sources believed to be reliable. However, such information has not been independently verified and no guarantee, representation or warranty, express or implied, is made as to its accuracy, completeness or correctness. Investments in Alternative Investment Strategies are suitable only for sophisticated investors who fully understand and are willing to assume the risks involved. Alternative Investments by their nature involve a substantial degree of risk and performance may be volatile.

   

   

  Share this:

  • LinkedIn
  • X
  • WhatsApp
  • Email
  • Print
  Posted in Quantitative Strategies, Uncategorized, Volatility Modeling, Volatility Trading | 2 Comments

• Why Python for quantitative trading?

  Posted at 12:41 pm by artursepp, on October 24, 2018

  

  “Today a new language is overtaking French as the most popular language taught in primary school. Its name is Python… 6 out of 10 parents want their kids to learn Python”, Joel Clark.

  Well, when I attended school, I learnt BASIC… But I must confess, I do share the excitement taking over the Python language.

  I have recently taken part in a webinar organised by Risk.net and Fincad where we discussed the advantages and challenges in using Python for developing quantitative trading applications. The panel included experts from various corners of the industry including myself and:

  1. Joel Clark, contributing editor, Risk.net (Moderator)
  2. Gary Collier, CTO, Man Group Alpha Technology
  3. Per Eriksson, senior executive, enterprise risk and valuation solutions, FINCAD
  4. Ronnie Shah, head of US quantitative research and quantitative investment solutions, Deutsche Bank

  The webinar was a success with over 500 participants. Since Python is on everyone’s mind, I wanted to highlight some interesting questions and thoughts from our discussion. The audio of the webinar is available here

   

  Why Python has become an increasingly popular programming language in financial markets?

  One of the major advantages of using Python is the ease to interconnect different systems with data feeds and databases, to process data, and to output results into user and trading applications.

  My first experience with Python came in 2012, when Bank of America Merrill Lynch, where I worked as a front office quant strategist, introduced the Quartz system developed in Python. The Quartz was supposed to be the bank-wide solution to share data and trading risks. The reason is that the insufficient centralization and aggregation of positions and risks across all trading books (traditionally differentiated by geographies and asset classes) was one of the key weaknesses shared by large investment banks during and in the aftermath of the 2008 financial crisis.  As a result, the Quartz and Python-based analytics were thought as a bridge to connect different parts of analytics, data centres, and development teams. A daunting task for any large organization employing hundreds of developers and users!

  Moving fast forward, Python has been widely applied by major financial institutions for developing tools to connect different parts of analytics and to increase collaboration within a firm. Over time, people have also started to do more core development in Python in addition to using Python as a glue language.

  New developments using the Python language have been leveraged thanks to a rich Python ecosystem with huge number of libraries for data analytics and visualization. For an example, Man AHL illustrated how they benefited by moving both research and production code to Python.

  Summarising their paper and our panel, Python has become increasingly popular because:

  1. Python enhances the communication between different teams.
  2. Python provides an advanced ecosystem with packages for numerical and statistical analysis, data handling and visualization.
  3. Python is easy to learn and it is flexible to apply, and it’s actually fun to program using the Python language. As a person with many years of doing quantitative modelling in C++ and Matlab, I fully support this view.

   

  How Python works among other languages for data analysis?

  Since data analytics is currently one of the key drivers across all industries including the finance and investment management, choosing the right ecosystem for development may have a crucial impact on the business development and success.

  Presently, the three development tools are widely applied for the data analytics.

  1. Python along with pandas for tabular data structures and multiple packages for data analysis (statsmodels for statistical analysis, matplotlib for data visualization, scikit-learn for machine learning, etc). The advantage is that Python provides a free and open-source solution with plentiful resources for data fetching, processing, and visualization. Python can be easily deployed on either a PC or a server to make scalable firm-wide solutions.
  2. Traditionally, Matlab has been widely applied in academic and research labs but it comes with a heavy cost for commercial firms. Matlab has numerous packages for data processing, analysis, and visualization, however each package is available at a separate price. Personally, I have used Matlab a lot along with its capabilities for the object-oriented programming. While I value some capabilities of Matlab, the major drawback of Matlab, apart from its licensing cost, is that the deployment of Matlab-based analytics is problematic and comes with separate fees. Matlab applications can be compiled and deployed on a server but the deployment process looks complex and not well documented and it may be costly if external consultancy is needed. In my opinion, the insufficient portability and scalability are major obstacles for developing firm-wide solutions using Matlab.
  3. R along with its multiple packages for statistical data analytics. While R is free and it has many packages to do various statistical analyses, the deployment of R across firm-wide platform may not be as efficient. In my opinion, the R language is suitable only for the development of stand-alone tools for statistical analyses. In fact, Jupiter Lab enables to apply R functionality within the Python ecosystem.

   

  How long would it take to convert Matlab production code to Python?

  Given the advantages of Python over Matlab, most firms would now employ Python to start any new development from scratch. How is about converting the legacy code and systems?

  Gary Collier gave one example of AHL converting a fairly complicated trading system for single stock equities to Python within 8-9 months.

  In fact, my friend Saeed Amen has just written a short overview paper on moving from Matlab to Python. The transition is feasible… While there will be short-term costs, the long-term benefit is to have a firm-wide solution developed in one multi-purpose language that everyone can understand and contribute to.

   

  Python everywhere?

  To conclude, the top figure shows the share of questions about various programming languages asked each month at Stack Overflow, which is the largest online community for developers. We clearly see the growing trend for Python against all other major programming languages. Perhaps soon enough the Python will overtake all other languages taught not only in primary school but e,plyed everywhere else…

  Share this:

  • LinkedIn
  • X
  • WhatsApp
  • Email
  • Print
  Posted in Python, Uncategorized | 1 Comment

• Receive notifications of new posts by email

  Email Address

  Subscribe

  Join 6,305 other subscribers

• Recent Posts

  • The Science and Practice of Trend-following Systems: paper and presentation
  • Lognormal Stochastic Volatility – Youtube Seminar and Slides
  • Optimal allocation to cryptocurrencies in diversified portfolios – update on research paper
  • Unified Approach for Hedging Impermanent Loss of Liquidity Provision – Research paper
  • Log-normal stochastic volatility with quadratic drift – open access publication
  • Stochastic Volatility for Factor Heath-Jarrow-Morton Framework – research paper
  • AD Derivatives podcast on volatility modeling and DeFi
  • What is a robust stochastic volatility model – research paper
  • Robust Log-normal Stochastic Volatility for Interest Rate Dynamics – research paper
  • Optimal Allocation to Cryptocurrencies in Diversified Portfolios – research paper
  • Log-normal Stochastic Volatility Model for Assets with Positive Return-Volatility Correlation – research paper
  • Developing systematic smart beta strategies for crypto assets – QuantMinds Presentation
  • Toward an efficient hybrid method for pricing barrier options on assets with stochastic volatility – research paper
  • Paper on Automated Market Making for DeFi: arbitrage-fee exchange between on-chain and traditional markets
  • Tail risk of systematic investment strategies and risk-premia alpha
  • Trend-Following CTAs vs Alternative Risk-Premia (ARP) products: crisis beta vs risk-premia alpha
  • My talk on Machine Learning in Finance: why Alternative Risk Premia (ARP) products failed
  • Why Python for quantitative trading?
  • Machine Learning for Volatility Trading
  • Trend-following strategies for tail-risk hedging and alpha generation

• Categories

  • Asset Allocation (10)
  • Crypto (8)
  • Decentralized Finance (5)
  • Python (4)
  • Quantitative Strategies (17)
  • Trend-following (4)
  • Uncategorized (20)
  • Volatility Modeling (18)
  • Volatility Trading (13)
• Search for: