• ## Robust Log-normal Stochastic Volatility for Interest Rate Dynamics – research paper

##### Posted at 5:49 pm by artursepp, on December 31, 2022

The volatility of interest rates in 2022 has been indeed extreme. In Figure 1, I show the dependence the between the MOVE index (which measures the implied volatility of one-month options on UST bond futures and which is constructed similarly to the VIX index for implied volatilities of the S&P index futures), realized 10y UST rate volatility over the 6 months rolling window, and the level of 10y UST rates. For understanding of historical patterns, we classify the historical period from 2002 to the end of 2022 into the 5 periods: 2002-2007 (hiking cycle), 2008-2010 (tightening), 2011-2017 (QE), 2018-2020 (tightening), 2021-2022 (hiking cycle).

We see that period of 2021-2022 was indeed unprecedented period when the rates rose from low levels of around 100 basis point (bp) to over 400bp, while the rates implied and realised volatilities rose from 50bps to over 150bps.

Figure1. (A) The MOVE implied volatility index vs 10y UST bond rate; (B) 6m realized volatility of 10y UST bond rate vs 10y UST bond rate.

The dependence between the rate and its volatility manifests in implied volatilities with positive skews as I show in Figure 2 (The market convention is to use Bachelier normal model for marking implied swaption volatilities).

Figure 2. Implied normal volatilities for $10Y$ swaption as function of option expiries in basis points observed in December 2022. Option delta is Bachelier normal model delta.

The dependence between the rate and volatility also manifests in strong level between the implied and realized volatilities and the volatility of volatility and the volatility beta (the change in 1bp of the volatility predicted by 1bp change in rates) which I show in Figure 3.

Figure 3. (A) Realized volatility-of-volatility vs move volatility index. (B) Realized volatility beta vs Move index.

Quantitative modeling of such dynamics is challenging. In my previous joint paper with Parviz Rakhmonov on the
log-normal stochastic volatility for assets with positive return-volatility correlation we show that conventional SV model are ill-equipped for such dynamics. The rate dynamics are no exception, and practitioners rely on either local volatility models or local SV models with zero correlation. Both approaches are ill-poised because the may lead to explosive behavior of interest rates.

In our extension with Parviz we apply the log-normal SV dynamic for modelling the interest rate volatility, which is available on SSRN: Robust Log-normal Stochastic Volatility for Interest Rate Dynamics

We show that the proposed rates model is robust both on the quantitative dynamics and its practical implementation. While rate models in general are notorious for their tractability and implementation, we derive a closed form analytic solution for valuation of swaptions and for model calibration. In Figure 4, I show the model implied distribution of the 10y swap rate in the annuity measure computed using our analytical methods compared to the Monte Carlo simulations. Our solution is very accurate and it allows for robust calibration of the model to market data.

Figure 4. Probability density functions computed using the first order affine expansion and the second-order expansion for the distribution of 10y swap rate in one year. The blue histogram is computed using realizations from MC simulations in model dynamics.

All the technical details are available in the paper: Robust Log-normal Stochastic Volatility for Interest Rate Dynamics. Happy reading.

• ## Optimal Allocation to Cryptocurrencies in Diversified Portfolios – research paper

##### Posted at 1:43 pm by artursepp, on September 13, 2022

Cryptocurrencies have been acknowledged as an emerging asset class with a relatively low correlation to traditional asset classes. One of the most important questions for allocators is how much to allocate to Bitcoin and to a portfolios cryptocurrency assets within a broad portfolio which includes equities, bonds, and other alternatives. I wrote a research paper addressing this questions. I will provide a short summary here and refer to my paper on SSRN for details.

I apply four quantitative methods for optimal allocation to Bitcoin cryptocurrency within alternative and balanced portfolios based on metrics of portfolio diversification, expected risk-returns, and skewness of returns distribution. Using roll-forward historical simulations, I show that all four allocation methods produce a persistent positive allocation to Bitcoin in alternative and balanced portfolios. I find that the median of optimisers’ average weights is 2.3% and 4.8% for 100% alternatives and for 75%/25% balanced/alternatives portfolios, respectively. I conclude that Bitcoin may provide positive marginal contribution to risk-adjusted performances of optimal portfolios. I emphasize the diversification benefits of cryptocurrencies as an asset class within broad risk-managed portfolios with systematic re-balancing.

I start by considering a few drivers that support the allocation to Bitcoin using on statistical properties of its returns (see Harvey et al (2022) for an excellent review of supporting fundamental factors).

### Rolling Performance of Bitcoin returns

Stellar performances of core cryptocurrencies, including Bitcoin and Ethereum, have been a major supporting factor for investing into cryptocurrencies. However, these performances are realized with high volatilities, so that risk-adjusted performance, for example measured by Sharpe ratio of average log-returns, is not very significant and have been declining over the past years.

In Subplot (A) of Figure (1) I show Sharpe ratios 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 corresponding to the period from 31 December 2017 to 1 September 2022 is 0.10. It is obvious that most of large gains are attributed to periods prior to the end of 2017, when Bitcoin was little known to investment community. As a result, any historical analysis covering the early years of Bitcoin performance should be taken with caution.

Figure (1). Realized performance 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

### Correlations

A low correlation with traditional asset classes has been a supporting factor for allocating to cryptocurrencies within broad portfolios. In Figure (2) I show correlation matrices of monthly returns for three different periods: prior to 2018, from 2018 to August 2022, and from 2020 to August 2022. We see that returns of Bitcoin were little correlated to 60/40 portfolio in the early period, however, the correlation between Bitcoin and equities and bonds increased over the past three years. Remarkably, Bitcoin’s correlation with returns on alternative assets has not changed significantly. Thus, the allocation to Bitcoin is still viable within a diversified portfolio of alternatives.

Figure (2). Correlation matrix of monthly log-returns between assets in the investable universe for three periods. HFs is HFRX Global Hedge Fund Index, SG Macro is SG Macro Trading Index, SG CTA is SG CTA Index, Gold is SPDR Gold ETF (NYSE ticker GLD).

### Positive skewness of distribution of Bitcoin returns

Positive skewness of returns of cryptocurrencies is a supporting factor for allocation to this asset class. Indeed, in a very interesting paper, Ang et al (2022) argue that for skewness-seeking investors the allocation to Bitcoin could be optimal even if cross-sectional mean return may be negative. However, we observe that the realized skewness of returns of Bitcoin has been declining, following the decline of its Sharpe ratio, as I show in Subplot (B) of Figure (1). While in early years Bitcoin returns are characterized by high positive skewness, the skewness became negative in recent years. Still, the realized skewness of Bitcoin returns is higher than that of traditional assets. Importantly, Ang et al (2022) apply a two-state Normal mixture model to describe the profile of returns on Bitcoin. Further they apply maximization of CARA utility for skewness-seeking investors using this mixture model. I extend the model of Ang et al to multi-asset universe with N assets including Bitcoin.

I apply Gaussian Mixture model with M clusters to describe the distribution of asset returns conditional on a few clusters. Within each cluster, the distribution of N-dimensional vector of asset returns is normal with vector of estimated means and covariance matrix. I employ Python module sklearn.mixture for the estimation of Gaussian Mixture model and, through cross-validation, I have concluded that using 3 clusters is most robust to model the distribution of monthly returns of assets in our universe. In Figure below, I show the scatterplot of Bitcoin returns vs returns of 60/40 benchmark portfolio and one-std ellipsoids of Gaussian distribution in estimated clusters for two periods.

Figure (3). Scatter plot and model clusters using estimated Gaussian mixture model. Subplots (A) and (B) show returns data from 19 July 2010 and from 18 December 2017, respectively, to 31 August 2022. Subplots (C) and (D) show corresponding cluster parameters for Bitcoin.

### Portfolio Allocation Methods

I consider four quantitative asset allocation methods for construction of optimal portfolios.

Risk-only based methods which include portfolios with equal risk contribution (denoted by ERC) and with maximum diversification (MaxDiv).

Risk-return based methods which include portfolios with maximum Sharpe ratio (MaxSharpe and with maximum CARA-utility.

For each allocation method, I evaluate the following portfolios:

1. 100% Alts w/o BTC is the portfolio including alternative assets excluding Bitcoin;
2. 100% Alts with BTC is the portfolio including alternative assets and Bitcoin;
3. 75%/25% Bal/Alts w/o BTC is the portfolio with fixed allocation to 75% of balanced 60/40 equity/bond portfolio and 25% allocation to alternative assets excluding Bitcoin;
4. 75%/25% Bal/Alts With BTC is the portfolio with fixed allocation to 75% of balanced 60/40 equity/bond portfolio and 25% allocation to alternative asset classes including Bitcoin.

Portfolios 1 and 2 enable us to analyze the marginal contribution of including Bitcoin to the investable universe of alternative portfolios. Portfolios 2 and 3 provide with insights into including Bitcoin to alternatives universe for constructing overlays for 60/40 equity/bond portfolio.

### Optimal weights

In table below, I show the statistics of time series of optimal weights to Bitcoin produced by the four implemented portfolio optimisers. First, it is notable that all four optimizers produced non-zero weights at all quarterly re-balancing (because the time series minimum is higher than zero) for both portfolios, except for the last quarterly rebalancing of the most diversified 75%/25% portfolio. The optimization of CARA utility produced the highest allocation to Bitcoin for both portfolios, because Bitcoin adds most to the skewness of portfolio returns that is favorable for CARA method. However, the CARA portfolios have the lowest historical allocation to Bitcoin because of declining skewness of its returns. The median of optimisers’ average weights is 2.3% and 4.8% for 100% alts and 75%/25% alts/balanced portfolios, respectively. As a result, including of Bitcoin to the investable universe is beneficial for diversification benefits of broad portfolios.

Figure (4). Minimum, average, maximum, and last weight (as of last quarterly re-balancing on 30 June 2022) to Bitcoin by allocation methods computed using roll-forward simulations from 30 June 2015 to 31 August 2022. Subplot (A) shows the weight in the 100% alternatives portfolio, Subplot (B) shows the weight in the 75%/25% balanced and alts portfolio. ERC is portfolio with equal risk contribution, MaxDiv is portfolio with maximum diversification, MaxSharpe is portfolio with maximum Sharpe ratio, CARA-3 is portfolio with maximum CARA utility under Gaussian mixture model with 3 clusters.

### Trailing performance

In below table I show trailing realized Sharpe ratios of simulated optimal portfolios. I add equally weighted portfolio as a benchmark. For 100% alts portfolio w/o and with Bitcoin, the weight of Bitcoin is fixed to 0% and 2%, respectively, while the rest is equally allocated to alternative assets. For 75%/25% balanced/alts portfolio w/o and with Bitcoin, the weight of Bitcoin is fixed to 0% and 0.5%, respectively, the weight of 60/40 portfolio is 75% and rest is equally allocated to alternatives.

First, comparing 100\% alts portfolio w/o and with Bitcoin, we see that adding Bitcoin to the investable universe increased Sharpe ratios over the past periods of 2, 3, 5, 7 years except for the portfolio with maximum Sharpe ratio. The performance over the last year is better for portfolios without Bitcoin. However, I emphasize a robust positive performance of risk-based portfolios with and without Bitcoin in comparison to a poor performance of the benchmark balanced portfolio.

Contrasting 75%/25% balanced/alts portfolio w/o and with Bitcoin, we see that including Bitcoin benefits most of portfolios over all trailing periods. The exceptions include, first, the portfolio with the maximum Sharpe ratio and, second, for the ERC portfolio which slightly under-performs when Bitcoin is added.

A poor relative performance of portfolios with maximum Sharpe ratio highlights the hazard of relying on past data for forecast of future returns. In contrast, out-performers include risk-based methods that rely on the dynamic update of covariance matrices using most recent data.

Figure (5) Sharpe ratios for trailing periods of 1, 2, 3, 5, 7 years starting from 31 August 2021, 2020, 2019, 2017, 2016, respectively, up to 31 August 2022. 60/40 is the benchmark equity/bond balanced portfolio, and EqualWeight w/o and with BTC are equally weighted portfolios with fixed 0% and 2% weights to Bitcoin, respectively.

### Conclusion

I present empirical evidence that it has been optimal to include Bitcoin to an investable universe for alternative and blended portfolios, using portfolio diversification metrics. Using roll-forward analysis with dynamic updates of portfolio inputs, I also find that adding Bitcoin have improved performances of optimal portfolios.

I conclude that adding Bitcoin, and more generally, a diversified basket of cryptocurrencies, to the investable universe of broad portfolios may be beneficial for both alternative portfolios and blended balanced/alternative portfolios. I emphasize the need for a robust portfolio allocation method with regular updates of portfolio inputs and re-balancing of portfolio weights.

My favorite allocation method is the optimiser of portfolio diversification metric along with the optimiser of the CARA utility under Gaussian mixture distribution for skewness-seeking investors.

Further details are provided in my paper on SSRN http://ssrn.com/abstract=4217841

Disclaimer

The views and opinions presented in this article and post are mine alone. This research is not an investment advice.

• ## Developing systematic smart beta strategies for crypto assets – QuantMinds Presentation

##### Posted at 3:09 pm by artursepp, on February 23, 2022

I am delighted to share the video from my QuantMinds presentation that I made in Barcelona in December 2021. Many thanks to QuantMinds organizers for allowing me to share this video. First, it was nice to attend the onsite conference in a while and to meet old friends and colleagues. I was positively surprised by how many people attended. Many thanks to organizers for making it happen during these uncertain times!

I presented a framework for the design of sector-based smart beta indices and products for diversified investing to crypto assets. There are thee challenges to account for when designing a systematic strategy on crypto assets.

First, the data quality is poor indeed. We need to tackle the enormous challenge to accommodate and filter data from multiple data providers. Unlike the traditional asset classes, the market data for public data (such as market cap and traded volumes) can be a source of alpha for systematic strategies.

Second, the time history of data is very short. For example, most of protocol tokens for Decentralized Finance (DeFi) applications were listed during the second half of 2020, which means that we have to ascertain the design and risk-reward profile of a strategy using one year of data.

Third, the liquidity of crypto assets may be insufficient when contrasted with traditional assets. Therefore, we need to carefully design strategies by screening and incorporating the liquidity into the process. One of the challenges is that most crypto exchanges (there are about 30 tier one exchanges) tend to over-estimate their traded volumes.

To overcome these challenges, I constructed a bootstrapping simulation engine which allows to generate joint paths of price and fundamental data for the empirical distributions without breaking the correlation and auto-correlation structure of dependencies in the data.

• ## Toward an efficient hybrid method for pricing barrier options on assets with stochastic volatility – research paper

##### Posted at 2:00 pm by artursepp, on February 23, 2022

I am excited to share the latest paper with Prof. Alexander Lipton.

https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4035813

We find the semi-analytical solution to one of the unsolved problems in Quantitative Finance, which is to compute survival probabilities and barrier option values for two-dimensional correlated dynamics of stock returns and stochastic volatility of returns.

An analytical solution to such a problem does not appear feasible because the valuation equation is asymmetric in the log-price variable when the correlation between returns and the volatility of returns is non-zero. In the case of zero correlation, an analytic closed-form solution is achievable involving a numerical integration in the Fourier space.

In this article, we combine one-dimensional Monte Carlo simulations and the semi-analytical one-dimensional heat potential method (MHP) to design an efficient technique for pricing barrier options on assets with correlated stochastic volatility. Our approach to barrier options valuation utilizes two loops. First, we run the outer loop by generating volatility paths via the Monte Carlo method. Second, we condition the price dynamics on a given volatility path and apply the method of heat potentials to solve the conditional problem in closed-form in the inner loop. Next, we illustrate the accuracy and efficacy of our semi-analytical approach by comparing it with the two-dimensional Monte Carlo simulation and a hybrid method, which combines the finite-difference technique for the inner loop and the Monte Carlo simulation for the outer loop. Finally, we apply our method to compute state probabilities (Green function), survival probabilities, and the value of call options with barriers.

As a byproduct of our analysis, we generalize Willard’s (1997) conditioning formula for valuation of path-independent options to path-dependent options. Additionally, we derive a novel expression for the joint probability density for the value of drifted Brownian motion and its running minimum or maximum in the case of time-dependent drift.

Our approach provides better accuracy and is orders of magnitude faster than the existing methods. The methodology is general and can equally efficiently manage all known stochastic volatility models. Besides, relatively simple extensions (will be described elsewhere) can also handle rough volatility models. With minimal changes, one can use the method to price popular double-no-touch options and other similar instruments.

• ## Machine Learning for Volatility Trading

##### Posted at 6:33 am by artursepp, on May 29, 2018

Recently I have been working on applying machine learning for volatility forecasting and trading. I presented some of my findings at QuantMinds Conference 2018 which I wanted to share in this post.

• ## Trend-following strategies for tail-risk hedging and alpha generation

##### Posted at 11:39 am by artursepp, on April 24, 2018

Because of the adaptive nature of position sizing, trend-following strategies can generate the positive skewness of their returns, when infrequent large gains compensate overall for frequent small losses. Further, trend-followers can produce the positive convexity of their returns with respect to stock market indices, when large gains are realized during either very bearish or very bullish markets. The positive convexity along with the overall positive performance make trend-following strategies viable diversifiers and alpha generators for both long-only portfolios and alternatives investments.

I provide a practical analysis of how the skewness and convexity profiles of trend-followers depend on the trend smoothing parameter differentiating between slow-paced and fast-paced trend-followers. I show how the returns measurement frequency affects the realized convexity of the trend-followers. Finally, I discuss an interesting connection between trend-following and stock momentum strategies and illustrate the benefits of allocation to trend-followers within alternatives portfolio.

Interested readers can download the pdf of my paper Trend following strategies for tail-risk hedging and alpha generation or access the paper through SSRN web

# 1. Risk-profile of quant strategies

The skewness and the convexity of strategy returns with respect to the benchmark are the key metrics to assess the risk-profile of quant strategies. Strategies with the significant positive skewness and convexity are expected to generate large gains during market stress periods and, as a result, convex strategies can serve as robust diversifiers. Using benchmark Eurekahedge indices on major hedge fund strategies, I show the following.

• While long volatility hedge funds produce the positive skewness, they do not produce the positive convexity.
• Tail risk hedge funds can generate significant skewness and convexity, however at the expense of strongly negative overall performance.
• Trend-following CTAs can produce significant positive convexity similar to the tail risk funds and yet trend-followers can produce positive overall performance delivering alpha over long horizons.
• On the other spectrum, short volatility funds exibit significant negative convexity in tail events.

• ## Lessons from the crash of short volatility ETPs

##### Posted at 6:50 am by artursepp, on February 15, 2018

Exchange traded products with the short exposure to the implied volatility of the S&P 500 index have been proliferating prior to “Volatility Black Monday” on the 5th of February 2018. To investigate the crash of short volatility products, I will analyse the intraday risk of these products to steep intraday declines in the S&P 500 index. As a result, I will demonstrate that these products have been poorly designed from the beginning having too strong sensitivity to a margin call on a short notice. In fact, I estimate that the empirical probability of such a margin call has been high. To understand the performance of product with the short exposure to the VIX, I will make an interesting connection between the short volatility strategy and leveraged strategies in the S&P 500 index and investment grade bonds. Finally, I will discuss some ways to reduce the drawdown risk of short volatility products.

# Key takeaways

• Exchange traded products (ETPs) for investing in volatility may not be appropriate for retail investors because, to deliver the lasting performance in the long-term, these products need risk controls and dynamic rebalancing to avoid steep drawdowns and to optimise the carry costs from the VIX futures curve.
• The convexity of VIX changes and the sensitivity of changes in the VIX futures to changes in the S&P 500 index is extremely high in regimes with low and moderate levels of the implied volatility. As a result, a margin call on short volatility ETPs is more likely to occur in periods with low to medium volatility rather than in periods with high volatility.
• Without proper risk-control on the notional exposure, ETPs with the short VIX exposure are too sensitive to the intraday margin calls on a very short notice. Empirically, in the regimes with medium volatility, an intraday decline of 7% in the S&P 500 index is expected lead to 80-100% spike in the VIX futures and, as a result, to margin calls for short volatility ETPs.
• Short volatility ETNs provide with a leveraged beta exposure to the performance of the S&P 500 index, there is no alpha in these strategies. This leveraged exposure can be replicated using either S&P 500 index with leverage of 4.2 to 1 or with investment grade bonds with leverage of 9.6 to 1. All these strategies perform similarly well in a bull market accompanied by a small realized volatility and significant roll yields, yet these leveraged strategies are subject to a margin call on daily basis.
• ## Diversifying Cyclicality Risk of Quantitative Investment Strategies: presentation slides and webinar Q&A

##### Posted at 5:21 pm by artursepp, on December 1, 2017

What is the most significant contributing factor to the performance of a quantitative fund: its signal generators or its risk allocators? Can we still succeed if we have good signal generators but poor risk management? How should we allocate to a portfolio of quantitative strategies?

I have developed a top-down and bottom-up model for portfolio allocation and risk-management of quantitative strategies. The interested readers can find  the slides of my presentation here  and can watch the webinar can be viewed on youtube.

• ## Volatility Modelling and Trading: Workshop presentation

##### Posted at 5:13 pm by artursepp, on November 1, 2017

During past years I have found the great value in using implied and realized volatilities for volatility trading and quantitative investment strategies. The ability to stay focused and to follow quantitative models for investment decisions is what sets you apart in these volatile markets and contributes to your performance. The implied volatility from option prices typically overestimates the magnitude of extreme events across all assets – see the figure above The volatility risk-premia can indeed be earned using a quantitative model.

Nevertheless, after many years of working on volatility models, I realize that there a lot of gaps and inconsistencies in existing models for measuring and trading volatility. Unsurprisingly, by designing a model that sets you apart from the existing ones, you can significantly improve the performance of your investment strategies.

In workshop presentation at Global Derivatives Conference 2016  I have discussed in depth the volatility risk premia. The beginning and largest part of the presentation is devoted to measuring and estimating historical volatilities. The historical volatility is the key to many of the quantitative strategies, so that the historical volatility an important starting point in all applications. Then I discuss delta-hedging, transaction costs, and macro-risk management. Finally, I discuss using volatility for systematic investment strategies.

• ## Allocation to systematic volatility strategies using VIX futures, S&P 500 index puts, and delta-hedged long-short strategies

##### Posted at 3:45 pm by artursepp, on September 20, 2017

I present a few systematic strategies for investing into volatility risk-premia and illustrate their back-tested performance. I apply the four factor Fama-French-Carhart model to attribute monthly returns on volatility strategies to returns on the style factors. I show that all strategies have insignificant exposure to the style factors, while the exposure to the market factor becomes insignificant when strategies are equipped with statistical filtering and delta-hedging. I show that, by allocating 10% of portfolio funds to these strategies within equity and fixed-income benchmarked portfolios, investors can boost the alpha by 1% and increase the Sharpe ratio by 10%-20%.