• ## 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.

• ## Log-normal Stochastic Volatility Model for Assets with Positive Return-Volatility Correlation – research paper

##### Posted at 3:04 pm by artursepp, on August 10, 2022

I am introducing my most recent research on log-normal stochastic volatility model with applications to assets with positive implied volatility skews, such as VIX index, short index ETFs, cryptocurrencies, and some commodities.

Together with Parviz Rakhmonov, we have extended my early work on the log-normal volatility model and we have written an extensive paper with an extra focus on modelling implied volatilities of assets with positive return-volatility correlation in addition to deriving a closed-form solution for option valuation under this model.

### Assets with positive implied volatility skews and return-volatility correlations

While it is typical to observe negative correlation between returns of an asset and changes in its implied and realized volatilities, there are in fact many assets with positive return-volatility correlation and, as a consequence, with positive implied volatility skews. In below Figure, I show some representative examples.

(A) The VIX index provides protection against corrections in the S&P 500 index, so that out-of-the-money calls on VIX futures are valuable and demand extra risk-premia than puts.

(B) Short and leveraged short ETFs on equity indices have positive implied volatility skews because of their anti-correlation with underlying equity indices. I use 3x Short Nasdaq ETF with NYSE ticker SQQQ, which is the largest short ETF in US equity market and which has very liquid listed options market.

(C) Cryptocurrencies, including Bitcoin and Ethereum, and “meme” stocks, such as AMC, have positive skews during speculative phases when positive returns feed speculative demand for upside. These self-feeding price dynamics increase the demand for calls following a period of rising prices. However, positive return-volatility correlation tend to reverse once “greed” regime is over and “risk-off” regime prevails.

(D) Gold and commodities in general may have positive volatility skews dependent on supply-demand imbalances, seasonality, etc.

Importantly, the valuation of options on these assets is not feasible using conventional stochastic volatility models applied in practice such as Heston, SABR, Exponential Ornstein-Uhlenbeck stochastic volatility models, because these models fail to be arbitrage-free (forwards and call prices are not martingals). Curiously enough, the topic of no-arbitrage for SV models with positive return-volatility correlation has not received attention in literature, despite a large number of assets with positive return-volatility correlation.

### Applications to Options on Cryptocurrencies

Additional, yet important application of our work is the pricing of options on cryptocurrencies, where call and put options with inverse pay-offs are dominant. The advantage of inverse pay-offs for cryptocurrency markets is that all option-related transactions can be handled using units of underlying cryptocurrencies, such as Bitcoin or Ethereum, without using fiat currencies. Critically, since both inverse options (traded on Deribit exchange) and vanilla (traded on CBOE) are traded for cryptocurrencies, a stochastic volatility must satisfy the martingale condition for both money-market-account and inverse measures to exclude arbitrage opportunities between vanilla and inverse options. We show that prices dynamics in our model are martingales under the both inverse and money-market-account measures.

In below Figure, I show the model fit to Bitcoin options observed on 21-Oct-2021 (the period with positive skew) for most liquid maturities of 2 weeks, 1 month, and 2 and 3 months. We see that the model calibrated to Bitcoin options data is able to capture the market implied skew very well across most liquid maturities with only 5 model parameters. The average mean squared error (MSE) is about 1% in implied volatilities, which is mostly within the quoted bid-ask spread. Calibration to ATM region can be further improved using a term structure of the mean volatility or augmenting the SV model with a local volatility part to fit accurately to the implied volatility surface.

### Model applications

The quality of model fit is similar for other assets with either positive or negative skews. The main strength of our model is that it can be used for the following purposes.

1. Cross-sectional no-arbitrage model for different exchanges and options referencing the same underlying.
2. Model for time series analysis of implied volatility surfaces.
3. Dynamic valuation model for structured products and option books.

### Further resources

Github project with the example of model implementation in Python: https://github.com/ArturSepp/StochVolModels

Youtube video with lecture I made at Imperial College for model applications for Bitcoin volatility surfaces: https://youtu.be/dv1w_H7NWfQ

Youtube podcast with introduction of the paper and review of Github project with Python analytics for model implementation: https://youtu.be/YHgw0zyzT14

Disclaimer

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

• ## 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.

• ## 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%.

• ## Why the volatility is log-normal and how to apply the log-normal stochastic volatility model in practice

##### Posted at 3:23 pm by artursepp, on August 27, 2017

Empirical studies have established that the log-normal stochastic volatility (SV) model is superior to its alternatives. Importantly, Christoffersen-Jacobs-Mimouni (2010) examine the empirical performance of Heston, log-normal and 3/2 stochastic volatility models using three sources of market data: the VIX index, the implied volatility for options on the S&P500 index, and the realized volatility of returns on the S&P500 index. They found that, for all three sources, the log-normal SV model outperforms its alternatives. Keep on Reading!

• ## Volatility Modeling and Trading: Q&A with Euan Sinclair

##### Posted at 4:18 pm by artursepp, on July 1, 2017

Q: Euan Sinclair
A: me 🙂
Q:  What is your educational background?
A: My educational background is a bit unusual. I have a PhD in Probability and Statistics which I obtained after obtaining a bachelor in mathematical economics and three master degrees in statistics, industrial engineering and mathematical finance. As a result, I like to think I am truly diversified when it comes to work and experience. It was not my intention to get many degrees, I was driven by curiosity and desire to learn new skills.

Q: (given that I know the answer to that) how did you get from a phd in statistics to direct involvement in the markets? Did you ever intend to be an academic?

A: Since my undergraduate studies, I have been attracted to the capital markets, first as an observer, then as a researcher, and finally as a professional and an investor. I enjoy academic research as a way to postulate the hypothesis based on some assumptions and then apply empirics to test it. In statistics, we always differentiate between a population and a sample. So, we can create a theoretical model for the population and test it using a sample, but not the other way around. I am interested in both developing models and also in testing them empirically. With a pinch of salt, I like to think that the finance is a unique field. On one hand, it is very challenging to create a theoretical model because of the number of assumptions we need to make. On the other hand, the testing of theoretical ideas is also challenging because of the limited data samples. At the same time, I believe people still under appreciate the power of quantitative research for financial applications. As an example, I suggest to read this fascinating article : “Buffett’s Alpha”. Warren Buffett created his wealth not because of stock picking but because of sticking to a quantitative strategy. Personally, I didn’t think of becoming an academic, I was pursuing my studies to have a deeper understanding of the theoretical background and then work on developing quantitative models for financial applications.