Marco Avellaneda has contributed tremendously to financial mathematics, and to volatility trading in particular, has passed away earlier this year. Here I will review one of his last papers on trading VIX futures. I think most readers of this blog have modeled VIX futures understand both the risks and potential profitability of roll strategies, and the most difficult part is balancing the two.

VIX futures strategies do not have the "level" dependency of, for example, index put selling strategies, and in general seem to be less complex. However regime shifting (from typical contango to infrequent backwardation) nature of these strategies too requires constant re-balancing to keep portfolio beta-neutral.

For the strategy Avellaneda et al consider a 2-ETF portfolio: 1 month, and 5 month ( think VXX and VXZ ) A simple regression suggest portfolio short 1 month roll, and long 5 month roll, but as one can image that such *static *strategy is highly risky.

To construct a dynamic strategy the first step is to construct a mode ( robust form of mean ) , "typical" curve, based on fixed maturity 1-5 month futures, and calculate a regular 1-day AR model using least squares, which is our forecasting model for VIX returns.

The interesting part comes how the trading signals are constructed. Let's denote **R** as returns simulated based on the AR model above, and **X** as the state vector, and **a** are portfolio weights. We are looking to maximize expected utility **E[U(R(a))|X]** by varying portfolio weights. To do this, we sample possible states X from our model, and calculate utility for a range of given portfolios, train neural network to map from **a **to **E[U] **

There is a general recent trend in financial mathematics to use neural networks to approximate hard-to-compute functions, for example instead of running MC in real-time, train NN to approximate the function in a batch mode, and then use NN for pricing in real-time, which is significantly faster.

It is an active area of research, and in the near-future I hope to review this excellent cutting-edge paper on the subject.

When you think of "evil" what comes to mind? A demon from Exorcist? A Demogorgon from Stranger Things?

8 years ago Russian soldiers shut down flight MH17 killing 300 people. Immediately Russian media claimed that they shot down Ukrainian fighter jet, Russian terrorists made photos of themselves with debris in the background, looted luggage - and then proudly posted about it on social media.

This is not the first time Russia has engaged in terrorism. They have shot down Korean Air Flight 007, and then denied any knowledge of it, and Korean Air Flight 902.

If you have Russian co-workers, ask them if they support the war. Do not take a generic "I am against all wars" as an answer. Ask if they consider Russia to be an aggressor, and Russia should de-occupy Donbas, Luhansk, and Crimea, and pay reparations for the war they started 8 years ago. If the answer is anything but yes you will know who are you working with.

And please, call on your government representatives to provide more arms to Ukraine. We cannot defend ourselves with olive branches, and any agreement to peace Russian Federation will sign is worthless.

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