The problem of building a portfolio of stocks, maximising expected returns for a given level of risk, was tackled by Markowitz as far back as 1952 in his seminal paper on portfolio selection. While this was the foundation stone for modern portfolio theory, subsequently used widely in the financial industry, and worthy of a Nobel Prize, it is not a universal mechanism as it fails to tackle very asymmetric or fat-tailed distributions; Post-modern portfolio theory was an attempt to improve it, in particular recognising that Equity asset returns are not symmetric and have fat tails. Quite remarkably however, there seems to be little literature on how to construct optimal portfolios, not of single assets, but of European options.
The main issue is that these options display highly asymmetric returns distributions, making previous theories shaky. Investing in options instead of single assets is a more high-risk strategy because of their ‘all or nothing’ payout structure. However, purchasing long options offers advantages that stocks do not, in particular limited downside and high leverage. Deep out-of-the-money (OTM) options can be purchased at a fraction of the price of the underlying stock, and given a large enough movement in the underlying, large returns are at hand. The key difference between investing in options and in stocks is the risk, in the former case, of losing it all, even without extreme events like default.
Unfortunately, this risk is not well represented by the variance of an option’s distribution: whereas high variance in a stock returns distribution suggests high likelihood of going either up and down, for deep OTM options, the loss is limited to−100% and high variance is mostly due to the right tail of the distribution. Therefore high variance may reflect a high possibility of large positive returns, which is certainly not a risk. In fact, low variance may reflect an option’s distribution with the majority of its mass centered around −100% returns and a high risk of an investor losing all their money. Minimising the variance of the portfolio in a Markowitz way may only be making things worse.
The picture is complicated even further when dealing with multivariate option payout distributions. Underlying assets, at least on Equity, have asymmetric and fat-tailed distributions, along with a covariance matrix linking them together. This in turn implies highly asymmetric multivariate distributions for the options, and explains why Modern Portfolio Theory struggles with option portfolios. A natural extension is to optimise instead for high skew and low kurtosis, or using a CRRA utility function. While this may work for small portfolios, the fact that skew and kurtosis are rank 3 and 4 tensors respectively makes the problem quickly run into the curse of dimensionality and becomes unfeasible.
In order to tackle these issues, two approaches have been followed in the past: the first optimises a utility function taking into account higher-order moments, such as CRRA, while the second optimises with respect to Greek preferences. In general, these papers show good results on metrics like annualised Sharpe ratio, but most seem incapable of dealing with high-dimensional options portfolios.
Our approach builds on [S. Malamud, Portfolio selection with options and transaction costs, Swiss] and aims at solving the high-dimensional issue while taking into account asymmetry and fat-tailed distributions of option prices. We introduce a dependency matrix, based on copulas to create bivariate dependency measures between options’ payoffs and use it to replace Markowitz’ covariance matrix in the optimisation problem. One of such matrix dependency examples is the one below:
Aitor Muguruza Gonzalez is Head of Quantitative Modelling and Data Analytics at Kaiju Capital Management. He will be speaking at QuantMinds International 2022 and this is a summary of the research on which his session will be based. Secure your spot for the conference below.
