Whether the financial markets are turbulent or calm, the subject of volatility has been of great interest to quants for decades. Some of the pioneering research was published in the mid-1990s, including work by Derman and Kani, Granger and Engle, and Dupire. One of the key questions about volatility is more defensive; how can one protect the downside? However, there are also opportunities to exploit sudden changes in market prices. In a recent talk at the BBQ (Bloomberg Quant) seminar, Jesper Andreasen of Saxo Bank discussed the topic of multi-asset volatility arbitrage. He evaluated a class of multi-asset local volatility models in which local correlation is specified through the local volatility of spreads. He provided a simple model and method for calibration via Monte Carlo simulation and showed how the model could be applied to foreign exchange and equities. He also offered thoughts on application to interest rates, with further research underway.
The minimal multi-asset model
The initial model focuses on stocks and assumes interest rates and dividends are zero for simplicity’s sake. Looking at the trading variance contracts on all spreads and creating a covariance matrix, by the prescription of no-arbitrage, the matrix must be positive definite for all discrete time steps. However, there could be weightings such that arbitrage is possible. It is also possible to decompose all pairs of the covariance matrix and see what direction should be taken in order to construct the arbitrage.
Exploring this in more detail, Andreasen introduced the minimal multi-asset model, a local volatility model where the local correlation is given from the volatility of the spread. The model fits initial option prices using the Dupire equation (1994), while using spread volatility instead of correlation draws on Austing (2011). Here the findings are that if the minimal model exists, then there will be no arbitrage. However, the reverse does not hold: the absence of arbitrage does not imply the existence of a minimal model.
Adjusting the model for the case of discrete time is useful for several reasons and helps in developing model extensions to address stochastic volatility and stochastic interest rates. In the local vol model, the local correlation is given from the volatility of the spread and is symmetric in all directions. Further, all spreads and underlyings have the same type of specification. To move to discrete time, Andreasen applies an Euler discretization and finds that, as in the continuous time case, the model is specified through spread volatility rather than by correlation. Following the discretization, on can undertake the Monte Carlo pricing and calibration; the pricing formula is exact within the discrete model and the calibration method can be used for both minimal and non-minimal correlation structures.
Turning to positive definiteness and a bootstrapping method, Andreasen explains that the correlation matrix is not necessarily positive definite. To make it so, he decomposes the matrix, takes out the negative eigenvalues, and rescales. Once the calibration has been completed for the first time step, additional simulations can be performed to calibrate for the next time steps. Andreasen notes that if work is being done on the correlation or covariance matrix, then the model will not hit the option prices at a particular expiry, but the bootstrap methodology will help to catch-up at the next expiry. The Monte Carlo pricing and calibration are similar to updating local volatility in this regard.
Applications to trading
The model can be applied to a number of asset classes, with several caveats. For foreign exchange, it is necessary to use the log normal form and complete currency translations.
In the case of equities, one can calibrate to basket options rather than spread options. It is also possible to use the notion of average correlation, and dividend models can be covered in this manner. For interest rates, multifactor Cheyette and LIBOR Market Models can be constructed. Interest rate models can also be calibrated to cap/swaption smiles, smiles of spread options, and/or smiles of mid-curve options simultaneously. The work also encompasses adjoint differentiation for risk and Greek calculations; an area where Jesper has made significant contributions for many years. Looking at the numerical performance, there are computational costs with some calculations, but TensorFlow/GPU acceleration is a promising solution. Andreasen anticipates future work on combining interest rate and price models, assessing non-minimal correlation structures, and using acceleration solutions to ensure optimal numerical performance.
During the Q&A session, Bruno Dupire asked about options on spreads for equities using baskets, an area that Andreasen is currently looking into. He has also been engaged in a long debate with a leading quant on whether or not the minimal model spans the universe of arbitrage-free FX options. He suggests looking at some of the literature from the early 2010s on this topic.
Lightning talks
Following the short Q&A session, event host Bruno Dupire kicked off a series of “lightning talks,” 5-minute presentations where industry experts, researchers, and academics present a wide range of subjects to stimulate fresh thinking and interaction between various disciplines. Each talk examines a way that the industry is evolving and serves as an essential exploratory aspect of the Bloomberg Quant Seminar series.
In this session, Peter Carr of NYU’s Tandon School of Engineering calculated the maximum loss in options and addressed hedging strategies through adding optionality. Lily Gu of Bloomberg LP examined the financial markets and the US Presidential election, explaining how different sectors and industries could be affected differently depending on who wins in November. Keith Lewis of KALX, LLC revealed an unexpected CAPM formula, and Anju Kambadur of Bloomberg LP offered a series of financial insights using the Bloomberg Knowledge Graph, which brings the power of machine learning to bear on disparate data sets.
To round out the session, Arturo Cifuentes of CLAPES and Columbia University explored the relationship between color preferences and auction prices for Joseph Albers’ Square Series and Mathieu Rosenbaum of the École Polytechnique discussed “Ad hoc Electronic Auction Design” (AHEAD) in which a hybrid mechanism in financial markets would allow for alternation between continuous market sessions and auctions, with implications for the speed of markets and high frequency trading.
About the Bloomberg Quant seminar series
The Bloomberg Quant (BBQ) seminar series is held each month and covers a wide range of topics in quantitative finance. The BBQ seminar is offered in a virtual format and scheduled to accommodate participants from across EMEA and the Americas. Each session is chaired by Bruno Dupire, head of Quantitative Research at Bloomberg L.P., and features a keynote speaker presenting on his or her current research. This presentation is followed by several “lightning talks” of 5 minutes each in quick succession. This format gives the audience the opportunity to be exposed to a wider variety of topics.
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