Exotic derivatives and Local Stochastic Volatility (LSV)

The ability to compute exotic greeks is important in explaining profit and loss statements, but what is the best way to calculate them effectively? In a virtual talk for the Bloomberg Quant (BBQ) seminar Adil Reghai, Head of Quantitative Research on Equity and Commodity Derivatives at Natixis, explained how to work in a production setting with exotic greeks, which are sensitivities of the price of path dependent options with respect to the input parameters and that they are also important for risk analysis and hedging.

Reghai presented a method for combining price adjustment techniques with a singular perturbation approach to design exotic scenarios and noted that it is possible to use these components in a simple formula to evaluate the impact of Local Stochastic Volatility (LSV) on exotic derivatives. He also demonstrated a numerical test of the formula on autocalls under real parameters and showed how it gave a very good fit.

LSV and the Profit and Loss (PnL)

Reghai began by asking how we can understand the impact of local stochastic volatility on the PnL. A considerable amount of quant work in the 1980s and ‘90s was focused on volatility and this work builds on the research from that era. Specifically, singular exotic perturbation is a blending of three techniques: singular perturbation, price adjustment for calibration, and exotic greeks to obtain accurate perturbation expansions for exotic products. This approach requires: 1) a good implied volatility calculator, 2) an exotic scenario generator, and 3) a local volatility model. The result is the transformation of a complex problem with a cumbersome calculation into a very light problem that can be calculated quickly, with only a small loss in accuracy.

Some of the main considerations in the development of the formula and methodology include analysis of the model versus the market and a decision on the approach to the price adjustment (calibration).  Here, Reghai mentioned modified Newton’s lemma and the volatility surface adjustment. He also addressed singular perturbations of Order 0 and Order 1, with comments on the effects, emphasizing that this technique showcases a new way to inject stochastic volatility impacts into existing production.

Comparing techniques for exotics

In the Q&A session, Bruno Dupire noted that he had also worked on the issue of stochastic volatility and derivatives using different techniques, including functional Ito calculus. Some of the findings were counterintuitive, he said, and he asked if Reghai and his research team had experienced the same phenomenon. Reghai agreed, adding that it can be difficult to present the observed results to a trader or risk manager because initially they will say the information is wrong. However, he has applied other methods including Monte Carlo Vega KT (the sensitivity to the volatility of a strike K and maturity T) using AAD (Adjoint Algorithmic Differentiation)

with 500,000 simulations and 1,000 timesteps and what was observed was not noise. Further research is underway and will contribute to a growing area of work on volatility and derivatives.

Lightning talks

 Following the short Q&A session, 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 (BBQ) Seminar series.

In this session, Yuri Saporito of the School of Applied Mathematics and the Getulio Vargas Foundation explored deep learning and path dependency, with a focus on neural nets. Cindy Liu of Bloomberg L.P. explained a system of synthetic controls, showing an example of California’s tobacco control program, and Marcos Carreira of the École Polytechnique described a relationship between Hawkes Processes and Market Microstructure. Rounding out the lightning talks, Achintya Gopal of Bloomberg L.P. discussed the trials and tribulations of model uncertainty, focusing on errors (inherent noise in the data due to missing information or measurement issues) and the limitations of the models themselves.

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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