Author: Frédéric Godin, Assistant Professor, Concordia University, Montreal
Frédéric Godin is an Assistant Professor at Concordia University (Montreal, Canada) in the Mathematics and Statistics Department. His expertise and areas of research are financial engineering, risk management, actuarial science and data science. He also holds the Fellow of the Society of Actuaries (FSA) and Associate of the Canadian Institute of Actuaries (ACIA) designations. He published several papers in various international mathematical finance and actuarial science journals such as Quantitative Finance, Journal of Risk and Insurance, ASTIN Bulletin, Scandinavian Actuarial Research, Insurance: Mathematics and Economics and Journal of Economic Dynamics and Control. His most active research topics are dynamic hedging procedures, variable annuities and derivatives pricing. Frédéric is speaking at the AI in Finance Summit in New York this September 5-6. Register today to save with the Early Bird Discount.
In my upcoming presentation at the RE•WORK AI in Finance Summit, I will be discussing derivatives hedging policies for financial institutions such as banks and insurers. Thanks to the recent development and hype around machine learning, more sophisticated hedging methodologies have recently receive attention from academics and practitioners. The out-performance of these new methods against traditional benchmarks in specific circumstances (e.g. the hedging of long-dated options, the presence of substantial transaction costs) make them promising alternatives worthy of considerations for financial institutions desiring increasing their profitability and reducing their level of exposure to risk.
Traditionally, derivatives hedging has been based on the calculation of local sensitivities of derivatives prices to shocks on various risk factors. Such sensitivities are commonly referred as Greeks; the Delta, Gamma, Rho, Vega of options are examples of Greeks frequently encountered in the vocabulary of financial engineers designing hedging strategies. The calculated Greeks are then use to determine the amount of hedging assets to be purchased so as to neutralize the sensitivity to risk of the overall portfolio. This traditional way of proceeding has numerous benefits which partly explain the very high popularity of Greeks-based hedging. First and foremost, the only requirement to calculate a Greek is a valuation model; the Greek, being a partial derivative with respect to one of the risk factors, can be calculated through shocking a single input in the valuation model, and then performing a finite difference with the original price. Therefore, often, no distinctive model needs to be constructed for hedging purposes as the valuation model can be sufficient for that purpose. Second, the Greeks easily aggregate across the overall portfolio, as the portfolio Greek is simply the sum of the Greek for each respective financial instruments comprised in the portfolio. Finally, the Black-Scholes theory implying the theoretical possibility to construct replicating portfolios fully offsetting the risks faced by a financial institution was instrumental in the wide acceptance of Greeks-based hedging as the composition of the replicating portfolio is often simply expressed in terms of the Greeks.
However, several theoretical pitfalls associated with the used of Greeks-hedging based strategies were identified in the literature. The main drawback of this methodology is that it ignores the interaction of hedging errors through time, see for instance Brandt (2003) and Trottier et al. (2018). The purchase of hedging assets can sometimes be very costly, especially when hedging short positions of equity puts where you need to short the equity risk premium. Sometimes, the time-diversification of hedging errors can give the possibility to under-hedge to generate additional profitability without substantially increasing risk. Furthermore, Greeks-based hedging is only meant to alleviate risk for small shocks on risk factors. In practice, severe shocks such as jumps in asset price paths can occur. The presence of such jumps has to be acknowledged in the construction of hedging strategies. Finally, the inclusion of transaction costs, although feasible, is not natural for Greeks-based hedging. Transaction costs can severely impair hedging efficiency, especially when assets traded exhibit a low level of liquidity.
More recently, another hedging paradigm based on optimization problems received attention from academics and practitioners. The hedging strategy is set up so as to optimize the risk (or risk-reward tradeoff) associated with the terminal hedging error. Such a problem, called a global hedging problem, is a particular case of a sequential decision problem where all rebalancing decisions are optimized jointly to achieve the desired outcome. Machine learning and artificial intelligence tools are tailor-made to calculate the solution to the problem, which is a highly non-trivial numerical endeavor. In low dimension, when a Markovian model for the dynamics of the option underlying asset is postulated, tools from dynamic programming (e.g. the Bellman Equation) can be adapted so as to calculate the solution efficiently. However, for more complex hedging problems including multiple features (e.g. transaction costs, volatility indices, market regimes, several underlying assets), reinforcement learning frameworks were shown to produce satisfactory results and improve on the performance provided by traditional hedging schemes, see for instance Buehler et al. (2019). The high-dimensionality of the considered problems can be handled through the use of a (deep) neural network to approximate the functional form of either the value function (value assigned to each possible rebalancing action) or the policy itself. Global hedging was shown in the literature (see Augustyniak et al. 2017) to be extremely performant for long-dated options, as it allows for the correction of accumulated hedging errors). Such methodology should therefore be of particular interest to insurers willing to hedging guarantees embedded in the long-dated insurance products such as variable annuities.
My objective as a researcher is to pursue the effort to develop these more modern hedging strategies by using recently developed machine learning tools and state-of-the-art technology. Before being readily implementable, global hedging methodologies should be more extensively investigated to understand their behavior more thoroughly and identify their potential weakness. For instance, the robustness of such methods with respect to assumptions on underlying asset dynamics (when a model is considered) or to the size of the dataset (for model-free implementations) should be assessed. Stability is a very important issue in implementation as one wants to avoid strategies that are overly sensitive to key inputs. Moreover, the best approach to translate global strategies applied on individual products into portfolio strategies is not clearly understood yet, as overall portfolio diversification might need to be reflected so as to improve on the strategies. Finally, derivatives pricing schemes might need to be adapted to reflect these more modern hedging strategies in a more coherent manner. Studying these interesting questions will be an important part of my work in upcoming years.
Frédéric Godin, Ph.D., FSA, ACIA
References:
Augustyniak, M., Godin, F., & Simard, C. (2017). Assessing the effectiveness of local and global quadratic hedging under GARCH models.Quantitative Finance, 17(9), 1305-1318.
Brandt, M. W. (2003). Hedging demands in hedging contingent claims. Review of Economics and Statistics, 85(1), 119-140.
Buehler, H., Gonon, L., Teichmann, J., & Wood, B. (2019). Deep hedging. Quantitative Finance, 1-21.
Godin, F. (2016). Minimizing CVaR in global dynamic hedging with transaction costs. Quantitative Finance, 16(3), 461-475.
Trottier, D. A., Godin, F., & Hamel, E. (2018). Local hedging of variable annuities in the presence of basis risk. ASTIN Bulletin: The Journal of the IAA, 48(2), 611-646.