Deep Hedging: Learning Risk-Neutral Implied Volatility Dynamics

The paper titled "Deep Hedging: Learning Risk-Neutral Implied Volatility Dynamics" presents a numerically efficient approach for learning a risk-neutral measure for paths of simulated spot and option prices. This approach is designed to handle convex transaction costs and convex trading constraints, allowing for the implementation of a stochastic implied volatility model. The authors propose a two-step process to implement the model. First, they train a market simulator for option prices using the discussed methodology. Then, they find a risk-neutral density, specifically the minimal entropy martingale measure. This resulting model can be utilized for risk-neutral pricing or Deep Hedging in scenarios involving transaction costs or trading constraints. To support their proposed approach, the authors demonstrate that market dynamics are devoid of "statistical arbitrage" when there are no transaction costs if and only if they follow a risk-neutral measure. Additionally, they provide a more comprehensive characterization in the presence of convex transaction costs and trading constraints. These findings can be seen as an analogue of the fundamental theorem of asset pricing for statistical arbitrage under trading frictions and are deemed to be of independent interest. In summary, this paper introduces an efficient method for learning risk-neutral implied volatility dynamics by training a market simulator and finding a risk-neutral density. The results have implications for risk-neutral pricing and Deep Hedging in the presence of transaction costs or trading constraints. The findings also shed light on market dynamics under different scenarios and contribute to the understanding of statistical arbitrage under trading frictions.

Deep Hedging: Learning Risk-Neutral Implied Volatility Dynamics

In this research paper, the authors present a numerically efficient approach for learning a risk-neutral measure for paths of simulated spot and option prices. This approach is designed to handle convex transaction costs and convex trading constraints, allowing for the implementation of a stochastic implied volatility model. The results have implications for risk-neutral pricing and Deep Hedging in the presence of transaction costs or trading constraints. Additionally, they provide insight into market dynamics under different scenarios and contribute to the understanding of statistical arbitrage under trading frictions.

Overview

The paper titled "Deep Hedging: Learning Risk-Neutral Implied Volatility Dynamics" introduces an efficient method for learning risk-neutral implied volatility dynamics by training a market simulator and finding a risk-neutral density. The authors propose a two-step process to implement the model. First, they train a market simulator for option prices using their discussed methodology. Then, they find a risk-neutral density, specifically the minimal entropy martingale measure. To support their proposed approach, they demonstrate that market dynamics are devoid of "statistical arbitrage" when there are no transaction costs if and only if they follow a risk-neutral measure. They also provide more comprehensive characterization in the presence of convex transaction costs and trading constraints which can be seen as an analogue of fundamental theorem of asset pricing for statistical arbitrage under trading frictions.

Methodology

The authors propose two steps to learn an optimal solution from data given by observed spot prices S_t (t=1,...T)and corresponding option prices V_i(S_t). The first step involves training deep neural networks (DNNs) with supervised learning algorithms such as backpropagation on historical data to obtain accurate estimates V^*_i(S_t)for all i = 1,...N options at each time t = 1,...T . In this step DNNs are used because it allows them to capture complex nonlinear relationships between S_tand V^* _i(S_t). After obtaining these estimates in first step ,the second step requires finding out optimal weights w_{ij}in order to construct portfolio P_{ij}which will minimize expected cost E[C]under certain conditions like convexity assumptions on C . Here w_{ij}are chosen such that P_{ij}will be consistent with minimal entropy martingale measure (MEMM).

Results

The authors demonstrate that market dynamics are devoid of "statistical arbitrage" when there are no transaction costs if and only if they follow MEMM which is obtained after optimization procedure described above . Additionally ,they provide more comprehensive characterization in presence of convex transaction costs or trading constraints which can be seen as an analogue fundamental theorem asset pricing applied on statistical arbitrage under trading frictions .

Conclusion

This paper introduces an efficient method for learning risk neutral implied volatility dynamics by training market simulator followed by finding out optimal weights w_{ij}. Results obtained from this method have implications on both risk neutral pricing as well as deep hedging in presence transactions cost or trading constraint while providing insight into underlying market dynamics under different scenarios contributing towards understanding statistical arbitrage under various types frictions .