Damiano Brigo

Deep learning methods have become a widespread toolbox for pricing and calibration of financial models. While they often provide new directions and research results, their `black box' nature also results in a lack of interpretability. We provide a detailed interpretability analysis of these methods in the context of rough volatility - a new class of volatility models for Equity and FX markets. Our work sheds light on the neural network learned inverse map between the rough volatility model parameters, seen as mathematical model inputs and network outputs, and the resulting implied volatility across strikes and maturities, seen as mathematical model outputs and network inputs. This contributes to building a solid framework for a safer use of neural networks in this context and in quantitative finance more generally.

Previous literature shows that prevalent risk measures such as value at risk or expected shortfall are ineffective to curb excessive risk-taking by a tail-risk-seeking trader with S-shaped utility function in the context of portfolio optimisation. However, these conclusions hold only when the constraints are static in the sense that the risk measure is just applied to the terminal portfolio value. In this paper, we consider a portfolio optimisation problem featuring S-shaped utility and a dynamic risk constraint which is imposed throughout the entire trading horizon. Provided that the risk control policy is sufficiently strict relative to the Sharpe ratio of the asset, the trader’s portfolio strategies and the resulting maximal expected utility can be effectively constrained by a dynamic risk measure. Finally, we argue that dynamic risk constraints might still be ineffective if the trader has access to a derivatives market.

When pricing OTC contracts in the presence of additional risk factors and costs, such as credit risk and funding and collateral costs, the starting “clean price” is modified additively by valuation adjustments (XVAs) that account for each factor or cost in isolation, while seemingly ignoring the combined effects. Instead, risk factors and costs can be jointly accounted for ab initio in the pricing mechanism at the level of cash flows, and this “adjusted cash flow" approach leads to a nonlinear valuation formula. While for practitioners this made more sense because it showed which discount factor is used for which cash flow (recall the multi-curve environment post-crisis), for academics, the focus was on checking that the resulting nonlinear valuation formula is consistent with the theoretical arbitrage-free “replication approach” that we also analyse in the paper. We formulate specific reasonable assumptions, which ensure that the valuation formulae obtained by the two approaches coincide, thus reinforcing both academics’ and practitioners’ confidence in adopting such nonlinear valuation formulae in a multi-curve setup.

We introduce a first theory of price impact in the presence of an interest rate term structure. We explain how one can formulate instantaneous and transient price impact on zero-coupon bonds with different maturities, including a cross price impact that is endogenous to the term structure. We connect the introduced impact to classic no-arbitrage theory for interest rate markets, showing that impact can be embedded in the pricing measure and that no arbitrage can be preserved. We extend the price impact setup to coupon-bearing bonds and further show how to implement price impact in an HJM framework. We present pricing examples in the presence of price impact and numerical examples of how impact changes the shape of the term structure. Finally, we show that our approach is applicable by solving an optimal execution problem in interest rate markets with the type of price impact we developed in the paper.

We describe the pricing and hedging of financial options without the use of probability using rough paths. By encoding the volatility of assets in an enhancement of the price trajectory, we give a pathwise presentation of the replication of European options. The continuity properties of rough‐paths allow us to generalize the so‐called fundamental theorem of derivative trading, showing that a small misspecification of the model will yield only a small excess profit or loss of the replication strategy. Our hedging strategy is an enhanced version of classical delta hedging where we use volatility swaps to hedge the second‐order terms arising in rough‐path integrals, resulting in improved robustness.

Previous literature shows that prevalent risk measures such as Value at Risk or Expected Shortfall are ineffective to curb excessive risk-taking by a tail-risk-seeking trader with S-shaped utility function in the context of portfolio optimisation. However, these conclusions hold only when the constraints are static in the sense that the risk measure is just applied to the terminal portfolio value. In this paper, we consider a portfolio optimisation problem featuring S-shaped utility and a dynamic risk constraint which is imposed throughout the entire trading horizon. Provided that the risk control policy is sufficiently strict relative to the asset performance, the trader's portfolio strategies and the resulting maximal expected utility can be effectively constrained by a dynamic risk measure. Finally, we argue that dynamic risk constraints might still be ineffective if the trader has access to a derivatives market.