3.14.2 Convex Risk Measures & the Deep-Hedging Objective

Basic Prerequisites: 01 · From Zero and ES · 03 Coherent Risk Measures.

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1. Intuition & Practical Objective

Page 01 established that an incomplete market leaves a residual. This page answers the only remaining question: what do you do with it? The answer is to rank residual distributions, and the ranking device is a convex risk measure. Three objects dominate practice, and they are not interchangeable:

risk measure	what it penalises	where it comes from	computable?
variance $\mathrm{Var}$	both tails, equally	Föllmer–Sondermann projection, Markowitz	yes - closed form (a regression)
CVaR${}_{\alpha }$	the worst $(1-\alpha )$ tail only	Rockafellar–Uryasev, Basel/ES regulation	yes - a convex program, but no PDE
entropic ${\rho }_{\gamma }$	all moments (exponential tilting)	exponential utility, indifference pricing	yes - a quadratic BSDE

The last row is the hinge of the whole folder: one of these preferences is analytically tractable, and it is the one that generates a backward SDE - which is why pages 03–04 exist. The one-sentence essence:

Convex risk measures are the preference ordering that replaces no-arbitrage in an incomplete market; the entropic measure is the unique one of the family with a closed-form dynamic (BSDE) representation, and the mean–variance hedge is its small-risk-aversion limit - so the entire BSDE/Deep-BSDE machinery is the “utility” special case of deep hedging, while CVaR and general convex measures remain purely numerical.

Three things to internalise:

1. The risk measure is the objective, not a tuning knob. §3 shows the same data, same payoff, same instrument set, giving optimal hedge ratios $0.56$, $0.59$, $0.66$ under variance, CVaR${}_{97.5\%}$ and entropic risk. Nothing else changed.
2. Entropic risk is exponential-utility indifference pricing. With $U(x)=-e^{-\gamma x}$ the certainty-equivalent is exactly ${\rho }_{\gamma }$; the indifference price of a liability $H$ is ${\rho }_{\gamma }(H)=\frac{1}{\gamma }\ln \mathbb{E}[e^{\gamma H}]$. That is the bridge to the BSDE of §03 and to every nonlinear pricing result in the literature.
3. Convexity is what makes learning legitimate. $\delta \mapsto L_T^{\delta }$ is affine and $\rho$ is convex, so the deep-hedging objective is convex in the strategy - no spurious local minima from the economics. What is non-convex is the parametrisation (a neural network), which is why the optimisation is a machine-learning problem at all.

The practical objective: be able to write down and distinguish the three risk measures, know their robust (dual) representations, know the exact entropic/indifference-pricing correspondence, and know that the choice of measure - not the optimiser - determines the hedge.

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2. Mathematical Ground Truth & Derivations

2.1 The axiomatic frame

A convex risk measure on a space of bounded random variables is a map $\rho :\mathcal{X}\to \mathbb{R}$ with

$$\text{(monotone) }X\le Y\Rightarrow \rho (X)\le \rho (Y);\text{(cash-additive) }\rho (X+c)=\rho (X)+c;\text{(convex) }\rho (\lambda X+(1-\lambda )Y)\le \lambda \rho (X)+(1-\lambda )\rho (Y).$$

Cash-additivity makes $\rho (X)$ read as “the capital that must be added to $X$ to make it acceptable”, which is exactly the desk meaning: the premium. Adding sub-additivity ($\rho (X+Y)\le \rho (X)+\rho (Y)$) upgrades convex to coherent (Artzner–Delbaen–Eber–Heath 1999) - the CVaR/ES family qualifies, the entropic family does not (it is convex but not coherent; it is not positively homogeneous).

The universal dual object is the robust representation

$$\boxed{\rho (X)=\underset{\mathbb{Q}\in \mathcal{Q}}{\sup }({\mathbb{E}}_{\mathbb{Q}}[-X]-\alpha (\mathbb{Q}))}$$

with $\mathcal{Q}$ a set of probability measures and $\alpha$ a penalty function (Föllmer–Schied). This is what turns “minimise a convex risk measure” into “minimise a worst case over a family of models” - the source of the distributionally-robust hedging of 06.

2.2 The three workhorse measures

(i) Variance (quadratic). $\rho (X)=\mathrm{Var}(X)=\mathbb{E}[(X-\mathbb{E}X{)}^2]$. It is not monotone (it ignores the mean), so in deep hedging it is used in the mean-demeaned form $\rho (R(\delta ))$ with $R=\delta \Delta S-H$ - the residual already absorbs the mean. The optimal hedge is the regression

$${\delta }^{\star }=\frac{\mathrm{Cov}(\Delta S,H)}{\mathrm{Var}(\Delta S)}.$$

(ii) CVaR (expected shortfall). With confidence $\alpha$,

$$\boxed{{\mathrm{CVaR}}_{\alpha }(X)=\underset{t\in \mathbb{R}}{\inf }\{t+\frac{1}{1-\alpha }\mathbb{E}[(X-t{)}^{+}]\}=\frac{1}{1-\alpha }{\int }_{\alpha }^1{\mathrm{VaR}}_u(X)du=\mathbb{E}[-X|-X\ge {\mathrm{VaR}}_{\alpha }]}$$

(Rockafellar–Uryasev 2000). It is coherent, and its robust representation uses the absolutely continuous set

$${\mathcal{Q}}_{\mathrm{CVaR}}=\{\mathbb{Q}\ll \mathbb{P}:\frac{d\mathbb{Q}}{d\mathbb{P}}\le \frac{1}{1-\alpha }\},\rho (X)=\underset{\mathbb{Q}\in {\mathcal{Q}}_{\mathrm{CVaR}}}{\sup }{\mathbb{E}}_{\mathbb{Q}}[-X].$$

For $X\sim N(0,1)$ there is a closed form: ${\mathrm{CVaR}}_{\alpha }(X)=\phi (z_{\alpha })/(1-\alpha )$ with $z_{\alpha }={\Phi }^{-1}(\alpha )$ - the check used in §3.

(iii) Entropic risk. For $\gamma >0$,

$$\boxed{{\rho }_{\gamma }(X)=\frac{1}{\gamma }\ln \mathbb{E}\left[e^{\gamma X}\right]}$$

which is monotone, cash-additive and convex, with $X\sim N(m,s^2)\Rightarrow {\rho }_{\gamma }(X)=m+\frac{\gamma }{2}{s}^2$ exactly. Its robust representation is the “reverse” one over all measures,

$${\rho }_{\gamma }(X)=\underset{\mathbb{Q}\ll \mathbb{P}}{\sup }({\mathbb{E}}_{\mathbb{Q}}[-X]-\frac{1}{\gamma }H(\mathbb{Q}\Vert \mathbb{P})),H=\text{relative entropy},$$

which is the Girsanov log-density in the diffusion case - the exact reason it produces a BSDE with a quadratic driver (§03).

2.3 Entropic risk = exponential-utility indifference pricing

Let $U(x)=-e^{-\gamma x}$ (CARA) and consider a seller of a liability $H$. The indifference price $p$ is the cash making her indifferent between selling (and hedging optimally) and not:

$$\underset{\delta }{\sup }\mathbb{E}\left[U\right(p+X_T^{\delta }-H\left)\right]=\underset{\delta }{\sup }\mathbb{E}\left[U\right(X_T^{\delta }\left)\right].$$

In the zero-hedge / complete-market case this collapses to

$$-e^{-\gamma p}\mathbb{E}\left[e^{\gamma H}\right]=-1⟹\boxed{p^{\mathrm{ind}}=\frac{1}{\gamma }\ln \mathbb{E}\left[e^{\gamma H}\right]={\rho }_{\gamma }(H)}.$$

The indifference price is the entropic risk measure of the liability. This identity is the entire justification for calling the risk-measure-minimising hedge “utility-based”, and it is what makes the entropic case solvable by a BSDE rather than by brute-force learning.

2.4 The mean–variance limit (the bridge back to page 01)

Expand the entropic transform in small risk aversion:

$$\ln \mathbb{E}[e^{-\gamma \xi }]=-\gamma \mathbb{E}[\xi ]+\frac{{\gamma }^2}{2}\mathrm{Var}(\xi )+O({\gamma }^3)⟹\boxed{Y_0=-\frac{1}{\gamma }\ln \mathbb{E}[e^{-\gamma \xi }]=\mathbb{E}[\xi ]-\frac{\gamma }{2}\mathrm{Var}(\xi )+O({\gamma }^2)}.$$

So quadratic (variance-optimal) hedging is the $\gamma \to 0$ limit of the entropic/BSDE problem. This is why the Föllmer–Sondermann projection of page 01 is not a different theory from the BSDE one - it is its first-order approximation, and the two agree exactly in the limit. For a Gaussian terminal it is exact: $\xi \sim N(m,s^2)\Rightarrow Y_0=m-\frac{\gamma }{2}{s}^2$ (§03 check C confirms this numerically).

2.5 Transaction-cost-aware objectives

Costs enter the objective, not the model: with proportional cost $\kappa$ on traded notional, the static one-period problem becomes

$$\underset{c,\delta }{\min }\mathrm{Var}(c+\delta \Delta S-H)+\underset{\text{cost}}{\underbrace{\kappa |\delta |S_0}},$$

whose solution is ${\delta }^{\star }(\kappa )=(\mathrm{Cov}(\Delta S,H)-\frac{\kappa S_0}{2})/\mathrm{Var}(\Delta S)$ - the hedge shrinks linearly in $\kappa$. Multi-period, the cost term is $\kappa {\sum }_i|{\delta }_{t_i}-{\delta }_{t_{i-1}}|S_{t_i}$: a path functional, so the objective is no longer a function of a single number and the deep-hedging formulation becomes unavoidable. §05 quantifies both effects (the optimum frequency, and Leland’s asymptotic correction in §06).

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3. Computational Implementation - the risk measure picks the hedge

We (a) verify the closed forms of CVaR and entropic risk on a standard-normal sample, and (b) minimise three different convex risk measures over the one-period hedge ratio applied to a jump-diffusion residual, showing that the optimal hedge is a function of the preference, not of the data.

Reading the output.

• (a) The closed forms are reproduced. CVaR: the empirical values $1.7589/2.0686/2.3450/2.6711$ sit $0.2$–$0.3\%$ above the exact $\phi (z_{\alpha })/(1-\alpha )=1.7550/2.0627/2.3378/2.6652$ - the residual is the discreteness of the empirical quantile ($k=(1-\alpha )n$ atoms in the tail), which biases the estimate slightly upward. Entropic: $0.2511/0.5006/1.0034$ against $\gamma /2=0.25/0.5/1.0$, i.e. within one or two Monte-Carlo standard errors ($0.0024$–$0.0098$). Both families are correct to sampling precision, and the standard errors are reported so that “within tolerance” is a measured statement, not a claim.
• (b) The preference, not the data, chooses the hedge. All three columns are computed from the same 60 000 residual paths: variance is minimised at $\delta =0.56$, CVaR${}_{97.5\%}$ at $0.59$, entropic ($\gamma =1$) at $0.66$. Moving from the variance hedge to the CVaR hedge costs $1.26\%$ of variance and buys $1.62\%$ of CVaR; the entropic hedge sacrifices $12\%$ of variance. There is no “correct” answer here - there is only the stated preference, and the spread between the three answers (a $18\%$ range in the hedge ratio, $0.56\to 0.66$) is the size of the modelling decision a desk makes when it names its risk measure.
• The jump does the work. With a symmetric residual all three measures would coincide at the same $\delta$ (the residual is a linear function of $\delta$, and for a symmetric payoff the optima nearly agree). The downward $5\%$ jump makes the loss distribution skewed, and it is exactly the asymmetric measures (CVaR, entropic) that then demand a larger hedge. This is the same leverage/skew mechanism as stochastic volatility, now applied to the hedge rather than the price.

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4. Failure Modes & First-Principles Breakdowns

1. Choosing the risk measure to make the hedge cheap. Because the objective is the preference, a desk can loosen $\alpha$ (CVaR) or lower $\gamma$ (entropic) until the “optimal” hedge is the one it wanted. The mitigation is governance, not mathematics: the risk measure belongs to the risk function, not the trading desk. Compare Model Risk & Validation.
2. Using variance as if it were a risk measure. Variance is not monotone: adding a deterministic loss leaves it unchanged, and it penalises upside and downside identically. For a short option book, upside and downside are not symmetric and variance systematically under-hedges the left tail. It survives in practice only as the small-risk-aversion limit (§2.4) and because it has a closed form.
3. Confusing CVaR’s two definitions. $\mathbb{E}[-X\mid -X\ge {\mathrm{VaR}}_{\alpha }]$ and ${\inf }_t\{t+\frac{1}{1-\alpha }\mathbb{E}[(X-t{)}^{+}]\}$ agree for continuous distributions but differ for atoms. Simulated P&Ls are always discrete, so the two estimators differ at the $O(1/n)$ level. State which one you implement (the infimum form is the convex program used in learning; the conditional-mean form is the reporting number). Here §3 reports the conditional-mean form after demeaning.
4. Forgetting that cash-additivity requires the premium to be exogenous. If $p$ is optimised jointly with $\delta$ under a non-mean-centred $\rho$ (e.g. raw CVaR), the objective is unbounded below in $p$ and the “optimum” is $-\infty$. The standard fix - used in §3 - is to demean the residual, i.e. optimise the shape of the risk and let cash handle the level.
5. Reading the entropic $\gamma$ as a market parameter. $\gamma$ is a preference (risk aversion), not a calibrated quantity, and it cannot be inferred from option prices. Calibrating $\gamma$ to make deep hedging reproduce a market price is circular: it fits the preference to a quantity that does not observe it.
6. Assuming the robust representation is innocent. $\rho (X)={\sup }_{\mathbb{Q}\in \mathcal{Q}}({\mathbb{E}}_{\mathbb{Q}}[-X]-\alpha (\mathbb{Q}))$ turns a risk-measure minimisation into a min–max problem, and the inner adversary has to be represented too (a second network, or an explicit finite set of measures as in §06). Deep hedging with CVaR is therefore an adversarial learning problem, with all the stability issues that implies (RL for Trading).
7. Treating transaction costs as a “small adjustment”. In the one-period problem the shift is $O(\kappa S_0/\mathrm{Var}(\Delta S))$ (tiny for small $\kappa$), which is why a single-period cost adjustment looks negligible. Multi-period, the cost is summed over the path and dominates the $\sqrt{\Delta t}$ gain from finer rebalancing (§05, §06). Never import the one-period intuition into the multi-period problem.
8. Ignoring that the objective is $L^2$-convex but the parametrisation is not. Convexity of the economics guarantees a unique optimum in strategy space; it says nothing about the loss surface in weight space once the strategy is a neural network. Multiple weight configurations represent (nearly) the same strategy and can have very different apparent loss (§05, training instability).

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5. References

• Buehler, H., Gonon, L., Teichmann, J., Wood, B. (2019), Deep Hedging, Quantitative Finance 19(8), 1271–1291
• Artzner, P., Delbaen, F., Eber, J.-M., Heath, D. (1999), Coherent measures of risk, Mathematical Finance 9(3), 203–228
• Rockafellar, R.T. & Uryasev, S. (2000), Optimization of conditional value-at-risk, Journal of Risk 2, 21–41
• Föllmer, H. & Sondermann, D. (1986), Hedging of non-redundant contingent claims; Schweizer, M. (2001), A guided tour through quadratic hedging approaches
• Musiela, M. & Zariphopoulou, T. (2004), A valuation algorithm for indifference prices in incomplete markets, Finance & Stochastics 8, 399–414
• Almgren, R. & Chriss, N. (2001), Optimal execution of portfolio transactions, Journal of Risk 3, 5–39
• Hull, J., Options, Futures, and Other Derivatives

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6. Connected Graph Bridges

• Back: 01 · From Zero · ES · 03 Coherent Risk Measures
• Forward: 03 · BSDEs & Nonlinear Feynman–Kac (why entropic risk is a quadratic BSDE) · Index Hub
• Risk measures in the risk pillar: VaR & Expected Shortfall · ES · 04 Expected Shortfall · ES · 05 Failure Modes
• Cost side: Constraints & Transaction Costs · P5 · 03 Transaction-Cost Models · MM · 03 Temporary vs Permanent Impact
• Utility/hedging classics: The Greeks & Dynamic Hedging · VS · 05 Failure Modes
• Robustness: Robust Optimization · Stress Testing & Scenario Analysis