3.14.6 Advanced Extensions

Basic Prerequisites: 05 · Failure Modes & Practice and 03 · BSDEs & Nonlinear Feynman–Kac.

---

1. Intuition & Practical Objective

Page 05 established the three hard constraints: model risk, training instability, costly rebalancing. Each has an extension, and each extension buys its fix at a price. This page is the ladder:

failure	extension	what it buys	what it costs
proportional costs make fine rebalancing uneconomic	Leland / asymptotic cost corrections, cost-in-the-objective	a modified volatility / a modified driver	only valid asymptotically; a first-order fix
the hedge is learned under one model	distributionally-robust / min–max hedging	worst-case protection over an uncertainty set	the set is a choice; more conservative hedges
$d$ large, grid impossible	Deep BSDE in high dimension	prices and hedges at $d\sim 100$	no error bounds; harder training
the risk measure is dynamic and path-dependent	risk-averse reinforcement learning	CVaR/entropic objectives in an MDP	the classic RL instability, amplified

The one-sentence essence:

Every extension of deep hedging exists to move a friction from the model into the objective - costs, model uncertainty, dimension, and dynamic risk preferences - and the price is always the same: a more conservative strategy whose conservatism is a choice, plus a harder optimisation problem with fewer guarantees.

Three things to carry out of this page:

1. Leland’s correction is a driver modification, not a hedge trick. Adding a transaction cost to the replication argument changes the effective volatility to ${\sigma }_L=\sigma \sqrt{1+\sqrt{2/\pi }\kappa /(\sigma \sqrt{\Delta t})}$ - i.e. it changes the driver of the BSDE. §3 shows it lowers cost and tail risk (${\mathrm{CVaR}}_{5\%}$) but not variance: a fixed-scheme correction optimises a different functional than the one you may be measured on.
2. Robustness is a min–max, hence an adversary. The robust representation $\rho (X)={\sup }_{\mathbb{Q}\in \mathcal{Q}}({\mathbb{E}}_{\mathbb{Q}}[-X]-\alpha (\mathbb{Q}))$ of §02 means a robust hedge is trained against a worst case, and the inner supremum has to be represented - by a finite set of scenarios (cheap, interpretable, the §3 approach) or by an adversary network (expressive, unstable).
3. Dimension and risk-aversion are where the method earns its keep. Pricing in $d=100$ and hedging under CVaR are the two problems where nothing classical works; they are the reason the field exists at all.

The practical objective: be able to state and implement the Leland correction and know its limitations, build a cheap min–max robust hedge, know what “high-dimensional” actually buys and does not buy, and recognise the deep-hedging problem inside the reinforcement-learning formalism.

---

2. Mathematical Ground Truth & Derivations

2.1 Transaction costs I: the modified driver

In continuous time with proportional cost $\kappa$ per unit traded notional, the self-financing condition is replaced by

$$d{Y}_t={\delta }_{t}d{S}_t-\kappa S_t|d{\delta }_t|-\left(\text{liability accrual}\right),$$

and $|d{\delta }_t|$ - the total variation of the strategy - is not approximable by a diffusion (it is of order $\sqrt{d⟨\delta {⟩}_t}$, i.e. a local time). Two classical ways to proceed:

(i) Leland’s heuristic (1985). Require the hedged P&L to have zero expected cost by widening the volatility:

$$\boxed{{\sigma }_L^2={\sigma }^2(1+\sqrt{\frac{2}{\pi }}\frac{\kappa }{\sigma \sqrt{\Delta t}})}\text{i.e.}{\delta }_t^{\text{Leland}}=\frac{\partial C_{BS}}{\partial S}{|}_{\sigma ={\sigma }_L}.$$

The correction vanishes as $\Delta t\to 0$ only if $\kappa \to 0$; for fixed $\kappa$ the adjusted volatility blows up as the rebalancing interval shrinks - the mathematical statement that finer rebalancing is increasingly costly. Leland’s derivation is asymptotic ($\kappa$ small, $\Delta t$ small, with $\kappa /\sqrt{\Delta t}$ fixed) and it is a first-order correction to expected cost; it is not optimal for variance or for CVaR, as §3 shows.

(ii) The BSDE/HJB route. The exact problem is a nonlinear pricing equation of the form

$$u_t+\frac{{\sigma }^2}{2}{S}^2{u}_{SS}+\frac{{\sigma }^2{S}^2{u}_S^2\kappa }{\dots }=0,$$

studied by Davis–Panas–Zariphopoulou (1993) and Whalley–Wilmott (1997), which in the BSDE language is a driver $f$ concave in $z$. This is the mathematically honest formulation and the one deep hedging attacks directly: put the cost term (a path functional) into the objective $\rho (L_T^{\delta })$ and let the network optimise it. The practitioner rule that falls out of both: the optimal rebalancing frequency is finite and is a function of $\kappa$ - never of machine speed.

2.2 Transaction costs II: the frequency trade-off

With discrete rebalancing at $N$ dates, §05’s law $\mathrm{SD}\propto N^{-1/2}$ and the cost $\propto N^{1/2}$ combine into

$$\underset{N}{\min }\frac{c_1}{\sqrt{N}}+c_2\kappa \sqrt{N}⟹N^{\star }=\frac{c_1}{c_2\kappa }\text{(the two terms balancing)},$$

which is the same structure as the Almgren–Chriss liquidation trade-off in Optimal Execution & Almgren–Chriss. The measured optimum in §05 at $\kappa =50$bp is $N^{\star }=16$ - and it is a business number (it depends on $\kappa$ and on the risk measure), not a numerical one.

2.3 Robust (min–max) hedging

Let $\mathcal{U}$ be a set of candidate models (here: volatilities $\sigma \in [0.15,0.25]$). The robust hedge is

$$\boxed{{\delta }^{\mathrm{rob}}=\arg \underset{\delta }{\min }\underset{\sigma \in \mathcal{U}}{\max }\rho (L_T^{\delta }(\sigma ))}.$$

This is the primal problem; the dual is the robust representation of the risk measure over a set of measures $\mathcal{Q}$, and the two coincide when $\mathcal{U}$ and $\mathcal{Q}$ are appropriately paired (a minimax theorem - convexity in $\delta$, concavity/compactness in $\sigma$). Three structural facts:

• The robust hedge is not any single-model hedge; it is the hedge whose worst case is smallest. In §3 it coincides with the most extreme model’s hedge - a general phenomenon when the worst case is attained at a boundary of $\mathcal{U}$.
• The uncertainty set is a modelling choice with a first-order effect. Enlarging $\mathcal{U}$ makes the hedge more conservative and the robust price higher - the same “price of robustness” as in Robust Optimization.
• Robustness is cheap when the objective is flat. §3’s worst-case SD improves only $7.5065\to 7.4852$ ($0.28\%$) across a $[15\%,25\%]$ vol set: for a one-period variance objective the min–max hedge buys very little, because the SD is not very sensitive to $\delta$ near the optimum. The larger the uncertainty set and the more asymmetric the risk measure (CVaR!), the more robustness is worth paying for.

2.4 High dimension, and what it does (and does not) buy

Deep BSDE’s headline result is $d=100$. The mechanism: with a grid the cost of a $d$-dimensional PDE is $O(N^d)$, whereas Monte Carlo averaging is $O(n^{-1/2})$ in any dimension and a neural network represents $u(t,\cdot )$ with a parameter count that grows far more slowly than the grid. Formally, the loss is

$$\mathbb{E}[(\xi -Y_T^{\theta }{)}^2],Y_T^{\theta }=Y_0^{\theta }-\sum _{i}f(t_i,Y_i^{\theta },Z_i^{\theta })\Delta t+\sum _i{Z}_i^{\theta }\cdot \Delta W_i,$$

with $Z_i^{\theta }\in {\mathbb{R}}^{1\times d}$ now a matrix-valued network output (a Jacobian), so the per-step network must learn the full gradient. Two honest caveats:

• Monte Carlo error is dimension-free in rate, not in constant. The variance of the terminal payoff grows with $d$ (a basket call is riskier than an ATM call), so the path count needed for a fixed accuracy grows with $d$. The correct claim is ”$O(n^{-1/2})$ regardless of $d$”, not “equally accurate at any $d$“.
• The hedging problem is harder than the pricing problem in high dimension. Pricing needs $Y_0$; hedging needs the whole $Z$ process (the Jacobian). The deep-hedging paper’s experiments are typically in low dimension for exactly this reason - and this is an honest gap between the pricing literature and the hedging literature.

2.5 Risk-averse reinforcement learning

The deep-hedging problem is a finite-horizon Markov decision process with a risk-sensitive objective:

$$\text{state}=(t,S_t,{\delta }_{t^{-}}),\text{action}={\delta }_t,\text{reward}=-\kappa |{\delta }_t-{\delta }_{t^{-}}|S_t,\text{terminal cost}=\rho (\cdot ).$$

and one optimises $\rho$ of the cumulative reward. Two consequences:

• Risk-averse RL (CVaR-RL, entropic-RL) is the same mathematics: a Bellman recursion with a nonlinear (risk) aggregator, whose continuous-time limit is a BSDE with a driver given by the risk measure. This is why BSDEs are not an alternative to RL but its continuous-time skeleton.
• The classical RL pathologies apply: reward normalisation, sparse terminal signals, non-stationarity, and a train/evaluation mismatch. Deep hedging inherits all of them, plus the fact that its “environment” is a simulator the modeller wrote - i.e. the model risk of §05 is built in. See RL · 04 Policy Gradient & Actor-Critic.

---

3. Computational Implementation - Leland’s correction and a min–max robust hedge

We (1) implement the Leland-adjusted delta and measure it against the plain Black–Scholes delta under a $2\%$ proportional cost, and (2) compute the min–max robust one-period delta over a volatility-uncertainty set.

Reading the output.

• (1) Leland lowers cost and tail risk, not variance - and the trade is regime-dependent. At $n=16$ the adjusted volatility is $22.97\%$ (from $20\%$), and the resulting hedge reduces expected cost $3.546\to 3.391$ ($-4.4\%$) and ${\mathrm{CVaR}}_{5\%}$ $3.875\to 3.558$ ($-8.2\%$), while the raw SD rises $1.709\to 1.769$ ($+3.5\%$). Two lessons: (i) Leland’s expansion is calibrated to expected cost, so it improves cost-control functionals and not dispersion - the objective determines whether “the fix worked”; (ii) at $n=64$ the variance penalty grows ($+30\%$), consistent with the corrected volatility growing as $1/\sqrt{\Delta t}$. Leland is a first-order tool whose sign of benefit depends on the loss functional you are measured on.
• (2) The min–max hedge collapses to the most extreme model - and buys very little here. All three single-model hedges have worst-case SD within $1\%$ of each other ($7.5563$, $7.5065$, $7.4852$), and the robust optimum is exactly the $25\%$ delta $0.5997$. Two honest observations: first, when the worst case is attained at a boundary of the uncertainty set, the robust hedge is that boundary’s hedge (no averaging, no “in-between” solution); second, for a symmetric, one-period variance objective the price of robustness is negligible ($0.28\%$). Robustness pays when the risk measure is asymmetric (CVaR) or the uncertainty set is large - otherwise the adversarial formulation is mostly a governance statement, not a P&L improvement.

---

4. Failure Modes & First-Principles Breakdowns

1. Treating Leland’s ${\sigma }_L$ as “the” cost-adjusted model. It is an asymptotic expansion valid for small $\kappa$ with $\kappa /\sqrt{\Delta t}$ fixed, derived for expected cost, and it breaks down (and becomes worse than plain Black–Scholes, as the $n=64$ row of §3 shows) outside its regime. The exact problem is a nonlinear HJB/BSDE with a concave driver - use it directly (deep hedging) when costs are material.
2. Calibrating the uncertainty set $\mathcal{U}$ to make the robust hedge palatable. The set is the conservatism. Choosing it after seeing the answer (“shrink $\mathcal{U}$ until the hedge looks reasonable”) makes the whole robustification decorative. $\mathcal{U}$ belongs to risk management, exactly as $\rho$ and $\alpha$ do.
3. Assuming the robust hedge is a hedge against everything. It is a min–max over the specified set. Model risk outside $\mathcal{U}$ (e.g. a jump the set does not contain) is not covered, and the robust hedge can be more fragile than the nominal one to such a misspecification because it is tuned to a boundary.
4. Importing ”$d=100$ works” from pricing to hedging. The high-dimensional results are for the value; the hedge needs the full Jacobian $Z\in {\mathbb{R}}^{1\times d}$ at every step, whose estimation is the harder and less-benchmarked problem. Do not present a pricing success as a hedging success.
5. Ignoring that the dimension enters the variance, not just the architecture. Monte-Carlo error is dimension-free in rate; a $d$-asset basket has a larger terminal variance, so the path count for fixed accuracy grows with $d$. Efficiency claims must be stated per unit of accuracy, not per unit of dimension.
6. Assuming risk-averse RL is a different subject. It is the same BSDE structure with a nonlinear (risk) aggregator in the Bellman recursion; the continuous-time limit of CVaR-RL is a BSDE with a driver determined by the risk measure. Treating them as separate literatures duplicates effort and loses the comparison/duality theory.
7. Forgetting that RL’s environment is a simulator you wrote. All of deep hedging’s model risk (page 05) is present, plus RL’s own instabilities. The composite failure mode - a risk-averse agent over-optimising a mis-specified simulator - is the most expensive mistake in this space and does not appear in single-period academic experiments.
8. Buying extensions for their asymptotics rather than their mechanism. Leland’s correction, robustness, high dimension and risk-averse RL each fix a specific friction. Adopting one because it is fashionable (or because it fits better) is exactly the failure mode of SABR · 06 §4: extensions are bought to fix dynamics/frictions, and the price is paid in identifiability, stability and interpretability.

---

5. References

• Leland, H.E. (1985), Option pricing and replication with transactions costs, Journal of Finance 40(5), 1283–1301
• Cont, R. (2006), Model uncertainty and its impact on the pricing of derivative instruments, Mathematical Finance 16(3), 519–547; Glasserman, P. & Xu, X. (2014), Robust risk measurement and model risk, Quantitative Finance 14(1), 29–58; Föllmer, H. & Schied, A. (2004), Stochastic Finance (robust representation)
• Han, J., Jentzen, A., E, W. (2018), Solving high-dimensional partial differential equations using deep learning, PNAS 115(34), 8505–8510
• Buehler, H., Gonon, L., Teichmann, J., Wood, B. (2019), Deep Hedging, Quantitative Finance 19(8), 1271–1291
• Gierjatowicz, P., Sabate-Vidales, M., Šiška, D., Szpruch, Ł., Žurič, Ž. (2020), Robust pricing and hedging via neural SDEs, and Horvath, B., Muguruza, A., Tomas, M. (2021), Deep learning volatility, Quantitative Finance
• Chow, Y., Tamar, A., Mannor, S., Pavone, M. (2015), Risk-sensitive and robust decision-making: a CVaR optimization approach, NeurIPS; Tamar, A., Glassner, Y., Mannor, S. (2015), Optimizing the CVaR via sampling; Mäki, H. et al. (2020), Risk-averse deep hedging

---

6. Connected Graph Bridges

• Back: 05 · Failure Modes & Practice · 04 · Deep BSDE & Deep Galerkin · 03 · BSDEs & Nonlinear Feynman–Kac · Index Hub
• Costs & execution: Optimal Execution & Almgren–Chriss · Market Impact & Depth · MM · 03 Temporary vs Permanent Impact · P5 · 03 Transaction-Cost Models
• Robustness & risk: Robust Optimization · P5 · 03 Robust Formulations · Stress Testing & Scenario Analysis · MRV · 06 Advanced Extensions
• RL & frontier: Reinforcement Learning for Trading · RL · 04 Policy Gradient & Actor-Critic · SABR · 06 Advanced Extensions (rough vol, LSV - the model side of the same frontier)
• Verification & governance: Model Risk & Validation · CMP · 06 Advanced Extensions · P8 · 04 Reproducibility