Scaling Limits for Exponential Hedging in Trinomial Models

Abstract: We study scaled trinomial models converging to the Black--Scholes model, and analyze exponential certainty-equivalent prices for path-dependent European options. As the number of trading dates $n$ tends to infinity and the risk aversion is scaled as $nl$ for a fixed constant $l>0$, we derive a nontrivial scaling limit. Our analysis is purely probabilistic. Using a duality argument for the certainty equivalent, together with martingale and weak-convergence techniques, we show that the limiting problem takes the form of a volatility control problem with a specific penalty. For European options with Markovian payoffs, we analyze the optimal control problem and show that the corresponding delta-hedging strategy is asymptotically optimal for the primal problem.

• The paper establishes that exponential utility-indifference pricing in trinomial models converges to a continuous-time stochastic control problem as trading frequency increases.
• The authors employ weak convergence, dual representations, and entropy penalization to link trinomial hedging strategies with dynamic programming and viscosity solutions of the associated HJB equation.
• The findings illustrate that resulting prices interpolate between the Black–Scholes value and worst-case super-replication, offering key insights for robust option pricing and model calibration.

Scaling Limits for Exponential Hedging in Trinomial Models

Motivation and Framework

The paper "Scaling Limits for Exponential Hedging in Trinomial Models" (2603.28948) addresses the convergence of discrete-time utility-based hedging and pricing problems in incomplete markets, specifically within the context of scaled trinomial models approaching the Black--Scholes regime. Classical complete market models, such as the Cox--Ross--Rubinstein binomial tree, admit perfect replication and thus unique arbitrage-free prices. In contrast, trinomial models are incomplete due to the introduction of a zero-increment state, resulting in residual, irreducible risk. The focus lies on investors with exponential utility, whose risk aversion is scaled proportionally with the trading frequency, and on the certainty-equivalent pricing of potentially path-dependent European claims.

The central objective is to determine the limiting behavior of exponential utility-indifference prices, including both the value function and optimal strategies, as the number of discrete trading periods $n$ diverges and both the risk aversion and local volatility scaling are suitably renormalized.

Model Specification and Main Theorem

Consider a trinomial model with $n$ steps and a time horizon $[0,1]$. Conditional increments are i.i.d. taking values $\{-1,0,1\}$, with probabilities $\{p/2, 1-p, p/2\}$, and stock price process dynamics are given by

$$S_{k}^{n}=S_{0} \prod_{j=1}^{k}\left(1+\frac{\bar{\sigma}}{\sqrt{n}}\xi_{j}\right), \quad k=0,\ldots,n,$$

where $\bar{\sigma}>0$ is a volatility parameter. The measure $\mathbb{P}$ is the reference (pre-model) measure. Hedging strategies consist of adapted share holdings $(\gamma_0,\ldots,\gamma_{n-1})$, and the exponential certainty-equivalent for a European payoff $F_n$ is defined as

$n$0

where $n$1 encapsulates the investor's absolute risk aversion, scaled linearly with $n$2.

The Main Theorem establishes that, under suitable regularity and as $n$3, $n$4 converges to a continuous-time stochastic control problem of the form: $n$5 Here, $n$6 is a stochastic exponential with time-dependent volatility $n$7 (bounded, progressively measurable, $n$8), and $n$9 is a specific relative entropy (Bernoulli divergence) given by

$[0,1]$0

The penalty accounts for the deviation in quadratic variation from the reference trinomial baseline.

Analytical Structure and Methodology

The proof methodology is entirely probabilistic, leveraging weak convergence of the stock process (under quantifiable martingale measures), duality representations of the certainty equivalent, and tightness arguments within the Skorokhod and $[0,1]$1 spaces. The paper avoids PDE-centric or functional analytic machinery, instead using martingale and stochastic control tools to pass to the scaling limits.

A critical insight is the analysis of the dual representation for the certainty equivalent in discrete time, which can be decomposed as an optimization over martingale measures (with potentially node-dependent variance choices), subject to entropy penalization. Under scaling, the effective continuous-time control becomes the volatility process itself, interpolating between risk-neutral and worst-case ("$[0,1]$2-expectation") pricing. The penalty term in the control problem is exactly the specific relative entropy between the candidate (quadratic variation) volatility law and the reference.

For convex payoffs, the monotonicity of the certainty equivalent price function in both $[0,1]$3 and $[0,1]$4 is established via dynamic programming and convex ordering arguments, ensuring well-posedness and stability with respect to model specification.

Markovian Payoff Case and the Associated HJB Equation

When $[0,1]$5 depends only on the final stock price (Markovian), the limiting control problem can be analyzed via viscosity solution techniques for associated HJB equations. The resulting PDE for the value function $[0,1]$6 takes the nonlinear, nondivergence form: $[0,1]$7 where $[0,1]$8 is a function inherited from the entropy penalization kernel.

The paper rigorously derives existence, uniqueness, regularity (Lipschitz and Hölder continuity), and comparison principles for the PDE, even for path-dependent payoffs under reasonable smoothness and growth criteria. Furthermore, the $[0,1]$9-based delta-hedging strategy is shown to be asymptotically optimal, i.e., its certainty equivalent value converges to the same limit as the original utility maximization problem.

Notable Results and Consequences

• In the limit $\{-1,0,1\}$0 (infinite risk aversion), the entropy penalty vanishes, and the limiting price agrees with the super-replication value under volatility uncertainty: a $\{-1,0,1\}$1-expectation over volatilities in $\{-1,0,1\}$2.
• In the opposite limit $\{-1,0,1\}$3 (risk neutrality), the limiting price recovers the Black--Scholes value with volatility $\{-1,0,1\}$4.
• For any fixed $\{-1,0,1\}$5, the certainty equivalent price lies strictly between the classical Black--Scholes price and the worst-case super-replication price, interpolating smoothly as a function of investor risk aversion and the incompleteness parameter.

The paper’s assertion that the optimal volatility control problem with entropy penalization provides a concrete, operational connection between discrete-time exponential utility-based hedging and the continuous-time volatility control paradigm is a significant claim. The limiting control structure is further recognized as central in model calibration under entropy penalties, providing connections to specific relative entropy and robust finance literature.

Practical and Theoretical Implications

The results provide new theoretical underpinnings for utility-based pricing and hedging in incomplete markets and suggest practical implications for model calibration, robust option pricing, and numerical approximation strategies for exponential utility indifference prices via trinomial trees. The approach clarifies how penalty-driven, continuous-time control problems emerge as scaling limits of discrete-time utility-based hedging, thus connecting stochastic control, divergence penalization, and probabilistic convergence.

The explicit construction of asymptotically optimal delta-hedging strategies for Markovian claims provides a basis for practical implementation and error quantification in high-frequency regimes. Additionally, the robust representation and penalty structure facilitate a probabilistic interpretation of model uncertainty and allow for natural extensions to superhedging, convex risk measures, and divergence-based model calibration methodologies.

Future Directions

Potential research directions involve generalization to multi-asset settings or more intricate payoff structures (e.g., American or path-dependent options), and detailed error analysis for finite-$\{-1,0,1\}$6 approximations. Extensions to models involving frictions, stochastic interest rates, and feedback between control and volatility uncertainty are also suggested by the theoretical framing. The methodology provides a natural bridge for stochastic analysis and PDE-based approaches in modern mathematical finance.

Conclusion

This paper elucidates the asymptotic limit of exponential utility-based hedging in incomplete trinomial models, connecting the discrete dynamic programming representation to a robust continuous-time volatility control problem with entropy penalization. Through a purely probabilistic argument, it establishes well-posedness, convergence, and optimality results and interprets the scaling limits in terms of both risk-neutral and worst-case pricing. The framework justifies the use of entropy-penalized control in robust pricing and model calibration and provides rigorous results on the structure and behavior of utility-indifference prices in the high-frequency, risk-averse regime.