Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2607.23161

Risk Aversion in the Small and in the Large: Beyond Arrow-Pratt A Wiener Chaos Hierarchy of Dynamic Risk Premia

Christian Oliver Ewald
View original source ↗Read source document ↗Export metadata ↓

Abstract

The Arrow-Pratt approximation is one of the cornerstones of expected utility theory, providing the classical local approximation of certainty equivalents and risk premia in terms of absolute risk aversion. Despite its widespread use, its mathematical scope and relationship to higher-order risk preferences remain only partially understood. This paper develops a new framework for the analysis of certainty equivalents and dynamic risk premia based on Malliavin calculus and Wiener chaos analysis. We first show that the classical Arrow-Pratt approximation is not asymptotically valid for arbitrary sequences of vanishing risks, thereby identifying precise limitations of the traditional theory. Motivated by this observation, we formulate certainty equivalents dynamically by considering the progressive revelation of uncertainty through a Brownian filtration. Combining It\^o calculus, the Clark--Ocone representation and the Wiener chaos decomposition, we derive a complete hierarchy of higher-order dynamic risk premia and obtain explicit representations of the corresponding coefficients in terms of Malliavin derivatives. For mixed Wiener chaos expansions, higher-order preference measures, including prudence and temperance, emerge naturally through interactions between chaos components and are characterised using Bell polynomial representations. Explicit results for quadratic Gaussian functionals and the Vasicek interest-rate model illustrate the theory and identify a broad class of regular Wiener functionals for which the classical Arrow-Pratt approximation is recovered as the leading-order term. The results establish a unified framework linking expected utility theory, stochastic analysis and Wiener chaos expansions, opening a new perspective on higher-order certainty equivalents and the dynamic measurement of risk.

Explore related subjects

q-fin.MFecon.GNq-fin.EC

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Christian Oliver Ewald. 2026-07-25. Risk Aversion in the Small and in the Large: Beyond Arrow-Pratt A Wiener Chaos Hierarchy of Dynamic Risk Premia. https://arxiv.org/abs/2607.23161

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Variance-Optimal Hedging in the Rough Hawkes--Heston Model

We study variance-optimal stock hedging and the convergence of approximate strategies in the rough Hawkes--Heston model. Starting from the model's affine conditional transform and the affine Volterra jump framework, we obtain semi-explicit hedges for European calls and a representation of the minimum quadratic error through the Galtchouk--Kunita--Watanabe projection. Our main approximation result keeps the original stock, variance driver, and information flow fixed while regularizing the kernel used to evaluate the hedge. To handle singular memory and common marked jumps, we construct the approximate holdings from histories available before trading and preserve the conditional transform's random modulus envelope. Riccati--Volterra stability and weighted truncation then yield convergence in the original stock's trading norm on compact Fourier intervals. For calls, a joint choice of kernel regularization and Fourier cutoff gives convergence of the initial capitals and strategies, uniform-in-time square-mean convergence of continuous-time gains, and convergence of the terminal mean-square error to the variance-optimal value. A numerical experiment with shifted fractional kernels illustrates the construction on common original-market paths.

q-fin.MF↗
arXiv2026

Numeraire Invariance of Entropy-Projected Martingale Measures

Let \(P\) be a fixed physical law and let \(Q\) be an equivalent martingale measure selected from the martingale-measure set associated with a chosen numeraire. A change of numeraire maps \(Q\) to \(T_LQ\), where \(d(T_LQ)=L\,dQ\) and \(L\) is the terminal likelihood ratio. The forward relative-entropy projection minimizing \(D_{\mathrm{KL}}(P\Vert Q)\) commutes with this transform because its objective changes only by the constant \(-E_P\log L\). The minimal entropy martingale measure (MEMM) orientation \(D_{\mathrm{KL}}(Q\Vert P)\) does not have this property, and a trinomial counterexample shows that independently recomputed MEMMs need not be likelihood compatible. We make two economic consequences explicit. First, the two entropy orientations are precisely the \(Q\)-dependent terms in the classical convex-dual objectives for logarithmic and exponential utility, respectively. Second, likelihood compatibility is equivalent to equality of the pricing functionals obtained in the two numeraires. Hence the forward selectors value every integrable claim consistently across numeraires, whereas the two MEMMs in the counterexample assign different prices to a nonreplicable digital claim. We also prove a finite-state class-level characterization: uniform invariance over the elementary one-period likelihood-ratio families forces a smooth convex \(f\)-divergence to be logarithmic, up to scaling and affine equivalence. Finally, in finite-state markets, the forward projection exists under the usual strictly positive feasible-point condition; its density \(dP/dQ^*\) is attainable log-optimal terminal wealth, and the minimum forward entropy equals maximal expected log growth.

q-fin.MF↗
arXiv2026

The Delta of a Variance Swap

We define the variance swap delta as the sensitivity of the price of variance to a change in underlying price. We use Carr-Madan spanning formulas to analyze this sensitivity when the implied volatility smile curve may depend on the underlying price. We show that the variance swap total delta is zero for the class of smile curves that are pure functions of (log) moneyness, which goes against the empirical observation that variance is up when the market is down. We propose a simple modification of the smile to correct this issue.

q-fin.MF↗