arXiv subjects

Eckhard Platen

Publications and source records attributed to Eckhard Platen.

At least 19 recordsLinked to original sources

arXiv2026

Real-world models for multiple term structures: a unifying HJM semimartingale framework

We develop a unified framework for modeling multiple term structures arising in financial, insurance, and energy markets, adopting an extended Heath-Jarrow-Morton (HJM) approach under the real-world probability. We study market viability and characterize the set of local martingale deflators. We conduct an analysis of the associated stochastic partial differential equation (SPDE), addressing existence and uniqueness of solutions, invariance properties and existence of affine realizations.

q-fin.MF↗

arXiv2026

Information-Theoretic Approach to Financial Market Modelling

The paper treats the financial market as a communication system, using four information-theoretic assumptions to derive an idealized model with only one parameter. State variables are scalar stationary diffusions. The model minimizes the surprisal of the market and the Kullback-Leibler divergence between the benchmark-neutral pricing measure and the real-world probability measure. The state variables, their sums, and the growth optimal portfolio of the stocks evolve as squared radial Ornstein-Uhlenbeck processes in respective activity times.

q-fin.MF↗

arXiv2025

Free Lunches with Vanishing Risks Most Likely Exist

The hypothesis that there do not exist free lunches with vanishing risk (FLVRs) in the real market underpins the popular risk-neutral pricing and hedging methodology in quantitative finance. The paper documents the fact that this hypothesis can be safely rejected. It performs extremely accurately the hedging of an extreme-maturity zero-coupon bond (ZCB). This hedge is part of a portfolio that starts with zero initial wealth and invests dynamically in a total return stock market index and the savings account to generate at the maturity date of the extreme-maturity ZCB a strictly positive amount with strictly positive probability, which represents an FLVR. The fact that FLVRs naturally exist in the real market can be accommodated theoretically under the benchmark approach.

q-fin.MF↗

arXiv2025

Information-minimizing stationary financial market dynamics

The paper derives the dynamics of a financial market from basic mathematical principles. It models the market dynamics using independent stationary scalar diffusions, assumes the existence of its growth optimal portfolio (GOP), interprets the market as a communication system, and minimizes, in an information-theoretical sense, the joint information of the risk-neutral pricing measure with respect to the real-world probability measure. In this information-minimizing market, its basic independent securities, their sums, minimum variance portfolio, and GOP, as well as the GOP of the entire market, represent squared radial Ornstein-Uhlenbeck processes with additivity and self-similarity properties.

q-fin.MF↗

arXiv2025

Benchmark-Neutral Risk-Minimization for insurance products and nonreplicable claims

In this paper we study the pricing and hedging of nonreplicable contingent claims, such as long-term insurance contracts like variable annuities. Our approach is based on the benchmark-neutral pricing framework of Platen (2024), which differs from the classical benchmark approach by using the stock growth optimal portfolio as the numéraire. In typical settings, this choice leads to an equivalent martingale measure, the benchmark-neutral measure. The resulting prices can be significantly lower than the respective risk-neutral ones, making this approach attractive for long-term risk-management. We derive the associated risk-minimizing hedging strategy under the assumption that the contingent claim possesses a martingale decomposition. For a set of nonreplicable contingent claims, these strategies allow monitoring the working capital required to generate their payoffs and enable an assessment of the resulting diversification effects. Furthermore, an algorithmic refinancing strategy is proposed that allows modeling the working capital. Finally, insurance-finance arbitrages of the first kind are introduced and it is demonstrated that benchmark-neutral pricing effectively avoids such arbitrages.

q-fin.MF↗

arXiv2025

Pricing under the Benchmark Approach

The paper summarizes key results of the benchmark approach with a focus on the concept of benchmark-neutral pricing. It applies these results to the pricing of an extreme-maturity European put option on a well-diversified stock index. The growth optimal portfolio of the stocks is approximated by a well-diversified stock portfolio and modeled by a drifted time-transformed squared Bessel process of dimension four. It is shown that the benchmark-neutral price of a European put option is theoretically the minimal possible price and the respective risk-neutral put price turns out to be significantly more expensive.

q-fin.MF↗

arXiv2024

Benchmark-Neutral Pricing

The paper introduces benchmark-neutral pricing and hedging for long-term contingent claims. It employs the growth optimal portfolio of the stocks as numeraire and the new benchmark-neutral pricing measure for pricing. For a realistic parsimonious model, this pricing measure turns out to be an equivalent probability measure, which is not the case for the risk-neutral pricing measure. Many risk-neutral prices of long-term contracts are more expensive than necessary. Benchmark-neutral pricing identifies the minimal possible prices of contingent claims, which is illustrated with remarkable accuracy for a long-term zero-coupon bond.

q-fin.MF↗

arXiv2023

Entropy-Maximizing Dynamics of Continuous Markets

By assuming the existence of the growth optimal portfolio (GOP), the stationarity of GOP-volatilities, and the maximization of relative entropy, the paper applies the benchmark approach to the modeling of the long-term dynamics of continuous markets. It reveals conservation laws, where the GOP is shown to follow a time-transformed squared Bessel process of dimension four. Moreover, it predicts the convergence of the averages of the GOP-volatilities with respect to the driving independent Brownian motions toward a common level.

math.GM↗

arXiv2022

Exploiting arbitrage requires short selling

We show that in a financial market given by semimartingales an arbitrage opportunity, provided it exists, can only be exploited through short selling. This finding provides a theoretical basis for differences in regulation for financial services providers that are allowed to go short and those without short sales. The privilege to be allowed to short sell gives access to potential arbitrage opportunities, which creates by design a bankruptcy risk.

q-fin.MF↗

arXiv2022

No arbitrage and multiplicative special semimartingales

Consider a financial market with nonnegative semimartingales which does not need to have a numéraire. We are interested in the absence of arbitrage in the sense that no self-financing portfolio gives rise to arbitrage opportunities, where we are allowed to add a savings account to the market. We will prove that in this sense the market is free of arbitrage if and only if there exists an equivalent local martingale deflator which is a multiplicative special semimartingale. In this case, the additional savings account relates to the finite variation part of the multiplicative decomposition of the deflator.

q-fin.MF↗

arXiv2021

No-arbitrage concepts in topological vector lattices

We provide a general framework for no-arbitrage concepts in topological vector lattices, which covers many of the well-known no-arbitrage concepts as particular cases. The main structural condition we impose is that the outcomes of trading strategies with initial wealth zero and those with positive initial wealth have the structure of a convex cone. As one consequence of our approach, the concepts NUPBR, NAA$_1$ and NA$_1$ may fail to be equivalent in our general setting. Furthermore, we derive abstract versions of the fundamental theorem of asset pricing (FTAP), including an abstract FTAP on Banach function spaces, and investigate when the FTAP is warranted in its classical form with a separating measure. We also consider a financial market with semimartingales which does not need to have a numéraire, and derive results which show the links between the no-arbitrage concepts by only using the theory of topological vector lattices and well-known results from stochastic analysis in a sequence of short proofs.

math.FA↗

arXiv2020

Robust Product Markovian Quantization

Recursive marginal quantization (RMQ) allows the construction of optimal discrete grids for approximating solutions to stochastic differential equations in d-dimensions. Product Markovian quantization (PMQ) reduces this problem to d one-dimensional quantization problems by recursively constructing product quantizers, as opposed to a truly optimal quantizer. However, the standard Newton-Raphson method used in the PMQ algorithm suffers from numerical instabilities, inhibiting widespread adoption, especially for use in calibration. By directly specifying the random variable to be quantized at each time step, we show that PMQ, and RMQ in one dimension, can be expressed as standard vector quantization. This reformulation allows the application of the accelerated Lloyd's algorithm in an adaptive and robust procedure. Furthermore, in the case of stochastic volatility models, we extend the PMQ algorithm by using higher-order updates for the volatility or variance process. We illustrate the technique for European options, using the Heston model, and more exotic products, using the SABR model.

q-fin.CP↗

arXiv2020

Existence of equivalent local martingale deflators in semimartingale market models

This paper offers a systematic investigation on the existence of equivalent local martingale deflators, which are multiplicative special semimartingales, in financial markets given by positive semimartingales. In particular, it shows that the existence of such deflators can be characterized by means of the modified semimartingale characteristics. Several examples illustrate our results. Furthermore, we provide interpretations of the deflators from an economic point of view.

q-fin.MF↗

arXiv2019

On the existence of sure profits via flash strategies

We introduce and study the notion of sure profit via flash strategy, consisting of a high-frequency limit of buy-and-hold trading strategies. In a fully general setting, without imposing any semimartingale restriction, we prove that there are no sure profits via flash strategies if and only if asset prices do not exhibit predictable jumps. This result relies on the general theory of processes and provides the most general formulation of the well-known fact that, in an arbitrage-free financial market, asset prices (including dividends) should not exhibit jumps of a predictable direction or magnitude at predictable times. We furthermore show that any price process is always right-continuous in the absence of sure profits. Our results are robust under small transaction costs and imply that, under minimal assumptions, price changes occurring at scheduled dates should only be due to unanticipated information releases.

q-fin.TR↗

arXiv2019

Real-world forward rate dynamics with affine realizations

We investigate the existence of affine realizations for Lévy driven interest rate term structure models under the real-world probability measure, which so far has only been studied under an assumed risk-neutral probability measure. For models driven by Wiener processes, all results obtained under the risk-neutral approach concerning the existence of affine realizations are transferred to the general case. A similar result holds true for models driven by compound Poisson processes with finite jump size distributions. However, in the presence of jumps with infinite activity we obtain severe restrictions on the structure of the market price of risk; typically, it must even be constant.

q-fin.MF↗

arXiv2019

Fair Pricing of Variable Annuities with Guarantees under the Benchmark Approach

In this paper we consider the pricing of variable annuities (VAs) with guaranteed minimum withdrawal benefits. We consider two pricing approaches, the classical risk-neutral approach and the benchmark approach, and we examine the associated static and optimal behaviors of both the investor and insurer. The first model considered is the so-called minimal market model, where pricing is achieved using the benchmark approach. The benchmark approach was introduced by Platen in 2001 and has received wide acceptance in the finance community. Under this approach, valuing an asset involves determining the minimum-valued replicating portfolio, with reference to the growth optimal portfolio under the real-world probability measure, and it both subsumes classical risk-neutral pricing as a particular case and extends it to situations where risk-neutral pricing is impossible. The second model is the Black-Scholes model for the equity index, where the pricing of contracts is performed within the risk-neutral framework. Crucially, we demonstrate that when the insurer prices and reserves using the Black-Scholes model, while the insured employs a dynamic withdrawal strategy based on the minimal market model, the insurer may be underestimating the value and associated reserves of the contract.

q-fin.PR↗

arXiv2018

A Penny Saved is a Penny Earned: Less Expensive Zero Coupon Bonds

In this paper we show how to hedge a zero coupon bond with a smaller amount of initial capital than required by the classical risk neutral paradigm, whose (trivial) hedging strategy does not suggest to invest in the risky assets. Long dated zero coupon bonds we derive, invest first primarily in risky securities and when approaching more and more the maturity date they increase also more and more the fraction invested in fixed income. The conventional wisdom of financial planners suggesting investor to invest in risky securities when they are young and mostly in fixed income when they approach retirement, is here made rigorous. The paper provides a strong warning for life insurers, pension fund managers and long term investors to take the possibility of less expensive products seriously to avoid the adverse consequences of the low interest rate regimes that many developed economies face.

q-fin.PR↗

arXiv2018

Quantization Under the Real-world Measure: Fast and Accurate Valuation of Long-dated Contracts

This paper provides a methodology for fast and accurate pricing of the long-dated contracts that arise as the building blocks of insurance and pension fund agreements. It applies the recursive marginal quantization (RMQ) and joint recursive marginal quantization (JRMQ) algorithms outside the framework of traditional risk-neutral methods by pricing options under the real-world probability measure, using the benchmark approach. The benchmark approach is reviewed, and the real-world pricing theorem is presented and applied to various long-dated claims to obtain less expensive prices than suggested by traditional risk-neutral valuation. The growth-optimal portfolio (GOP), the central object of the benchmark approach, is modelled using the time-dependent constant elasticity of variance model (TCEV). Analytic European option prices are derived and the RMQ algorithm is used to efficiently and accurately price Bermudan options on the GOP. The TCEV model is then combined with a $3/2$ stochastic short-rate model and RMQ is used to price zero-coupon bonds and zero-coupon bond options, highlighting the departure from risk-neutral pricing.

q-fin.CP↗