Arbitrage-free Market Research Articles

Valuation of guaranteed minimum accumulation benefits (GMABs) with physics-inspired neural networks

Abstract Guaranteed minimum accumulation benefits (GMABs) are retirement savings vehicles that protect the policyholder against downside market risk. This article proposes a valuation method for these contracts based on physics-inspired neural networks (PINNs), in the presence of multiple financial and biometric risk factors. A PINN integrates principles from physics into its learning process to enhance its efficiency in solving complex problems. In this article, the driving principle is the Feynman–Kac (FK) equation, which is a partial differential equation (PDE) governing the GMAB price in an arbitrage-free market. In our context, the FK PDE depends on multiple variables and is difficult to solve using classical finite difference approximations. In comparison, PINNs constitute an efficient alternative that can evaluate GMABs with various specifications without the need for retraining. To illustrate this, we consider a market with four risk factors. We first derive a closed-form expression for the GMAB that serves as a benchmark for the PINN. Next, we propose a scaled version of the FK equation that we solve using a PINN. Pricing errors are analyzed in a numerical illustration.

A Black\u2013Scholes inequality: applications and generalisations

The space of call price curves has a natural noncommutative semigroup structure with an involution. A basic example is the Black–Scholes call price surface, from which an interesting inequality for Black–Scholes implied volatility is derived. The binary operation is compatible with the convex order, and therefore a one-parameter sub-semigroup gives rise to an arbitrage-free market model. It is shown that each such one-parameter semigroup corresponds to a unique log-concave probability density, providing a family of tractable call price surface parametrisations in the spirit of the Gatheral–Jacquier SVI surface. An explicit example is given to illustrate the idea. The key observation is an isomorphism linking an initial call price curve to the lift zonoid of the terminal price of the underlying asset.

Comonotonic asset prices in arbitrage-free markets

For an arbitrage-free market with a single underlying asset, we investigate conditions under which the consecutive price levels are comonotonic. Furthermore, for an arbitrage-free market with n assets we investigate the consequences of assuming comonotonicity of the vector containing the price levels of each asset at a single future date T . Although being of a theoretical nature, the results of this paper give insight in the reachability of the comonotonic upper bounds for Asian options and for basket options that can be found in Simon et al. (2000), Dhaene et al. (2002) [1], Hobson et al. (2005) and Chen et al. (2008).

Semi-nonparametric approximation and index options

In an arbitrage-free securities market, all state-contingent claims and the stochastic discount factors can be approximated appropriately by index options with a semi-nonparametric method. These index options are constructed by efficient algorithms and uniform approximation error under these efficient algorithms are derived. This paper suggests a method to examine state-contingent claims and stochastic discount factors using index options in financial market regardless the market is complete or not.

Comonotonic Asset Prices in Arbitrage-Free Markets

For an arbitrage-free market with a single underlying asset, we investigate conditions under which the consecutive price levels are comonotonic. Furthermore, for an arbitrage-free market with n assets we investigate the consequences of assuming comonotonicity of the vector containing the price levels of each asset at a single future date T. Although being of a theoretical nature, the results of this paper give insight in the reachability of the comonotonic upper bounds for Asian options and for basket options that can be found in Simon et al. (2000), Dhaene et al. (2002b), Hobson et al. (2005) and Chen et al. (2008).

Natural Hedging with Fix and Floating Strike Guarantees

The paper analyzes minimum return rate guarantees (MRRGs) including fixed guarantee rates prevailing for the whole contract horizon as well as floating guarantee rates which are linked to the interest rate evolution. In a complete arbitrage free market where the asset and bond price dynamics are given by Gaussian processes, we obtain closed form pricing solutions for both guarantee schemes. Differences in the guarantee costs are then explained by the difference of the arbitrage free values of the fix and floating rate guarantees and the difference between cumulated volatilities resulting from forward and simple volatilities. We then consider the perspective of the asset liability management, i.e. we analyze the sensitivities of the asset and liability side against changes in the interest rate. We show that a combination of fix price and floating strike guarantees enables a natural hedge against changes in the interest rate.

Long-run heterogeneity in an exchange economy with fixed-mix traders

We consider an exchange economy where agents have heterogeneous beliefs and assets are long-lived, and investigate the coupled dynamics of asset prices and agents’ wealth. We assume that agents hold fixed-mix portfolios and invest on each asset proportionally to its expected dividends. We prove the existence and uniqueness of a sequence of arbitrage-free market equilibrium prices and provide sufficient conditions for an agent, or a group of agents, to survive or dominate. Our main finding is that long-run coexistence of agents with heterogeneous beliefs, leading to asset prices endogenous fluctuations, is a generic outcome of the market selection process.

Optimization of Constrained Liquidity Management

A consistent framework for optimal liquidity management is presented. This framework optimizes the cost of covering expected cashflow gaps without violating regulatory and business constraints. Anticipated economic value loss, cashflow loss, and adverse market impact are the major drivers of cost. The notion of a deployable liquidity resource, which is subsequently extend to the notion of a dated liquidity strategy, is introduced. A formalization of LCR as a typical regulatory constraints is presented and included in the formulation. The formulation includes a general arbitrage free market impact function. A decoupling between liquidity risk management and that of market and credit risks is assumed. Both linear and quadratic programming approaches for solving the resulting optimization problem are derived. This is followed by the introduction of a novel mapping algorithm which transforms the linear program to a network flow problem that is more efficiently solvable via the network simplex algorithm. Next an algorithm for generating a plausible starting point for the iterative optimization problem, is given. This is shown to be already optimal under the risk neutral measure. Finally, heuristics that can help speed up the satisfaction of regulatory constraints are discussed. Throughout the presentation attention is given to algorithmic complexity issues.

There are No Predictable Jumps in Arbitrage-Free Markets

We model asset prices in the most general sensible form as special semimartingales. This approach allows us to also include jumps in the asset price process. We show that the existence of an equivalent martingale measure, which is essentially equivalent to no-arbitrage, implies that the asset prices cannot exhibit predictable jumps. Hence, in arbitrage-free markets the occurrence and the size of any jump of the asset price cannot be known before it happens. In practical applications it is basically not possible to distinguish between predictable and unpredictable discontinuities in the price process. The empirical literature has typically assumed as an identification condition that there are no predictable jumps. Our result shows that this identification condition follows from the existence of an equivalent martingale measure, and hence essentially comes for free in arbitrage-free markets.

Quantile Hedging in Arbitrage-Free Market. Cases of Complete and Incomplete Market

Quantile Hedging in Arbitrage-Free Market. Cases of Complete and Incomplete Market

Customizable Pecuniary Risk and Market Incompleteness Premium

We examine the optimal customization of a financial derivative in the presence of a background risk. This problem includes the model of finding the optimal constant amount of a given pecuniary risk as a degenerated case. We show the importance of this perspective with a preference-free solution for the pecuniary background risk case and a general solution for any given utility function. The latter solution allows us to measure the incompleteness premium demanded by risk-averse agent for a market where only constant amount of risk is allowed in comparison with a complete and arbitrage-free market where the customization is unrestricted.

A Tale in Three Cities: Comparison between GVV, SVI and IRV (Presentation Slides)

This presentation was based on the work of Jian Sun and Qiankun Niu, and presented at Morgan Stanley Global Modeling Meeting by Qiankun Niu on August 27, 2014. We thank Peter Carr for his invitation.Market Model is classified as the approach to model the implied volatility surface directly, which based on the observation that vallina options are liquidity traded in the market and the prices are driven by supply and demand factors. No-arbitrage imposes strict and complicated shape constraints on the implied volatility surface. Existed solution for this issue is to use another assistant quantity but sacrifice the simplicity of market models. To the best of our knowledge, a full arbitrage-free market model for the evolution of the implied volatility surface has yet to be developed.This work is an extension of the recent paper Carr and Sun (2013). The main thrust of this paper is to first follow the market model approach to establish a no arbitrage model named IRV model with simple closed form formula. Then, we will demonstrate the consistency between our IRV model and other two empirical models: SVI model in Gatheral and Jacquier (2014) and GVV model in Arslan et al. (2009). Potential application will be discussed in later sections. Applying the new model to the S&P 500 index options market and swaption market, the calibration results show that the model proposed in this paper works very well.

BUBBLES AND MULTIPLE-FACTOR ASSET PRICING MODELS

This paper derives a multiple-factor asset pricing model with asset price bubbles in an arbitrage-free, competitive, and frictionless market. As such it generalizes existing asset pricing models, all of which implicitly assume asset price bubbles do not exist. This generalization leads to two new empirical implications. The first is that positive alphas can exist in an arbitrage-free market due to the existence of asset price bubbles. These positive alphas do not represent abnormal profit opportunities. The second is that bubble risk factors can exist with positive risk premiums. The testing of these new empirical implications awaits subsequent research.

Bubbles and Multiple-Factor Asset Pricing Models

This paper derives a multiple-factor asset pricing model with asset price bubbles in an arbitrage-free, competitive, and frictionless market. As such it generalizes existing asset pricing models, all of which implicitly assume asset price bubbles do not exist. This generalization leads to two new empirical implications. The first is that positive alphas can exist in an arbitrage-free market due to the existence of asset price bubbles. These positive alphas do not represent abnormal profit opportunities. The second is that bubble risk factors can exist with positive risk premiums. The testing of these new empirical implications awaits subsequent research.

Russian

The article, based on materials, examines one of the methods for determining the access of the arbitrage-free market buyers, for example, the distribution Furry.

Optimal Investment with Nonconcave Utilities in Discrete-Time Markets

We consider an arbitrage-free, discrete-time and frictionless market. We prove that an investor maximizing the expected utility of his or her terminal wealth can always find an optimal investment strategy provided that his or her dissatisfaction of infinite losses is infinite and his or her utility function is nondecreasing, continuous, and bounded above. The same result is shown for cumulative prospect theory preferences, under additional assumptions.

Partial Hedging of American Claims in a Discrete Market

The article examines various formulations of the problem of partial hedging of American claims in an arbitrage-free and incomplete market. Market behavior scenarios are modeled by a finite tree without self-intersections. The problems are considered from the seller's viewpoint without information about buyer behavior. The main emphasis is on simplifying the constraints entering the explicit form of the buyer strategy. We find that in all problems the assumption of completely uncertain stopping time can be replaced with simpler restrictions.

Financial economics without probabilistic prior assumptions

The treatment of uncertainty in general equilibrium theory in the style of Arrow and Debreu does not require a prior probability on the state space. Finance models nevertheless treat payoffs as random variables, implicitly or explicitly using a known probability distribution. In the light of Knightian uncertainty, we might challenge such an assumption on the probabilistic sophistication of our market model. The present paper shows that one can still develop a sound model of arbitrage pricing under complete Knightian uncertainty as long as certain continuity conditions are met. The pricing functional given by an arbitrage-free market can be identified with a full support martingale measure (instead of equivalent martingale measure). We relate the no-arbitrage theory to economic equilibrium by establishing a variant of the Harrison–Kreps theorem on viability and no arbitrage. Finally, we consider (super) hedging of contingent claims and embed it in a classical infinite-dimensional linear programming problem.

Konstrukce výnosových křivek v pokrizovém období

Market value of derivatives after crisis requires discounting with interest rates that take into account the credit risk of the involved counterparties of the trade. The increase of credit risk is evidenced by the presence of basis swap spreads. Using one curve to both estimating the forward rates and discounting future cash flows is not plausible given the prerequisite of arbitrage free market. The aim of this paper is to derive discount curves which are consistent with market quotes. In the concluding part, we estimate CZK OIS rates, which can be then used to discount derivatives denominated in CZK and collateralized with CZK cash.

Transaction Costs, Shadow Prices, and Duality in Discrete Time

For portfolio choice problems with proportional transaction costs, we discuss whether or not there exists a shadow price, i.e., a least favorable frictionless market extension leading to the same optimal strategy and utility. By means of an explicit counterexample, we show that shadow prices may fail to exist even in seemingly perfectly benign situations, i.e., for a log-investor trading in an arbitrage-free market with bounded prices and arbitrarily small transaction costs. We also clarify the connection between shadow prices and duality theory. Whereas dual minimizers need not lead to shadow prices in above “global” sense, we show that they always correspond to a “local” version.