No-arbitrage Principle Research Articles

Gold’s Value as an Investment

Abstract This paper presents an approach for pricing gold from investors’ perspective. The model is based on no-arbitrage principles with minimal structural assumptions. There is no need to specify investor preferences. When fitted to match 10-year real U.S. Treasury rates, the model can replicate the salient fluctuations in the time series of gold prices since 2007. The model is also able to capture key patterns of CME Comex gold futures prices. The model implies that the majority of the value of gold is due to its role as an investment asset. (JEL G12, G13)

Numerical Methods for Option Pricing

Nowadays, financial markets have become more complex and have given rise to more research opportunities, one of which focuses on research related to the pricing of various financial instruments used in options trading. In this essay, a gradual process of reflection is used to deepen the option pricing theory. Firstly, an effective mathematical method for pricing option contracts is using the binomial tree model, a relatively straightforward way of calculating the value of an option. However, it has only two upward and downward trends and has limitations that are the most different from the actual market conditions. That is why the study will have to continue in-depth to get as close as possible to the real situation in the market. The next step is to make the model more in line with market reality by adding a possible rate of change, resulting in the Trinomial Tree Model. The model incorporates the possibility of future price increases, decreases, or stabilization (the third change). The Trinomial Tree Model follows the no-arbitrage principle and removes the assumption of a risk-free market opportunity. This paper derives a method for constraining market prices under risk-neutral conditions. This is crucial for investors seeking profitable outcomes in options trading. The main objective of this research-based paper is to develop a complete theory of option pricing in a one-step trinomial tree model.

Pricing geometric average Asian options in the mixed sub-fractional Brownian motion environment with Vasicek interest rate model

<p>Considering the characteristics of long-range correlations in financial markets, the issue of valuing geometric average Asian options is examined, assuming that the variations of the underlying asset follow the mixed sub-fractional Brownian motion, and the dynamics of short-term interest rate satisfies the mixed sub-fractional Vasicek model. Based on the principle of no arbitrage, the definite solution of PDE of a zero-coupon bond for geometric average Asian options under the circumstance of the mixed sub-fractional is given by the delta hedging technique. The derivation of the explicit pricing formula for geometric average Asian options with fixed strike price is achieved through the utilization of multiple variable substitutions. Furthermore, we perform numerical calculations to analyze the performance of the model.</p>

Tighter Robust Upper Bounds for Options via No-Regret Learning

Classic option pricing models, such as the Black-Scholes formula, often depend on some rigid assumptions on the dynamics of the underlying asset prices. These assumptions are inevitably violated in practice and thus induce the model risk. To mitigate this, robust option pricing that only requires the no-arbitrage principle has attracted a great deal of attention among researchers. In this paper, we give new robust upper bounds for option prices based on a novel η-momentum trading strategy. Our bounds for European options are tighter for most common moneyness, volatility, and expiration date setups than those presented in the existing literature. Our bounds for average strike Asian options are the first closed-form robust upper bounds for those options. Numerical simulations demonstrate that our bounds significantly outperform the benchmarks for both European and Asian options.

Nonequilibrium Geometric No-Arbitrage Principle and Asset Pricing Theorem

We find a novel and intimate correspondence in the present paper between the martingale and one-parameter transformation group and develop a nonequilibrium geometric no-arbitrage principle to a frictional financial market via this correspondence. Further, we achieve a fundamental pricing theorem via a geometric pricing transform (generator). Finally, we derive that the nonequilibrium geometric no-arbitrage is equivalent to NFLVR in a frictionless financial market. In addition, we apply the nonequilibrium geometric no-arbitrage condition to a frictional financial market. At the end of this paper, a numerical example confirms the effectiveness of the nonequilibrium geometric no-arbitrage condition.

Pricing of Basket CDS with Bilateral Default Risk under Vasicek Model with Circular Contagion

In this study, on the basis of basket CDS, we take into account the complex correlations among market participants and choose to use the contagion model to price basket CDS. This study assumes that a basket of reference assets is defaultable and that defaults between two counterparties are contagious. Without taking into account the default of the CDS buyer, we assume that both the interest rate and the default intensity of the reference assets follow the Vasicek model. We obtain the related probability density function by the PDE method. Under the no‐arbitrage principle, we set up several scenarios to get the basket CDS pricing formula. In the numerical analysis part, we verify the validity of the formula by Monte Carlo simulation, and discuss the influence of the main parameters on the formula.

Information Entropy Theory and Asset Valuation: A Literature Survey

The purpose of this study is to review the empirical work applied to market efficiency, portfolio selection and asset valuation, focusing on the presentation of the comprehensive theoretical framework of Information Entropy Theory (IET). In addition, we examine how entropy addresses the shortcomings of traditional models for valuing financial assets, including the market efficiency hypothesis, the capital asset pricing model (CAPM), and the Black and Scholes option pricing model. We thoroughly reviewed the literature from 1948 to 2022 to achieve our objectives, including well-known asset pricing models and prominent research on information entropy theory. Our results show that portfolio managers are particularly attracted to valuations and strive to achieve maximum returns with minimal risk. The entropy-based portfolio selection model outperforms the standard model when return distributions are non-Gaussian, providing more comprehensive information about asset and distribution probabilities while emphasising the diversification principle. This distribution is then linked to the entropic interpretation of the no-arbitrage principle, especially when extreme fluctuations are considered, making it preferable to the Gaussian distribution for asset valuation. This study draws important conclusions from its extensive analysis. First, entropy better captures diversification effects than variance, as entropy measures diversification effects more generically than variance. Second, mutual information and conditional entropy provide reasonable estimates of systematic and specific risk in the linear equilibrium model. Third, entropy can be used to model non-linear dependencies in stock return time series, outperforming beta in predictability. Finally, information entropy theory is strengthened by empirical validation and alignment with financial views. Our findings enhance the understanding of market efficiency, portfolio selection and asset pricing for investors and decision-makers. Using Information Entropy Theory as a theoretical framework, this study sheds new light on its effectiveness in resolving some of the limitations in traditional asset valuation models, generating valuable insights into the theoretical framework of the theory.

Envelopes of equivalent martingale measures and a generalized no-arbitrage principle in a finite setting

We consider a one-period market model composed by a risk-free asset and a risky asset with n possible future values (namely, a n-nomial market model). We characterize the lower envelope of the class of equivalent martingale measures in such market model, showing that it is a belief function. Then, we reformulate a general one-period pricing problem in the framework of belief functions: this allows to model frictions in the market and can be justified in terms of partially resolving uncertainty according to Jaffray. We provide a generalized no-arbitrage condition for a generic one-period market model under partially resolving uncertainty and show that the “risk-neutral” belief function arising in the one-period n-nomial market model does not allow to satisfy such condition. Finally, we derive a generalized arbitrage-free lower pricing rule through an inner approximation of the “risk-neutral” belief function arising in the one-period n-nomial market model.

Analytically Pricing Formula for Contingent Claim with Polynomial Payoff under ECIR Process

Contingent claims, such as bonds, swaps, and options, are financial derivatives whose payoffs depend on uncertain future real values of underlying assets which emphasize various real-world applications. In general, valuations for contingent claims can be derived from the conditional expectations of underlying assets. For a simple process, the moments are usually directly obtained from its transition probability density function (PDF). However, if the transition PDF is unavailable in simple form, the derivations of the moments and the contingent claim prices will not be accessible in closed forms. This paper provides a closed-form formula for pricing contingent claims with nonlinear payoff under a no-arbitrage principle when underlying assets follow the extended Cox–Ingersoll–Ross (ECIR) process with the symmetry properties of the Brownian motion. The formula proposed here is a consequence of successfully solving an explicit solution for a system of recurrence partial differential equations in which its solution subtly depends on the conditional moments. For the particular CIR process, we obtain simple closed-form formulas by solving the Riccati differential equation. Furthermore, we carry out a complete investigation of the convergent case for those formulas. In case such as that of the unsolvable Riccati differential equation, ECIR case, a numerical method for numerically evaluating the mentioned analytical formulas and numerical validations for the formulas are examined. The validity and efficiency of the formulas are numerically demonstrated by comparison with results from Monte Carlo simulations for various modeling parameters. Finally, the proposed formula is applied to the value contingent claims such as coupon bonds, interest rate swaps, and arrears swaps.

European Option Pricing Formula in Risk-Aversive Markets

In this study, using the method of discounting the terminal expectation value into its initial value, the pricing formulas for European options are obtained under the assumptions that the financial market is risk-aversive, the risk measure is standard deviation, and the price process of underlying asset follows a geometric Brownian motion. In particular, assuming the option writer does not need the risk compensation in a risk-neutral market, then the obtained results are degenerated into the famous Black–Scholes model (1973); furthermore, the obtained results need much weaker conditions than those of the Black–Scholes model. As a by-product, the obtained results show that the value of European option depends on the drift coefficient μ of its underlying asset, which does not display in the Black–Scholes model only because μ = r in a risk-neutral market according to the no-arbitrage opportunity principle. At last, empirical analyses on Shanghai 50 ETF options and S&P 500 options show that the fitting effect of obtained pricing formulas is superior to that of the Black–Scholes model.

Arbitrage-Based Recovery

We develop a novel recovery theorem based on no-arbitrage principles. Our Arbitrage-Based Recovery Theorem does not require assuming time homogeneity of either the physical probabilities, the Arrow-Debreu prices, or the stochastic discount factor; and it requires the observation of Arrow-Debreu prices only for one single maturity. We perform several different density tests and mean prediction tests using 25 years of S&P 500 options data, and we find evidence that our method can correctly recover the probability distribution of the S&P 500 index level on a monthly horizon.

Gold's Value as an Investment

For investors, gold is an asset without a yield that is attractive in times of low and negative real interest rates. Gold also has an embedded put option because investors can sell it to those who value its use as jewelry or as a productive input. This paper presents an approach for pricing gold from investors' perspective based on no-arbitrage principles. There is no need to specify investor preferences and the main input for pricing gold is a term structure model for real interest rates. The model implies that on average more than half of the value of gold is due to its role as an investment asset. When fitted to match 10-year real US Treasury rates the model can replicate the salient fluctuations in the time series of gold prices since 2007.

No-Arbitrage Principle in Conic Finance

In a one price economy, the Fundamental Theorem of Asset Pricing (FTAP) establishes that no-arbitrage is equivalent to the existence of an equivalent martingale measure. Such an equivalent measure can be derived as the normal unit vector of the hyperplane that separates the attainable gain subspace and the convex cone representing arbitrage opportunities. However, in two-price financial models (where there is a bid–ask price spread), the set of attainable gains is not a subspace anymore. We use convex optimization, and the conic property of this region to characterize the “no-arbitrage” principle in financial models with the bid–ask price spread present. This characterization will lead us to the generation of a set of price factor random variables. Under such a set, we can find the lower and upper bounds (supper-hedging and sub-hedging bounds) for the price of any future cash flow. We will show that for any given cash flow, for which the price is outside the above range, we can build a trading strategy that provides one with an arbitrage opportunity. We will generalize this structure to any two-price finite-period financial model.

Pricing of Power Exchange Option with Jumps under the Double Risk of Exchange and Default

Power exchange option is an exotic option which combines power option and exchange option. In this paper, we consider the pricing of the power exchange option under exchange rate volatility risk and issuing company bankruptcy risk. Meanwhile, considering the major events between the two countries, we add the Poisson jump process to the option model in order to reflect the impact of sudden factors on the price of transnational derivatives in the international market. According to the no-arbitrage principle, a mathematical model for pricing such problems is established, and explicit solutions are obtained. The numerical examples show that the model established in this paper is effective.

Perfect Market, Arbitrage, and Value Creation in the MM Proposition

The real contributions of the MM Proposition are its central assumptions of perfect capital markets and the associated arbitrage argument. In this text, we review the perfect market assumptions and no-arbitrage principle. Then, to explain the evolution of the understanding of arbitrage from a deterministic world into an uncertain world, we restate and comment on the proofs of the MM Proposition in current perspectives. With the no-arbitrage principle in mind, we clearly read the circular justification in the MM Proposition and the misleading concept of cost of equity. From the perspective of the distribution of a corporation's value creation, we find that shareholders prefer a maximum degree of debt. However, we believe that the capital structures are mainly constrained by industry characteristics and the market timing of equity and debt financing.

OPTION PRICING WITH HEAVY-TAILED DISTRIBUTIONS OF LOGARITHMIC RETURNS

A growing body of literature suggests that heavy tailed distributions represent an adequate model for the observations of log returns of stocks. Motivated by these findings, here, we develop a discrete time framework for pricing of European options. Probability density functions of log returns for different periods are conveniently taken to be convolutions of the Student’s [Formula: see text]-distribution with three degrees of freedom. The supports of these distributions are truncated in order to obtain finite values for the options. Within this framework, options with different strikes and maturities for one stock rely on a single parameter — the standard deviation of the Student’s [Formula: see text]-distribution for unit period. We provide a study which shows that the distribution support width has weak influence on the option prices for certain range of values of the width. It is furthermore shown that such family of truncated distributions approximately satisfies the no-arbitrage principle and the put-call parity. The relevance of the pricing procedure is empirically verified by obtaining remarkably good match of the numerically computed values by our scheme to real market data.

Pricing of Perpetual American Put Option with Sub-Mixed Fractional Brownian Motion

Abstract The pricing problem of perpetual American put options is investigated when the underlying asset price follows a sub-mixed fractional Brownian motion process. First of all, the sub-mixed fractional Black-Scholes partial differential equation is established by using the delta hedging method and the principle of no arbitrage. Then, by solving the free boundary problem, we get the pricing formula of the perpetual American put option.

Derivative pricing as a transport problem: MPDATA solutions to Black–Scholes-type equations

We discuss applications of the Multidimensional Positive Definite Advection Transport Algorithm (MPDATA) to numerically solve partial differential equations arising from stochastic models in quantitative finance. In particular, we develop a framework for solving Black–Scholes-type equations by transforming them into advection–diffusion problems. The equations are then numerically integrated backward in time using an iterative explicit finite-difference approach, in which the Fickian term is represented as an additional advective term. We discuss the correspondence between transport phenomena and financial models, uncovering the possibility of expressing the no-arbitrage principle as a conservation law. We show second-order accuracy in time and space of the embraced numerical scheme. This is done via a convergence analysis comparing MPDATA numerical solutions with classic Black–Scholes analytical formulæ for the valuation of European options. We demonstrate in addition a way of applying MPDATA to solve the free boundary problem (leading to a linear complementarity problem) for the valuation of American options. We finally comment on the potential of MPDATA methods with respect to more complex models typically used in quantitative finance.

Spot Market Mechanism Design for the Electricity Market in China Considering the Impact of a Contract Market

To promote the reformation of the electricity market in China, a market mechanism that can support collaboration between the contract market and the upcoming spot market was designed in this paper. The focus of this paper was to develop a mechanism to institutionally stabilize the market by way of disciplining market power abuse through limiting arbitrage opportunities generated from multi-markets. To quantitatively describe the arbitrage opportunity, the arbitrage opportunity function (AOF) was defined. Based on inferences of the no-arbitrage principle and the AOF, a cost-based decomposition algorithm for contracts that could improve contract coverage was proposed. The incentive compatible settlement rule for the uncovered generation on the spot market was designed to properly manipulate the arbitrage opportunity. The decomposition algorithm and the settlement rule constituted the designed market mechanism. To verify the applicability and effectiveness of the proposed mechanism, the principles of incentive compatibility, individual rationality, and payment cost minimization were employed to test the designed market mechanism based on the concept of dominant policy equilibrium. This test was conducted on a fictitious case based on the IEEE-14 system. The analysis and results may provide valuable insights on market design in China based on the functional correlation between the contract market and the spot market.

Valuing an investment project using no-arbitrage and the alpha-maxmin criteria: From Knightian uncertainty to risk

We consider a two-period irreversible investment decision problem in which the firm can either invest in period 0 or in period 1. The firm is assumed to be able to specify a set of three scenarios or more but not a probability measure. Assuming the option to wait is valued with the no-arbitrage principle, when the firm makes use of the criteria α-maxmin, we show the firm ends up with a known probability measure that assigns a positive probability to three or four scenarios only.