No-arbitrage Assumption Research Articles

CAPM in Real World: Risk-Friendly Investments

The aim of this paper is to show how the existence of equilibrium in CAPM may be obtained when individuals/investors are risk friendly. This assumption is closer to the real world, since risk aversion is rare and the portfolios implying a greater payoff are the ones which increase the variance of the payoff itself. Specifically, we assume that the wage vectors of the individuals/investors do not lie in the market space, which is the usual assumption for CAPM, namely, the payoff of any portfolio is replicated by the wage of the individuals/investors. The interpretation of such an assumption is that the wealth of the individuals/investors may not invest in financial markets as a whole. The second main result is that the perception of risk by the objective probabilities for the states of the world does not affect the equilibrium existence. This is a consequence of the No-Arbitrage assumption. This condition may be stated in terms of the existence of the objective probabilities for the states of the world. Since CAPM is usually considered as a way to compare the return of the so-called “market portfolio” and the return of any other portfolio, in the final section of this paper, we do provide the regression form of this altered form of CAPM.

ARM: The Analytic Recovery Method

The analytic recovery method (ARM) recovers arbitrage-free density functions from a given set of option prices with maximum accuracy and speed. For arbitrage-free option prices, ARM provides extremely fast convergence and arbitrary accuracy. In the presence of noise, the closest arbitrage-free approximation is identified. Option prices and densities, as well as their moments and other parameters, are easy-to-handle analytic functions defined for arbitrary strike prices. ARM reveals inconsistencies between quoted option prices, particularly for longer durations. ARM is essentially based on the no-arbitrage assumptions; it is not related to a specific model. It has been tested for a selection of S&P 500, EuroStoxx 50, and DAX data. Excellent no-arbitrage fit to call and put prices is obtained; extrapolations are in line with the market.

American options pricing under regime-switching jump-diffusion models with meshfree finite point method

In an incomplete market construction and by no-arbitrage assumption, the American options pricing problem under the jump-diffusion regime-switching process is formulated by a variational form of coupled partial integro-differential equations. In this paper, a valuation algorithm is developed for American options when the dynamics of underlying assets follow the regime-switching jump-diffusion processes. Using the fact that the price of an American option under jump-diffusion regime-switching processes is formulated by a collection of coupled variational partial integro-differential equations with the free boundary characteristic, we combine the moving least-squares approximation with an operator splitting method to treat American constraints. Numerical experiments with American options under three, five, and seven regimes demonstrate the efficiency and effectiveness of our computational scheme for pricing American options under the regime-switching models.

Explainable neural network for pricing and universal static hedging of contingent claims

We present here a regress-later based Monte Carlo approach that uses neural networks for pricing multi-asset discretely-monitored contingent claims. The choice of specific architecture of the neural networks used in the proposed algorithm provides for interpretability of the model, a feature that is often desirable in the financial context. Specifically, the interpretation leads us to demonstrate that any discretely monitored contingent claim —possibly high-dimensional and path-dependent— under Markovian and no-arbitrage assumptions, can be semi-statically hedged using a portfolio of short maturity options. We also show, for Bermudan style derivatives, how the method can be used to obtain an upper and lower bound to the true price, where the lower bound is obtained by following a sub-optimal policy, while the upper bound is found by exploiting the dual formulation. Unlike other duality based upper bounds where one typically has to resort to nested simulation for constructing super-martingales, the martingales in the current approach come at no extra cost, without the need for any sub-simulations. We demonstrate through numerical examples the simplicity and efficiency of the method for both pricing and semi-static hedging of path-dependent options.

Applying Heath-Jarrow-Morton Model to Forecasting the US Treasury Daily Yield Curve Rates

The Heath-Jarrow-Morton (HJM) model is a powerful instrument for describing the stochastic evolution of interest rate curves under no-arbitrage assumption. An important feature of the HJM approach is the fact that the drifts can be expressed as functions of respective volatilities and the underlying correlation structure. Aimed at researchers and practitioners, the purpose of this article is to present a self-contained, but concise review of the abstract HJM framework founded upon the theory of interest and stochastic partial differential equations in infinite dimensions. To illustrate the predictive power of this theory, we apply it to modeling and forecasting the US Treasury daily yield curve rates. We fit a non-parametric model to real data available from the US Department of the Treasury and illustrate its statistical performance in forecasting future yield curve rates.

Trading Time Seasonality in Commodity Futures: An Opportunity for Arbitrage in the Natural Gas and Crude Oil Markets?

For fixed maturity, under the no-arbitrage assumption, futures prices should follow a martingale with respect to the trading time, at least under the pricing measure. Therefore, a prominent display of trading time seasonality under the physical measure raises warning signs and can only occur by means of strong seasonality in the pricing kernel. We show that for natural gas and crude oil, trading time seasonality is present to an extent where it may violate the no-arbitrage assumption. We provide three layers of evidence. The first layer is descriptive only, the second involves the Kruskal--Wicksell test for establishing trading time seasonality, and the third is in the form of a trading strategy, which exploits trading date seasonality. This strategy can produce statistically significant positive alphas in the CAPM context, thereby indicating the possibility of an arbitrage.

Finding the nearest covariance matrix: the foreign exchange market case

We consider the problem of finding a valid covariance matrix in the foreign exchange market given an initial nonpositively semidefinite (non-PSD) estimate of such a matrix. The common no-arbitrage assumption imposes additional linear constraints on such matrixes, inevitably making them singular. As a result, even the most advanced numerical techniques will predictably balk at a seemingly standard optimization task. The reason is that the problem is ill posed, while its PSD solution is not strictly feasible. In order to deal with this issue we describe a low-dimensional face of the PSD cone that contains the feasible set. After projecting the initial problem onto this face, we come out with a reduced problem, which is both well posed and of a smaller scale. We show that, after solving the reduced problem, the solution to the initial problem can be recovered uniquely in one step. We run numerous numerical experiments to compare the performance of different algorithms in solving the reduced problem and to demonstrate the advantages of dealing with the reduced problem as opposed to the original one. The smaller scale of the reduced problem implies that its solution can effectively be found by the application of virtually any numerical method.

The conditional expected market return

We derive lower and upper bounds on the conditional expected excess market return that are related to risk-neutral volatility, skewness, and kurtosis indexes. The bounds can be calculated in real time using a cross section of option prices. The bounds require a no-arbitrage assumption, but they do not depend on distributional assumptions about market returns or past observations. The bounds are highly volatile, positively skewed, and fat-tailed. They imply that the term structure of expected excess holding period returns is decreasing during turbulent times and increasing during normal times and that the expected excess market return is on average 5.2%.

A High-Frequency Analysis of Bitcoin Markets

We study trading of Bitcoin against US dollar (BTCUSD) on exchanges in three continents, Bitfinex, Bitstamp and Coinbase Pro. We use a high frequency dataset that contains transactions and order book information. The BTCUSD market is highly liquid in terms of bid-ask spreads and order book depth. While spreads are even lower than in equity markets, prices are not integrated across exchanges. Persistent differences exist between the three exchanges in terms of trade prices and posted prices often violating no-arbitrage assumptions. The liquidity of the Bitcoin exchanges is predominantly determined by local factors and is essentially independent of liquidity in equity and FX markets. This suggests that despite the virtual nature of Bitcoin, local jurisdictional factors affect the flow of capital between low and high price jurisdictions.

Measure-invariance of copula functions as tool for testing no-arbitrage assumption

Copulas, which are invariant under margins’ transforms induced by some change of measure, are investigated. It is emphasized that this particular kind of transforms induced by some change of measure, largely used in pricing techniques, preserves the invariance of the aggregation operator and a sufficient condition to assure it is proved. The discussion is extended to the time-preserving of measure-invariance; a characterization of its stability in time for multivariate stationary processes, based on the dynamic copula representation (see Cherubini et al., 2011), is provided. Finally a measure invariance-based statistical test for the absence of arbitrage opportunity assumption and its preservation in time is proposed and an empirical experiment based on quotes of S&P 500 futures and options traded on the Chicago Mercantile Exchange (CME) is discussed.

The Conditional Expected Market Return

We derive lower and upper bounds on the conditional expected excess market return that are related to risk-neutral volatility, skewness, and kurtosis indexes. The bounds can be calculated in real time using a cross section of option prices. The bounds require a no-arbitrage assumption, but do not depend on distributional assumptions about market returns or past observations. The bounds are highly volatile, positively skewed, and fat tailed. They imply that the term structure of expected excess holding period returns is decreasing during turbulent times and increasing during normal times, and that the expected excess market return is on average 5.2%.

Option Pricing Under Short-Lived Arbitrage: Theory and Tests

Models in financial economics derived from no-arbitrage assumptions have found great favor among theoreticians and practitioners. We develop a model of option prices where arbitrage is short lived. The arbitrage process is Ornstein-Uhlenbeck with zero mean and rapid adjustment of deviations. We find that arbitrage correlated with the underlying can have sizeable impact on option prices. We use data from five large capitalization firms to test implications of the model. Consistent with the existence of arbitrage, we find that liquidity factors significantly effect arbitrage model parameters.

Option pricing under short-lived arbitrage: theory and tests

Models in financial economics derived from no-arbitrage assumptions have found great favour among theoreticians and practitioners. We develop a model of option prices where arbitrage is short lived. The arbitrage process is Ornstein–Uhlenbeck with zero mean and rapid adjustment of deviations. We find that arbitrage correlated with the underlying can have sizeable impact on option prices. We use data from five large capitalization firms to test implications of the model. Consistent with the existence of arbitrage, we find that idiosyncratic factors significantly effect arbitrage model parameters.

EXPLICIT FORMULAE FOR PARAMETERS OF STOCHASTIC MODELS OF A DISCOUNTED EQUITY INDEX USING MAXIMUM LIKELIHOOD ESTIMATION WITH APPLICATIONS

A discounted equity index is computed as the ratio of an equity index to the accumulated savings account denominated in the same currency. In this way, discounting provides a natural way of separating the modeling of the short rate from the market price of risk component of the equity index. In this vein, we investigate the applicability of maximum likelihood estimation to stochastic models of a discounted equity index, providing explicit formulae for parameter estimates. We restrict our consideration to two important index models, namely the Black–Scholes model and the minimal market model of Platen, each having an explicit formula for the transition density function. Explicit formulae for estimates of the model parameters and their standard errors are derived and are used in fitting the two models to US data. Further, we demonstrate the effect of the model choice on the no-arbitrage assumption employed in risk neutral pricing.

Long-Term Yield in an Affine HJM Framework on $$S_{d}^{+}$$ S d +

We develop the HJM framework for forward rates driven by affine processes on the state space of symmetric positive semidefinite matrices. In this setting we find an explicit representation for the long-term yield in terms of the model parameters. This generalises the results of El Karoui et al. (Rev Deriv Res 1(4):351–369, 1997) and Biagini and Hartel (Int J Theor Appl Financ 17(3):1–24, 2012), where the long-term yield is investigated under no-arbitrage assumptions in a HJM setting using Brownian motions and Levy processes respectively.

Optimal contracts for central bankers: Calls on inflation

We consider a framework featuring a central bank, private and financial agents as well as a financial market. The central bank’s objective is to maximize a functional, which measures the classical trade-off between output and inflation plus income from the sales of inflation linked calls minus payments for the liabilities that the inflation linked calls produce at maturity. Private agents have rational expectations and financial agents are averse against inflation risk. Following this route, we explain demand for inflation linked calls on the financial market from a no-arbitrage assumption and derive pricing formulas for inflation linked calls, which lead to a supply–demand equilibrium. We then study the consequences that the sales of inflation linked calls have on the observed inflation rate and price level. Similar as in Walsh (1995) we find that the inflationary bias is significantly reduced, and hence that markets for inflation linked calls provide a mechanism to implement inflation contracts as discussed in the classical literature.

Discretisation-Invariant Swaps

We derive a general multivariate theory for realised characteristics of `model-free discretisation-invariant swaps', so-called because the standard no-arbitrage assumption of martingale forward prices is sufficient to derive fair-value swap rates for such characteristics which have no jump or discretisation errors. This theory underpins specific examples for swaps based on higher moments of a single log return distribution where exact replication is possible via option-implied `fundamental contracts' like the log contact. The common factors determining the S&P 500 risk premia associated with these higher-moment characteristics are investigated empirically at the daily, weekly and monthly frequencies.

Time-Varying Risk Premium in Large Cross-Sectional Equity Data Sets

We develop an econometric methodology to infer the path of risk premia from a large unbalanced panel of individual stock returns. We estimate the time-varying risk premia implied by conditional linear asset pricing models where the conditioning includes both instruments common to all assets and asset-specific instruments. The estimator uses simple weighted two-pass cross-sectional regressions, and we show its consistency and asymptotic normality under increasing cross-sectional and time series dimensions. We address consistent estimation of the asymptotic variance by hard thresholding, and testing for asset pricing restrictions induced by the no-arbitrage assumption. We derive the restrictions given by a continuum of assets in a multi-period economy under an approximate factor structure robust to asset repackaging. The empirical analysis on returns for about ten thousand U.S. stocks from July 1964 to December 2009 shows that risk premia are large and volatile in crisis periods. They exhibit large positive and negative strays from time-invariant estimates, follow the macroeconomic cycles, and do not match risk premia estimates on standard sets of portfolios. The asset pricing restrictions are rejected for a conditional four-factor model capturing market, size, value, and momentum effects.

The long memory and the transaction cost in financial markets

In the present work, we investigate the fractal dimensions of 30 important stock markets from 2006 to 2013; the analysis indicates that the Hurst exponent of emerging markets shifts significantly away from the standard Brownian motion. We propose a model based on the Hurst exponent to explore the considerable profits from the predictable long-term memory. We take the transaction cost into account to justify why the market inefficiency has not been arbitraged away in the majority of cases. The empirical evidence indicates that the majority of the markets are efficient with a certain transaction cost under the no-arbitrage assumption. Furthermore, we use the Monte Carlo simulation to display “the efficient frontier” of the Hurst exponent with different transaction costs.

Finding the Nearest Valid Covariance Matrix: A FX Market Case

We consider the problem of finding a valid covariance matrix in the FX market given an initial non-PSD estimate of such a matrix. The standard no-arbitrage assumption implies additional linear constraints on such matrices, which automatically makes them singular. As a result, one cannot just take the given estimate plug it into the standard optimization problem and solve it by applying even the most advanced numerical techniques developed recently. The reason is that such a problem is not well-posed while the PSD-solution is not strictly feasible. In order to deal with this issue, we described a low-dimensional face of the PSD cone that contains the feasible set. After projecting the initial problem onto this face, we come out with a reduced problem, which turns out to be well posed and of a smaller scale. We show that after solving the reduced problem the solution to the initial problem can be uniquely recovered in one step. We run numerous numerical experiments to compare performance of different algorithms in solving the reduced problem and to demonstrate the advantages of dealing with the reduced problem as opposed to the original one. The smaller scale of the reduced problem implies that virtually any numerical method can be applied effectively to find its solution.