Arrow-Debreu Prices Research Articles

Prices and Taxes in a Ramsey Climate Policy Model under Heterogeneous Beliefs and Ambiguity

In a Ramsey policy regime, heterogeneity in beliefs about the potential costs of climate change is shown to produce policy ambiguities that alter carbon prices and taxation. Three sources of ambiguity are considered: (i) the private sector is skeptical, with beliefs that are unknown to the government, (ii) private agents have pessimistic doubts about the model, or (iii) the policy authority itself does not trust the extant scientific climate model and fears the worst. These three sources of ambiguity give rise to four potential belief regimes characterized by differentials between the government’s and the private sector’s inter-temporal rates of substitutions, with implications for the prices of carbon and capital, framed in terms of distorted Arrow–Debreu pricing theory that establishes an equivalence between the optimal carbon tax and the permit price of an underlying asset—the government-imposed limit on emissions in economies with cap and trade. This paper shows that in most instances, skeptical beliefs and resulting ambiguities justify higher carbon taxes and lower capital taxes to offset the private sector’s increased myopia compared with rational expectations. Conversely, ambiguities created by worst-case fears in either the private sector or in government tend produce forces in the opposite direction.

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.

Arrow-Debreu Meets Kyle

We consider an extension of the Kyle (1985) setting where Arrow-Debreu securities are traded and the informed trader has private information regarding arbitrary higher moments of the asset payoff distribution.In this setting we analyze price discovery and informed demand of Arrow-Debreu securities---equivalently, options. Our model admits a simple equilibrium solution. The informed trader strategy in our model is consistent with options trading strategies used to trade on higher moments in practice. We find that information efficiency of Arrow-Debreu prices decreases with respect to the dispersion of the informed trader's private signals.

Universal Arbitrage-Free Estimation of State Price Density

Given the valuable information content of Arrow-Debreu prices, the recovery of a well-behaved state price density is of considerable importance. However, this is a nontrivial task because of data limitation and the complex arbitrage-free constraints. In this article, we develop a more effective linear programming support vector machine estimator for state price density, which incorporates no-arbitrage restrictions and bid-ask spread. This method does not depend on a particular approximation function and framework and is, therefore, universally applicable. In a parallel empirical study, we apply the method to options on the S&P 500, showing it to be accurate and smooth. TOPICS:Derivatives, options Key Findings ▪ Recovery of a well-behaved state price density is an important but nontrivial because of data limitation and the complex arbitrage-free constraints. ▪ We develop a universally applicable linear programming support vector machine estimator for state price density that incorporates no-arbitrage restrictions and bid-ask spread. ▪ We apply the method empirically to options on the S&P 500, showing it to be accurate and smooth.

Six Types of Ambiguity in a Ramsey Climate Policy Model

While the scientific community is in general agreement about the nature and threat of global warming, the public and many political leaders are not. This paper casts the lack of full agreement as an instance of belief heterogeneity giving rise to six potential types of ambiguity with differing implications for carbon taxation and fiscal policy. Treating the Ramsey planner as a Stackelberg leader and private agents as followers, I distinguish three types of policy regime: (i) a political regime that strategically adopts the private sector's beliefs, (ii) a paternalistic regime that trust its own model only, and (iii) a pessimistic government with its own doubts about the climate model. The beliefs of private agents are represented by martingale multiplier perturbations M to a common approximating probability function and may be of two kinds: for pessimists}, M is a device to construct robust Euler conditions; for optimists, M is arbitrary and unknown to the planner, hence a source of ambiguity. The six proposed types of ambiguity differ in what the planner is ambiguous about, what private agents and the government know, and what the government believes private beliefs to be. In all instances, ambiguity induces robustness in policy and/or private decision rules affecting the social cost of carbon, and carbon and capital taxation. This paper shows further that, crucially, the carbon tax is not independent of other elements in fiscal policy. The results are framed in terms of distorted Arrow-Debreu pricing theory to show that the optimal carbon tax is equivalent to the permit price of an underlying asset---the government-imposed limit on emissions in economies with cap and trade.

Implicit Entropic Market Risk-Premium from Interest Rate Derivatives

Implicit in interest rate derivatives are Arrow–Debreu prices (or state price densities, SPDs) that contain funda- mental information for risk and portfolio management in interest rate markets. To extract such information from interest rate derivatives, we propose a nonparametric method to estimate state prices based on the minimization of the Cressie–Read (Entropic) family function between potential SPDs and the empirical probability measure. An empirical application of the method, in the US interest rates and derivatives market, shows that the entropic based risk-neutral density measure highlight potential risks previous to the 2007/2008 financial crisis, and the potential arbitrage burden during the Quantitative Easing period.

Closed-form Arrow-Debreu pricing for the Hull-White short rate model

We consider the Hull-White short rate model and provide a systematic derivation of an Arrow-Debreu pricing formula for European-style options in closed form, applying it to cap/floor pricing and CMS (constant maturity swap) index calculation. We present in the process a useful exponential expansion formula which facilitates the analytic solution of pricing equations. We propose that the methodology here described is of interest insofar as it is applicable to a wider range of short rate modelling problems potentially involving lognormal rates/credit volatility, local-stochastic volatility and/or multiple stochastic factors or underlyings.

APPROXIMATION METHODS FOR INHOMOGENEOUS GEOMETRIC BROWNIAN MOTION

We present an accurate and easy-to-compute approximation of the transition probabilities and the associated Arrow-Debreu (AD) prices for the inhomogeneous geometric Brownian motion (IGBM) model for interest rates, default intensities or volatilities. Through this procedure, dubbed exponent expansion, transition probabilities and AD prices are obtained as a power series in time to maturity. This provides remarkably accurate results — for time horizons up to several years — even when truncated after the first few terms. For farther time horizons, the exponent expansion can be combined with a fast numerical convolution to obtain high-precision results.

On Calibration and Simulation of Local Volatility Model with Stochastic Interest Rate

Local volatility model is a relatively simple way to capture volatility skew/smile. In spite of its drawbacks, it remains popular among practitioners for derivative pricing and hedging. For long-dated options or interest rate/equity hybrid products, in order to take into account the effect of stochastic interest rate on equity price volatility stochastic interest rate is often modelled together with stochastic equity price. Similar to local volatility model with deterministic interest rate, a forward Dupire PDE can be derived using Arrow-Debreu price method, which can then be shown to be equivalent to adding an additional correction term on top of Dupire forward PDE with deterministic interest rate. Calibrating a local volatility model by the forward Dupire PDE approach with adaptively mixed grids ensures both calibration accuracy and efficiency. Based on Malliavin calculus an accurate analytic approximation is also derived for the correction term incorporating impacts from both interest rate volatility and correlation, which integrates along a more likely straight line path for better accuracy. Eventually, the hybrid local volatility model can be calibrated in a two-step process, namely, calibrate local volatility model with deterministic interest rate and add adjustment for stochastic interest rate. Due to the lack of analytic solution and path-dependency nature of some products, Monte Carlo is a simple but flexible pricing method. In order to improve its convergence, we develop a scheme to combine merits of different simulation schemes and show its effectiveness.

Perturbation Expansion for Arrow-Debreu Pricing with Hull-White Interest Rates and Black-Karasinski Credit Intensity

We consider a rates-credit hybrid model with the rates governed by a Hull-White short-rate model and the credit intensities by a Black-Karasinski short-rate model. We provide a systematic derivation of a pricing kernel (also known as an Arrow-Debreu formula) for European-style options and/or protection payments, using operator formalism combined with exponential expansion formulae. Our approach gives rise to an analytic expression involving an infinite series in powers of the credit spread (not of its lognormal volatility). We propose that this can be used to provide results to a chosen level of accuracy by truncating the power series at a suitable point and give explicit expressions for all terms up to second order, which level we suggest should in practice suffice. We apply our first-order result to calculate the impact of rates-credit correlation on the pricing of credit default swaps (CDS), extinguishing interest rate swaps, survival-contingent capped Libor flows and contingent CDS with interest rate swap underlying. Very simple analytic expressions are obtained in all cases. Highly favourable comparison is found between even the first order approximations and Monte Carlo simulations.

Closed-Form Arrow-Debreu Pricing for the Hull-White Short Rate Model

We consider the Hull-White short rate model and provide a systematic derivation of an Arrow-Debreu pricing formula for European-style options using operator formalism combined with an exponential expansion formula. This gives rise to a closed-form expression. We propose that the methodology here described is of interest insofar as it is applicable to a wider range of short rate modelling problems involving lognormal rates volatility, local volatility and/or multiple stochastic factors or underlyings.

Exact Arrow-Debreu Pricing for the Black-Karasinski Short Rate Model

We consider the Black-Karasinski short rate model and provide a systematic derivation of an Arrow-Debreu pricing formula for European-style options using operator formalism combined with exponential expansion formulae. Our approach gives rise to an analytic expression involving an infinite series in powers of the interest rate (not of its lognormal volatility). We propose that this can be used to provide results to a chosen level of accuracy by truncating the power series at a suitable point; and further that the use of terms up to second or third order should in practice suffice.

A New Science of Finance for Addressing the Challenge of Uncertainty: Thoughts on the Future

To effectively cope with an unexpected, large, and negative income shock, I propose a life-cycle model for income risk management. I analyze the intertemporal consumption-investment problem in an incomplete market. I suggest a hybrid of the martingale approach and the dynamic programming approach for solving the problem and deriving closed-form solutions. The model has strong empirical implications on anomalies in finance. Finally, I provide a general equilibrium analysis by generalizing the Arrow-Debreu price with income risk premium.

Closed-Form Arrow-Debreu Pricing for FX and Inflation Options with Hull-White Stochastic Rates

We model the joint distribution of an FX rate and two interest rates assuming a Hull-White short rate model for foreign and domestic rates and a simple time-dependent lognormal volatility for the FX rate. We derive an Arrow-Debreu pricing formula under the domestic currency num\'{e}raire for multi-currency rates derivatives using operator formalism combined with an exponential expansion formula. The result is a closed-form expression, comprising essentially of the zero coupon bond price multiplied by the joint probability distribution function of the rates and the FX forward rate. We note also that the problem we consider is exactly analogous to the Jarrow-Yildirim inflation option pricing model and that the expression we derive for the vanilla option price is equivalent to theirs.

Closed-Form Arrow-Debreu Pricing for Equity Options with Hull-White Stochastic Rates

Closed-Form Arrow-Debreu Pricing for Equity Options with Hull-White Stochastic Rates

Bond pricing under the generalised Black-Karasinski models

Due to the lognormality of the short rate under the Black-Karasinski interest model, closed-form expressions for zero-coupon bond prices are not available. Existing methods for computing approximate prices include perturbation methods for solving the reaction-diffusion equation satisfied by the bond-price and the exponent expansion for computing the bond price via Arrow-Debreu prices. Perturbation methods are accurate for small volatility problems whereas the exponent expansion is accurate for small maturities. This work proposes a high-order computational method that works for all parameter settings. Several numerical examples are described to illustrate the high accuracy and rapid computation of bond prices.

An Effective Approximation for Zero-Coupon Bonds and Arrow-Debreu Prices in the Black-Karasinski Model

We present an accurate and easy-to-compute approximation of zero-coupon bonds and Arrow-Debreu (AD) prices for the Black-Karasinski model of interest rates or default intensities. Through this procedure, dubbed exponent expansion, AD prices are obtained as a power series in time to maturity. This provides remarkably accurate results - for time horizons up to several years - even when truncated to the first few terms. For larger time horizons the exponent expansion can be combined with a fast numerical convolution to obtain extremely accurate results.

A Simple Approximation Algorithm for Computing Arrow-Debreu Prices

We consider the Arrow-Debreu market with linear utilities in which there is a set G of divisible goods and a set B of buyers. Each buyer starts with an initial endowment of goods. The buyer's utility function is a linearly separable function of the goods that the buyer purchases. We develop a simple and efficient algorithm for determining an approximate market equilibrium. Our algorithm finds an ϵ-approximate solution in O(n/ϵ(|B||G|)) time, where n = |B| + |G|. The running time can be further improved to O(n/ϵ(m + |B|log|B|)) where m is the number of pairs (i, j) such that buyer i has some utility for purchasing good j.

In This Issue

In This Issue

Pricing rules and Arrow–Debreu ambiguous valuation

This paper considers pricing rules of single-period securities markets with finitely many states and without arbitrage opportunities. Our main result characterize those pricing rules C that are super-replication prices of a frictionless incomplete asset structure. This characterization relies on the equivalence between the sets of frictionless securities and undominated securities priced by C. The former captures securities without bid-ask spreads while the second captures the class of securities where, if some of its delivers is replaced by a higher payoff, then the resulting security is characterized by a higher value priced by C. We also analyze the special case of pricing rules revealing securities markets admitting a structure of basic assets paying one in some event and nothing otherwise. In this case we show that any security can be priced against a capacity. This risk-neutral capacity, or Arrow-Debreu ambiguous state price, can be viewed as a generalization for incomplete markets of Arrow Debreu price valuation, and the corresponding pricing rule is determined by an integral w.r.t. a subadditive capacity. For instance, a special class of Choquet integral is related to frictionless incomplete markets of Arrow securities and a riskless asset.