A Multiperiod Equilibrium Pricing Model

We propose an equilibrium pricing model in a dynamic multiperiod stochastic framework with uncertain income. There are one tradable risky asset (stock/commodity), one nontradable underlying (temperature), and also a contingent claim (weather derivative) written on the tradable risky asset and the nontradable underlying in the market. The price of the contingent claim is priced in equilibrium by optimal strategies of representative agent and market clearing condition. The risk preferences are of exponential type with a stochastic coefficient of risk aversion. Both subgame perfect strategy and naive strategy are considered and the corresponding equilibrium prices are derived. From the numerical result we examine how the equilibrium prices vary in response to changes in model parameters and highlight the importance of our equilibrium pricing principle

Option-pricing in incomplete markets: the hedging portfolio plus a risk premium-based recursive approach

Consider a non-spanned security $C_{T}$ in an incomplete market. We study the risk/return tradeoffs generated if this security is sold for an arbitrage-free price $\hat{C_{0}}$ and then hedged. We consider recursive "one-period optimal" self-financing hedging strategies, a simple but tractable criterion. For continuous trading, diffusion processes, the one-period minimum variance portfolio is optimal. Let $C_{0}(0)$ be its price. Self-financing implies that the residual risk is equal to the sum of the one-period orthogonal hedging errors, $\sum_{t\leq T} Y_{t}(0) e^{r(T -t)}$. To compensate the residual risk, a risk premium $y_{t}\Delta t$ is associated with every $Y_{t}$. Now let $C_{0}(y)$ be the price of the hedging portfolio, and $\sum_{t\leq T}(Y_{t}(y)+y_{t}\Delta t)e^{r(T-t)}$ is the total residual risk. Although not the same, the one-period hedging errors $Y_{t}(0) and Y_{t}(y)$ are orthogonal to the trading assets, and are perfectly correlated. This implies that the spanned option payoff does not depend on y. Let $\hat{C_{0}}-C_{0}(y)$. A main result follows. Any arbitrage-free price, $\hat{C_{0}}$, is just the price of a hedging portfolio (such as in a complete market), $C_{0}(0)$, plus a premium, $\hat{C_{0}}-C_{0}(0)$. That is, $C_{0}(0)$ is the price of the option's payoff which can be spanned, and $\hat{C_{0}}-C_{0}(0)$ is the premium associated with the option's payoff which cannot be spanned (and yields a contingent risk premium of sum $y_{t}\Delta$t$e^{r(T-t)}$ at maturity). We study other applications of option-pricing theory as well

Viability and Equilibrium in Securities Markets with Frictions

In this paper we study some foundational issues in the theory of asset pricing with market frictions. We model market frictions by letting the set of marketed contingent claims (the opportunity set) be a convex set, and the pricing rule at which these claims are available be convex. This is the reduced form of multiperiod securities price models incorporating a large class of market frictions. It is said to be viable as a model of economic equilibrium if there exist price-taking maximizing agents who are happy with their initial endowment, given the opportunity set, and hence for whom supply equals demand. This is equivalent to the existence of a positive linear pricing rule on the entire space of contingent claims - an underlying frictionless linear pricing rule - that lies below the convex pricing rule on the set of marketed claims. This is also equivalent to the absence of asymptotic free lunches - a generalization of opportunities of arbitrage. When a market for a non marketed contingent claim opens, a bid-ask price pair for this claim is said to be consistent if it is a bid-ask price pair in at least a viable economy with this extended opportunity set. If the set of marketed contingent claims is a convex cone and the pricing rule is convex and sublinear, we show that the set of consistent prices of a claim is a closed interval and is equal (up to its boundary) to the set of its prices for all the underlying frictionless pricing rules. We also show that there exists a unique extended consistent sublinear pricing rule - the supremum of the underlying frictionless linear pricing rules - for which the original equilibrium does not collapse, when a new market opens, regardless of preferences and endowments. If the opportunity set is the reduced form of a multiperiod securities market model, we study the closedness of the interval of prices of a contingent claim for the underlying frictionless pricing rules

Minding the gap : central bank estimates of the unemployment natural rate

A time-varying parameter framework is suggested for use with real-time multiperiod forecast data to estimate implied forecast equations. The framework is applied to historical briefing forecasts prepared for the Federal Open Market Committee to estimate the U.S. central bank’s ex ante perceptions of the natural rate of unemployment. Relative to retrospective estimates, empirical results do not indicate severe underestimation of the natural rate of unemployment in the 1970s.Unemployment

Option-Pricing in Incomplete Markets: The Hedging Portfolio plus a Risk Premium-Based Recursive Approach

Consider a non-spanned security C_{T} in an incomplete market. We study the risk/return trade-offs generated if this security is sold for an arbitrage-free price C₀ and then hedged. We consider recursive "one-period optimal" self-financing hedging strategies, a simple but tractable criterion. For continuous trading, diffusion processes, the one-period minimum variance portfolio is optimal. Let C₀(0) be its price. Self-financing implies that the residual risk is equal to the sum of the one-period orthogonal hedging errors, ∑_{t≤T}Y_{t}(0)e^{r(T-t)}. To compensate the residual risk, a risk premium y_{t}Δt is associated with every Y_{t}. Now let C₀(y) be the price of the hedging portfolio, and ∑_{t≤T}(Y_{t}(y)+y_{t}Δt)e^{r(T-t)} is the total residual risk. Although not the same, the one-period hedging errors Y_{t}(0) and Y_{t}(y) are orthogonal to the trading assets, and are perfectly correlated. This implies that the spanned option payoff does not depend on y. Let C₀=C₀(y). A main result follows. Any arbitrage-free price, C₀, is just the price of a hedging portfolio (such as in a complete market), C₀(0), plus a premium, C₀-C₀(0). That is, C₀(0) is the price of the option's payoff which can be spanned, and C₀-C₀(0) is the premium associated with the option's payoff which cannot be spanned (and yields a contingent risk premium of ∑y_{t}Δte^{r(T-t)} at maturity). We study other applications of option-pricing theory as wellOption Pricing; Incomplete Markets

Option-Pricing in Incomplete Markets: The Hedging Portfolio plus a Risk Premium-Based Recursive Approach

Consider a non-spanned security CT in an incomplete market. We study the risk/return tradeoffs generated if this security is sold for an arbitrage-free price bC0 and then hedged. We consider recursive “one-period optimal” self-financing hedging strategies, a simple but tractable criterion. For continuous trading, diffusion processes, the one-period minimum variance portfolio is optimal. Let C0(0) be its price. Self-financing implies that the residual risk is equal to the sum of the oneperiod orthogonal hedging errors, Pt=T Yt(0)er(T -t). To compensate the residual risk, a risk premium yt.t is associated with every Yt. Now let C0(y) be the price of the hedging portfolio, and Pt=T (Yt(y) + yt.t) er(T -t) is the total residual risk. Although not the same, the one-period hedging errors Yt(0) and Yt(y) are orthogonal to the trading assets, and are perfectly correlated. This implies that the spanned option payoff does not depend on y. Let bC0 = C0(y). A main result follows. Any arbitrage-free price, bC0, is just the price of a hedging portfolio (such as in a complete market), C0(0), plus a premium, bC0 - C0(0). That is, C0(0) is the price of the option’s payoff which can be spanned, and bC0 - C0(0) is the premium associated with the option’s payoff which cannot be spanned (and yields a contingent risk premium of Pyt.ter(T -t) at maturity). We study other applications of option-pricing theory as well.

Option Pricing: Real and Risk-Neutral Distributions

The central premise of the Black and Scholes [Black, F., Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy 81, 637–659] and Merton [Merton, R. (1973). Theory of rational option pricing. Bell Journal of Economics and Management Science 4, 141–184] option pricing theory is that there exists a self-financing dynamic trading policy of the stock and risk free accounts that renders the market dynamically complete. This requires that the market be complete and perfect. In this essay, we are concerned with cases in which dynamic trading breaks down either because the market is incomplete or because it is imperfect due to the presence of trading costs, or both. Market incompleteness renders the risk-neutral probability measure non unique and allows us to determine the option price only within a range. Recognition of trading costs requires a refinement in the definition and usage of the concept of a risk-neutral probability measure. Under these market conditions, a replicating dynamic trading policy does not exist. Nevertheless, we are able to impose restrictions on the pricing kernel and derive testable restrictions on the prices of options.We illustrate the theory in a series of market setups, beginning with the single period model, the two-period model and, finally, the general multiperiod model, with or without transaction costs.We also review related empirical results that document widespread violations of these restrictions.Option; Pricing

Deposit insurance and the cost of capital

The impacts of deposit insurance and forbearance on the costs and value of uninsured deposits and equity capital are shown under three regimes.Deposit insurance ; Bank capital