Minimum option prices under decreasing absolute risk aversion

We establish bounds on option prices in an economy where the representative investor has an unknown utility function that is constrained to belong to the family of nonincreasing absolute risk averse functions. For any distribution of terminal consumption, we identify a procedure that establishes the lower bound of option prices. We prove that the lower bound derives from a particular negative exponential utility function. We also identify lower bounds of option prices in a decreasing relative risk averse economy. For this case, we find that the lower bound is determined by a power utility function. Similar to other recent findings, for the latter case, we confirm that under lognormality of consumption, the Black Scholes price is a lower bound. The main advantage of our bounding methodology is that it can be applied to any arbitrary marginal distribution for consumption. This revised version was published online in June 2006 with corrections to the Cover Date.     

Term structure movements implicit in Asian option prices

In this paper we implement dynamic term structure models that adopt bonds and Asian options in the estimation process. The goal is to analyse the pricing and hedging implications of term structure movements when options are (or are not) included in the estimation process. We investigate how options affect the shape, risk premium and hedging structure of the dynamic factors. We find that the inclusion of options affects the loadings of the slope and curvature factors, and considerably changes the risk premium and hedging structure of all dynamic factors.     

Smoothing the payoff for efficient computation of Basket option prices

We consider the problem of pricing basket options in a multivariate Black–Scholes or Variance-Gamma model. From a numerical point of view, pricing such options corresponds to moderate and high-dimensional numerical integration problems with non-smooth integrands. Due to this lack of regularity, higher order numerical integration techniques may not be directly available, requiring the use of methods like Monte Carlo specifically designed to work for non-regular problems. We propose to use the inherent smoothing property of the density of the underlying in the above models to mollify the payoff function by means of an exact conditional expectation. The resulting conditional expectation is unbiased and yields a smooth integrand, which is amenable to the efficient use of adaptive sparse-grid cubature. Numerical examples indicate that the high-order method may perform orders of magnitude faster than Monte Carlo or Quasi Monte Carlo methods in dimensions up to 35.     

Dynamic risk measures: Time consistency and risk measures from BMO martingales

Time consistency is a crucial property for dynamic risk measures. Making use of the dual representation for conditional risk measures, we characterize the time consistency by a cocycle condition for the minimal penalty function. Taking advantage of this cocycle condition, we introduce a new methodology for the construction of time-consistent dynamic risk measures. Starting with BMO martingales, we provide new classes of time-consistent dynamic risk measures. These families generalize those obtained from backward stochastic differential equations. Quite importantly, starting with right-continuous BMO martingales, this construction naturally leads to paths with jumps.      

Modeling time series information into option prices: An empirical evaluation of statistical projection and GARCH option pricing model

This paper compares the empirical performances of statistical projection models with those of the Black–Scholes (adapted to account for skew) and the GARCH option pricing models. Empirical analysis on S&P500 index options shows that the out-of-sample pricing and projected trading performances of the semi-parametric and nonparametric projection models are substantially better than more traditional models. Results further indicate that econometric models based on nonlinear projections of observable inputs perform better than models based on OLS projections, consistent with the notion that the true unobservable option pricing model is inherently a nonlinear function of its inputs. The econometric option models presented in this paper should prove useful and complement mainstream mathematical modeling methods in both research and practice.     

Bounds for option prices and expected payoffs

This article sharpens Lo's upper bounds for option prices using an alternative approach with the assumption that the underlying asset price is continuously distributed. The increased sharpness is obtained using additional information about the distribution of the underlying assets. It is shown in this article that a large portion of Lo's upper bounds is the maximum of our bounds over all possible distributions.     

Market illiquidity and bounds on European option prices

The paper analyses the impact of illiquidity of a stock paying no dividends on the pricing of European options written on that stock. In particular, it is shown how illiquidity generates price bounds on an option on this stock, even in the absence of other imperfections, such as transaction costs and trading constraints, or the assumption of stochastic volatility. Moreover, price bounds are shown to be asymmetric with respect to the option price under perfect liquidity. This fact explains, under some conditions, the appearance of a smile effect when the implied volatility is estimated from the mid-quote.     

Insider employee stock option trading and stock prices

The question of which factors determine corporate bonds pricing is investigated by analysing the spreads of eurobonds issued by major G-10 companies during the 1991–2001 period. Three main results emerge from the analysis. First, bond ratings appear as the most important determinant of yield spreads, with investors’ reliance on rating agencies judgments increasing over time. Second, the primary market efficiency and the expected secondary market liquidity are not relevant explanatory factors of spreads cross-sectional variability. Finally, rating agencies adopt a different, ‘through the cycle’, evaluation criteria of default risk with respect to the forward looking one adopted by bond investors.     

地方政府债券发行溢价研究

本文通过构建债券发行溢价模型,研究我国地方政府债券发行定价偏离问题。结果表明,地方政府债券一级市场定价较二级市场定价存在严重背离,发行利率偏低,主要受发行要素、宏观经济要素、行政干预要素等多方面因素影响;二级市场定价相对合理。为促进地方政府债券发行定价市场化,本文提出地方政府债券发行利率应严格参考二级市场定价、拓展地方政府债券投资主体、降低商业银行投资地方政府债券风险权重等建议。     

Arbitrage violations and implied valuations: the option market

The ideas presented in this paper are those of the authors and not necessarily reflect the views of the National bank of Canada. Both authors thank the National Bank of Canada and the SSHRC of Canada for their help. Thanks are also due to Professor Y. Tian for his comments, and for participating, together with students of the Financial Engineering program at York University, in the data preparation and the execution of the Matlab programs. In this paper, we propose a necessary and sufficient condition for bid and ask prices of European options to be free of arbitrage, and derive from it an efficient numerical methodology to determine its satisfaction by a given set of prices. If the bid and ask prices satisfy the no-arbitrage (NA) condition, our methodology produces a vector of NA prices that lie between the bid and ask prices. Otherwise, our methodology generates a vector of arbitrage-free prices that is as close as possible, in some sense, to the bid–ask strip. The arbitrage-free prices detected by our methodology render the commonly used practice of using mid-points and then ‘cleaning’ arbitrage from them as unnecessary. Moreover, a vector of ‘cleaned’ prices obtained from mid-point prices may be outside the bid–ask spread even in an arbitrage-free market and, hence, in this case will not be representative of the current market. A new procedure of estimating implied valuation operators is also suggested here. This procedure is rooted in the economic properties of put and call prices and is based on Phillips and Taylor's approximation of a convex function. This approach is superior to common estimation techniques in that it produces an analytical expression for the implied valuation operator and is not data intensive as some other studies. Empirical findings for the new methods are documented and their economic implications are discussed.     

Expected returns, risk premia, and volatility surfaces implicit in option market prices

This article presents a pure exchange economy that extends Rubinstein (1976) to show how the jump-diffusion option pricing model of Merton (1976) is altered when jumps are correlated with diffusive risks. A non-zero correlation between jumps and diffusive risks is necessary in order to resolve the positively sloped implied volatility term structure inherent in traditional jump diffusion models. Our evidence is consistent with a negative covariance, producing a non-monotonic term structure. For the proposed market structure, we present a closed form asset pricing model that depends on the factors of the traditional jump-diffusion models, and on both the covariance of the diffusive pricing kernel with price jumps and the covariance of the jumps of the pricing kernel with the diffusive price. We present statistical evidence that these covariances are positive. For our model the expected stock return, jump and diffusive risk premiums are non-linear functions of time.     

国内人民币期权产品概述及应用

在岸人民币期权产品的推出,既丰富了境内人民币外汇市场的交易品种,又为投资者提供了一种对冲人民币汇率风险的避险工具。与发展已久的离岸人民币期权市场相比,在岸人民币期权市场尚处在初期阶段,不但需要培育相应的期权产品客户,还需要随着人民币期权市场的不断发展,进一步完善期权的定价和交易规则。帮助企业了解人民币期权、挖掘人民币期权的实际需求对于目前人民币期权市场的发展具有重要意义。     

Paramater estimation bias and volatility scaling in Black-Scholes option prices

When asset returns conform to a Gaussian distribution, the moments of the distribution over long return intervals may be estimated by scaling the moments of shorter return intervals. While it is well known that asset returns are not normally distributed, a key empirical question concerns the effect that scaling the volatility of dependent processes will have on the pricing of related financial assets. This study investigates the return properties of the most important currencies traded in spot markets against the U.S. dollar: the Japanese yen, the British pound, and the Swiss franc during the period November 1983 to April 2004. The novelty of this paper is that the volatility properties of the series are tested utilising statistical procedures developed from fractal geometry, with the economic impact determined within an option-pricing framework.     

Perfect option hedging for a large trader

Standard derivative pricing theory is based on the assumption of agents acting as price takers on the market for the underlying asset. We relax this hypothesis and study if and how a large agent whose trades move prices can replicate the payoff of a derivative security. Our analysis extends prior work of Jarrow to economies with continuous security trading. We characterize the solution to the hedge problem in terms of a nonlinear partial differential equation and provide results on existence and uniqueness of this equation. Simulations are used to compare the hedging strategies in our model to standard Black-Scholes strategies.     

Exact solutions for bond and option prices with systematic jump risk

A variety of realistic economic considerations make jump-diffusion models of interest rate dynamics an appealing modeling choice to price interest-rate contingent claims. However, exact closed-form solutions for bond prices when interest rates follow a mixed jump-diffusion process have proved very hard to derive. This paper puts forward two new models of interest-rate dynamics that combine infrequent, discrete changes in the interest-rate level, modeled as a jump process, with short-lived, mean reverting shocks, modeled as a diffusion process. The two models differ in the way jumps affect the central tendency of interest rates; in one case shocks are temporary, in the other shocks are permanent. We derive exact closed-form solutions for the price of a discount bond and computationally tractable schemes to price bond options.     

Examining the Equilibrium Relationship Between the Shanghai 50 Stock Index Futures and the Shanghai 50 ETF Options Markets

Based on the put-call-futures parity model, this article studies the equilibrium relationship between the Shanghai 50 stock index futures and the Shanghai 50 Exchange-Traded Fund (ETF) options markets by analyzing the arbitrage opportunities and profits between these two derivative markets. This article reveals that the cost spread, option volatility, days from the expiration date, moneyness of options, trading strategy, and policy factors all have a great impact on the arbitrage profits and opportunities. In addition, significant arbitrage profits and opportunities indicate violations of put-call-futures parity. Although no equilibrium relationship exists between the Shanghai 50 stock index futures and the Shanghai 50 ETF options markets, efficiency in these markets has gradually improved.     

Advisors and asset prices: A model of the origins of bubbles

We develop a model of asset price bubbles based on the communication process between advisors and investors. Advisors are well-intentioned and want to maximize the welfare of their advisees (like a parent treats a child). But only some advisors understand the new technology (the tech-savvies); others do not and can only make a downward-biased recommendation (the old-fogies). While smart investors recognize the heterogeneity in advisors, naive ones mistakenly take whatever is said at face value. Tech-savvies inflate their forecasts to signal that they are not old-fogies, since more accurate information about their type improves the welfare of investors in the future. A bubble arises for a wide range of parameters, and its size is maximized when there is a mix of smart and naive investors in the economy. Our model suggests an alternative source for stock over-valuation in addition to investor overreaction to news and sell-side bias.     

Conditions on option prices for absence of arbitrage and exact calibration

Under the assumption of absence of arbitrage, European option quotes on a given asset must satisfy well-known inequalities, which have been described in the landmark paper of Merton [Merton, R., 1973. Theory of rational option pricing. Bell Journal of Economics and Management Science 4 (1), 141–183]. If we further assume that there is no interest rate volatility and that the underlying asset continuously pays deterministic dividends, cross-maturity inequalities must also be satisfied by the bid and ask option prices.