ANALYTIC APPROXIMATIONS FOR MULTI‐ASSET OPTION PRICING

We derive general analytic approximations for pricing European basket and rainbow options on N assets. The key idea is to express the option’s price as a sum of prices of various compound exchange options, each with different pairs of subordinate multi‐ or single‐asset options. The underlying asset prices are assumed to follow lognormal processes, although our results can be extended to certain other price processes for the underlying. For some multi‐asset options a strong condition holds, whereby each compound exchange option is equivalent to a standard single‐asset option under a modified measure, and in such cases an almost exact analytic price exists. More generally, approximate analytic prices for multi‐asset options are derived using a weak lognormality condition, where the approximation stems from making constant volatility assumptions on the price processes that drive the prices of the subordinate basket options. The analytic formulae for multi‐asset option prices, and their Greeks, are defined in a recursive framework. For instance, the option delta is defined in terms of the delta relative to subordinate multi‐asset options, and the deltas of these subordinate options with respect to the underlying assets. Simulations test the accuracy of our approximations, given some assumed values for the asset volatilities and correlations. Finally, a calibration algorithm is proposed and illustrated.     

Currency option pricing: Mean reversion and multi‐scale stochastic volatility

This paper investigates the valuation of currency options when the underlying currency follows a mean‐reverting lognormal process with multi‐scale stochastic volatility. A closed‐form solution is derived for the characteristic function of the log‐asset price. European options are then valued by means of the Fourier inversion formula. The proposed model enables us to calibrate simultaneously to the observed currency futures and the implied volatility surface of the currency options within a unified framework. The fractional fast Fourier transform (FFT) is adopted to implement the Fourier inversion, thus ensuring that the grid spacing restriction of the standard FFT can be relaxed, which results in a more efficient computation. Using Monte Carlo simulation as a benchmark, our numerical examples show that the derived option pricing formula is accurate and efficient for practical use. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 30:938–956, 2010     

HEDGING STRATEGIES AND MINIMAL VARIANCE PORTFOLIOS FOR EUROPEAN AND EXOTIC OPTIONS IN A LÉVY MARKET

This paper presents hedging strategies for European and exotic options in a Lévy market. By applying Taylor’s theorem, dynamic hedging portfolios are constructed under different market assumptions, such as the existence of power jump assets or moment swaps. In the case of European options or baskets of European options, static hedging is implemented. It is shown that perfect hedging can be achieved. Delta and gamma hedging strategies are extended to higher moment hedging by investing in other traded derivatives depending on the same underlying asset. This development is of practical importance as such other derivatives might be readily available. Moment swaps or power jump assets are not typically liquidly traded. It is shown how minimal variance portfolios can be used to hedge the higher order terms in a Taylor expansion of the pricing function, investing only in a risk‐free bank account, the underlying asset, and potentially variance swaps. The numerical algorithms and performance of the hedging strategies are presented, showing the practical utility of the derived results.     

Option‐Expiration Effects in Small Markets: The Spanish Stock Exchange

This study analyzes the effect of the expiration of the Ibex‐35 Index derivatives, as well as the first four stock options traded in the Spanish Equity Derivatives Exchange, on the return, conditional volatility, and trading volume of the underlying assets. The analysis covers the period from the introduction of the various derivatives to December 1995. This period has been divided into two subperiods in order to determine if there are changes in the conclusions. The expiration of the Ibex‐35 index derivatives is associated with an increase in the trading volume of the underlying asset, but it has no significant effect on either the underlying asset prices or on the level of volatility on the expiration day. However, the expiration of the stock options has significant impact on their underlying assets. We observed a downward pressure on prices and a reduction of volatility level in the week before the expiration date and a significant increase in trading volume on the expiration day. The absence of futures contracts on individual stocks, among other possible causes, may explain these differences. © 2001 John Wiley & Sons, Inc. Jrl Fut Mark 21:905–928, 2001     

Decomposing the Income Insurance Channel across OECD and Emerging Markets

This paper investigates the underlying forces driving income insurance channels for the Organisation for Economic Co‐operation and Development (OECD) and emerging markets. We find income insurance channels across countries to be driven by different subchannels. For the OECD, income insurance is mostly governed by payments for financial liabilities; for the emerging markets, income flows from nationals working abroad constitute the main income smoother. Despite the growth in cross‐border financial asset trading over the years, we could not find evidence of income smoothing via foreign assets receipts for the OECD. For the majority of emerging markets, neither receipts of foreign assets nor foreign liability payments are strong enough to insure income as well.     

Option bid‐ask spread and scalping risk: Evidence from a covered warrants market

This study develops and empirically tests a simple market microstructure model to capture the main determinants of option bid‐ask spread. The model is based on option market making costs (initial hedging, rebalancing, and order processing costs), and incorporates a reservation bid‐ask spread that option market makers apply to protect themselves from scalpers. The model is tested on a sample of covered warrants, which are optionlike securities issued by banks, traded on the Italian Stock Exchange. The empirical analysis validates the model. The initial cost of setting up a delta neutral portfolio has been found to be an important determinant of option bid‐ask spread, as well as rebalancing costs to keep the portfolio delta neutral. This result provides evidence of a further link between options and underlying assets: the spread of the option is positively related to the spread of its underlying asset. Empirical evidence also indicates that the reservation bid‐ask spread, computed as the product of option delta and underlying asset tick, plays a very important role in explaining the bid‐ask spread of options. © 2006 Wiley Periodicals, Inc. Jrl Fut Mark 26:843–867, 2006     

Risk Management of Nonstandard Basket Options with Different Underlying Assets

Basket options are among the most popular products of the new generation of exotic options. They are particularly attractive because they can efficiently and simultaneously hedge a wide variety of intrinsically different financial risks and are flexible enough to cover all the risks faced by firms. Oddly, the existing literature on basket options considers only standard baskets where all underlying assets are of the same type and hedge the same kind of risk. Moreover, the empirical implementation of basket‐option models remains in its early stages, particularly when the baskets contain different underlying assets. This study focuses on various steps for developing sound risk management of basket options. We first propose a theoretical model of a nonstandard basket option on commodity price with stochastic convenience yield, exchange rate, and domestic and foreign zero‐coupon bonds in a stochastic interest rate setting. We compare the hedging performance of the extended basket option containing different underlying assets with that of a portfolio of individual options. The results show that the basket strategy is more efficient. We apply the maximum likelihood method to estimate the parameters of the basket model and the correlations between variables. Monte Carlo simulations are conducted to examine the performance of the maximum likelihood estimator in finite samples of simulated data. A real‐data study for a nonfinancial firm is presented to illustrate ways practitioners could use the extended basket option. © 2012 Wiley Periodicals, Inc. Jrl Fut Mark 33:299‐326, 2013     

The limits to stock index arbitrage: Examining S&P 500 futures and SPDRS

This study examines factors affecting stock index spot versus futures pricing and arbitrage opportunities by using the S&P 500 cash index and the S&P 500 Standard and Poor's Depository Receipt (SPDR) Exchange‐Traded Fund (ETF) as “underlying cash assets.” Potential limits to arbitrage when using the cash index are the staleness of the underlying cash index, trading costs, liquidity (volume) issues of the underlying assets, the existence of sufficient time to execute profitable arbitrage transactions, short sale restrictions, and the extent to which volatility affects mispricing. Alternatively, using the SPDR ETF as the underlying asset mitigates staleness and trading cost problems as well as the effects of volatility associated with the staleness of the cash index. Minute‐by‐minute prices are compared over different volatility levels to determine how these factors affect the limits of S&P 500 futures arbitrage. Employing the SPDR as the cash asset examines whether a liquid tradable single asset with low trading costs can be used for pricing and arbitrage purposes. The analysis examines how long mispricing lasts, the impact of volatility on mispricing, and whether sufficient volume exists to implement arbitrage. The minute‐by‐minute liquidity of the futures market is examined using a new transaction volume futures database. The results show that mispricings exist regardless of the choice of the underlying cash asset, with more negative mispricings for the SPDR relative to the S&P 500 cash index. Furthermore, mispricings are more frequent in high‐ and mid‐volatility months than in low‐volatility months and are associated with higher volume during high‐volatility months. © 2008 Wiley Periodicals, Inc. Jrl Fut Mark 28:1182–1205, 2008     

The quality of volatility traded on the over‐the‐counter currency market: A multiple horizons study

Previous studies of the quality of market‐forecasted volatility have used the volatility that is implied by exchange‐traded option prices. The use of implied volatility in estimating the market view of future volatility has suffered from variable measurement errors, such as the non‐synchronization of option and underlying asset prices, the expiration‐day effect, and the volatility smile effect. This study circumvents these problems by using the quoted implied volatility from the over‐the‐counter (OTC) currency option market, in which traders quote prices in terms of volatility. Furthermore, the OTC currency options have daily quotes for standard maturities, which allows the study to look at the market's ability to forecast future volatility for different horizons. The study finds that quoted implied volatility subsumes the information content of historically based forecasts at shorter horizons, and the former is as good as the latter at longer horizons. These results are consistent with the argument that measurement errors have a substantial effect on the implied volatility estimator and the quality of the inferences that are based on it. © 2003 Wiley Periodicals, Inc. Jrl Fut Mark 23:261–285, 2003     

Option pricing under extended normal distribution

This article proposes a closed pricing formula for European options when the return of the underlying asset follows extended normal distribution, that is, any different degrees of skewness and kurtosis relative to the normal distribution induced by the Black‐Scholes model. The moment restriction is suggested, so that the pricing model under any arbitrary distribution for an underlying asset must satisfy the arbitrage‐free condition. Numerical experiments and comparison of empirical performance of the proposed model with the Black‐Scholes, ad hoc Black‐Scholes, and Gram‐Charlier distribution models are carried out. In particular, an estimation of implied parameters such as standard deviation, skewness, and kurtosis of the return on the underlying asset from the market prices of the KOSPI 200 index options is made, and in‐sample and out‐of‐sample tests are performed. These results not only support the previous finding that the actual density of the underlying asset shows skewness to the left and high peaks, but also demonstrate that the present model has good explanatory power for option prices. © 2005 Wiley Periodicals, Inc. Jrl Fut Mark 25:845–871, 2005     

The Valuation of American Options on Multiple Assets

In this paper we provide valuation formulas for several types of American options on two or more assets. Our contribution is twofold. First, we characterize the optimal exercise regions and provide valuation formulas for a number of American option contracts on multiple underlying assets with convex payoff functions. Examples include options on the maximum of two assets, dual strike options, spread options, exchange options, options on the product and powers of the product, and options on the arithmetic average of two assets. Second, we derive results for American option contracts with nonconvex payoffs, such as American capped exchange options. For this option we explicitly identify the optimal exercise boundary and provide a decomposition of the price in terms of a capped exchange option with automatic exercise at the cap and an early exercise premium involving the benefits of exercising prior to reaching the cap. Besides generalizing the current literature on American option valuation our analysis has implications for the theory of investment under uncertainty. A specialization of one of our models also provides a new representation formula for an American capped option on a single underlying asset.     

Valuing Seller‐Defaultable Options

This study analyzes seller‐defaultable options that allow option writers to have a free‐will right to default, along with some prespecified default mechanisms. We analytically and numerically examine the pricing, hedging, defaulting, and profitability of the seller‐defaultable options, considering three possible scenarios for seller default. Analyzing the essential implications of seller‐defaultable options, we show that the option price is positively correlated with the default fine, underlying asset price, and volatility. The seller‐defaultable option's Greeks appear more complicated than those of the plain vanilla options. The likelihood of sellers defaulting increases with the underlying asset price, interest rate, volatility, and maturity time. Subject to the default mechanism, the buyers’ trading involves a trade‐off between profits and costs. © 2012 Wiley Periodicals, Inc. Jrl Fut Mark 33:129–157, 2013     

NO‐ARBITRAGE IN A NUMÉRAIRE‐INDEPENDENT MODELING FRAMEWORK

The classic approach to modeling financial markets consists of four steps. First, one fixes a currency unit. Second, one describes in that unit the evolution of financial assets by a stochastic process. Third, one chooses in that unit a numéraire, usually the price process of a positive asset. Fourth, one divides the original price process by the numéraire and considers the class of admissible strategies for trading. This approach has one fundamental drawback: Almost all concepts, definitions, and results, including no‐arbitrage conditions like NA, NFLVR, and NUPBR depend by their very definition, at least formally, on initial choices of a currency unit and a numéraire. In this paper, we develop a new framework for modeling financial markets, which is not based on ex‐ante choices of a currency unit and a numéraire. In particular, we introduce a “numéraire‐independent” notion of no‐arbitrage and derive its dual characterization. This yields a numéraire‐independent version of the fundamental theorem of asset pricing (FTAP). We also explain how the classic approach and other recent approaches to modeling financial markets and studying no‐arbitrage can be embedded in our framework.     

Minimum variance cross hedging under mean‐reverting spreads,stochastic convenience yields,and jumps: Application to the airline industry

Exchange traded futures contracts often are not written on the specific asset that is a source of risk to a firm. The firm may attempt to manage this risk using futures contracts written on a related asset. This cross hedge exposes the firm to a new risk, the spread between the asset underlying the futures contract and the asset that the firm wants to hedge. Using the specific case of the airline industry as motivation, we derive the minimum variance cross hedge assuming a two‐factor diffusion model for the underlying asset and a stochastic, mean‐reverting spread. The result is a time‐varying hedge ratio that can be applied to any hedging horizon. We also consider the effect of jumps in the underlying asset. We use simulations and empirical tests of crude oil, jet fuel cross hedges to demonstrate the hedging effectiveness of the model. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 29:736–756, 2009     

Robust Hedging of Barrier Options

This article considers the pricing and hedging of barrier options in a market in which call options are liquidly traded and can be used as hedging instruments. This use of call options means that market preferences and beliefs about the future behavior of the underlying assets are in some sense incorporated into the hedge and do not need to be specified exogenously. Thus we are able to find prices for exotic derivatives which are independent of any model for the underlying asset. For example we do not need to assume that the underlying assets follow an exponential Brownian motion.
We find model-independent upper and lower bounds on the prices of knock-in and knock-out puts and calls. If the market prices the barrier options outside these limits then we give simple strategies for generating profits at zero risk. Examples illustrate that the bounds we give can be fairly tight.     

THE SQUARED ORNSTEIN-UHLENBECK MARKET

We study a complete market containing J assets, each asset contributing to the production of a single commodity at a rate that is a solution to the squared Ornstein-Uhlenbeck (Cox-Ingersoll-Ross) SDE. The assets are owned by K agents with CRRA utility functions, who follow feasible consumption/investment regimes so as to maximize their expected time-additive utility from consumption. We compute the equilibrium for this economy and determine the state-price density process from market clearing. Reducing to a single (representative) agent, and exploiting the relation between the squared-OU and squared-Bessel SDEs, we obtain closed-form expressions for the values of bonds, assets, and options on the total asset value. Typical model parameters are estimated by fitting bond price data, and we use these parameters to price the assets and options numerically. Implications for the total asset price itself as a diffusion are discussed. We also estimate implied volatility surfaces for options and bond yields.     

Double Lookbacks

A new class of options, double lookbacks , where the payoffs depend on the maximum and/or minimum prices of one or two traded assets is introduced and analyzed. This class of double lookbacks includes calls and puts with the underlying being the difference between the maximum and minimum prices of one asset over a certain period, and calls or puts with the underlying being the difference between the maximum prices of two correlated assets over a certain period. Analytical expressions of the joint probability distribution of the maximum and minimum values of two correlated geometric Brownian motions are derived and used in the valuation of double lookbacks. Numerical results are shown, and prices of double lookbacks are compared to those of standard lookbacks on a single asset.     

BLACK–SCHOLES REPRESENTATION FOR ASIAN OPTIONS

Asian options are securities with a payoff that depends on the average of the underlying stock price over a certain time interval. We identify three natural assets that appear in pricing of the Asian options, namely a stock S, a zero coupon bond B^T with maturity T, and an abstract asset A (an “average asset”) that pays off a weighted average of the stock price number of units of a dollar at time T. It turns out that each of these assets has its own martingale measure, allowing us to obtain Black–Scholes type formulas for the fixed strike and the floating strike Asian options. The model independent formulas are analogous to the Black–Scholes formula for the plain vanilla options; they are expressed in terms of probabilities under the corresponding martingale measures that the Asian option will end up in the money. Computation of these probabilities is relevant for hedging. In contrast to the plain vanilla options, the probabilities for the Asian options do not admit a simple closed form solution. However, we show that it is possible to obtain the numerical values in the geometric Brownian motion model efficiently, either by solving a partial differential equation numerically, or by computing the Laplace transform. Models with stochastic volatility or pure jump models can be also priced within the Black–Scholes framework for the Asian options.     

On approximating deep in‐the‐money Asian options under exponential Lévy processes

This note proposes a new approach of valuing deep in‐the‐money fixed strike and discretely monitoring arithmetic Asian options. This new approach prices Asian options whose underlying asset price evolves according to the exponential of a Lévy process as a weighted sum of European options. Numerical results from experimenting on three different types of Lévy processes—a diffusion process, a jump diffusion process, and a pure jump process—illustrate the accuracy of the approach. © 2010 Wiley Periodicals, Inc. Jrl Fut Mark     

我国房地产价格与通货膨胀关系的实证分析

近年来,我国房地产等资产的价格高涨所引起的“资产泡沫型通货膨胀”现象比较明显。通过观察我国近年来房屋销售价格指数和居民消费价格指数的走势发现,房地产高涨与通货膨胀之间具有很大的联动作用。因此,研究房地产价格与通货膨胀的关系,寻找破解房地产价格高涨、缓解通货膨胀压力的方法,具有重要的学术研究价值和现实意义。本文利用Granger因果关系检验模型,验证了我国房地产价格与通货膨胀关系之间存在Granger因果关系,并围绕高房价是怎样影响通货膨胀的逻辑思路,提出破解高房价、缓解“资产泡沫型通货膨胀”压力的解决方法。