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.     

ON ARBITRAGE AND DUALITY UNDER MODEL UNCERTAINTY AND PORTFOLIO CONSTRAINTS

We consider the fundamental theorem of asset pricing (FTAP) and the hedging prices of options under nondominated model uncertainty and portfolio constraints in discrete time. We first show that no arbitrage holds if and only if there exists some family of probability measures such that any admissible portfolio value process is a local super‐martingale under these measures. We also get the nondominated optional decomposition with constraints. From this decomposition, we obtain the duality of the super‐hedging prices of European options, as well as the sub‐ and super‐hedging prices of American options. Finally, we get the FTAP and the duality of super‐hedging prices in a market where stocks are traded dynamically and options are traded statically.     

Mean‐variance efficiency of the market portfolio and futures trading

We derived an intertemporal capital asset pricing model in which the mean‐variance efficiency of the market portfolio is neither a necessary nor a sufficient condition. We obtained this result by modeling a frictionless, continuously open financial market in which nonredundant futures contracts are available for trade, in addition to cash assets. Introducing such contracts modifies the way investors optimally allocate their wealth. Their portfolios then comprise the riskless asset, a perturbed mean‐variance‐efficient portfolio of cash assets, and a perturbed mean‐variance‐efficient portfolio of futures contracts. Furthermore, a (3 + K) mutual fund separation is obtained, with K being the number of economic state variables, in lieu of the usual (2 + K) fund separation. Mean‐variance efficiency of the market portfolio is a necessary condition only when cash assets are the sole traded assets. © 2001 John Wiley & Sons, Inc. Jrl Fut Mark 21:329–346, 2001     

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     

PRICING AND SEMIMARTINGALE REPRESENTATIONS OF VULNERABLE CONTINGENT CLAIMS IN REGIME‐SWITCHING MARKETS

Using a suitable change of probability measure, we obtain a Poisson series representation for the arbitrage‐free price process of vulnerable contingent claims in a regime‐switching market driven by an underlying continuous‐time Markov process. As a result of this representation, along with a short‐time asymptotic expansion of the claim's price process, we develop an efficient novel method for pricing claims whose payoffs may depend on the full path of the underlying Markov chain. The proposed approach is applied to price not only simple European claims such as defaultable bonds, but also a new type of path‐dependent claims that we term self‐decomposable, as well as the important class of vulnerable call and put options on a stock. We provide a detailed error analysis and illustrate the accuracy and computational complexity of our method on several market traded instruments, such as defaultable bond prices, barrier options, and vulnerable call options. Using again our Poisson series representation, we show differentiability in time of the predefault price function of European vulnerable claims, which enables us to rigorously deduce Feynman‐Ka? representations for the predefault pricing function and new semimartingale representations for the price process of the vulnerable claim under both risk‐neutral and objective probability measures.     

THE EFFECT OF ESTIMATION IN HIGH‐DIMENSIONAL PORTFOLIOS

We study the effect of estimated model parameters in investment strategies on expected log‐utility of terminal wealth. The market consists of a riskless bond and a potentially vast number of risky stocks modeled as geometric Brownian motions. The well‐known optimal Merton strategy depends on unknown parameters and thus cannot be used in practice. We consider the expected utility of several estimated strategies when the number of risky assets gets large. We suggest strategies which are less affected by estimation errors and demonstrate their performance in a real data example. Strategies in which the investment proportions satisfy an L₁ ‐constraint are less affected by estimation effects.     

THE GENERAL STRUCTURE OF OPTIMAL INVESTMENT AND CONSUMPTION WITH SMALL TRANSACTION COSTS

We investigate the general structure of optimal investment and consumption with small proportional transaction costs. For a safe asset and a risky asset with general continuous dynamics, traded with random and time‐varying but small transaction costs, we derive simple formal asymptotics for the optimal policy and welfare. These reveal the roles of the investors' preferences as well as the market and cost dynamics, and also lead to a fully dynamic model for the implied trading volume. In frictionless models that can be solved in closed form, explicit formulas for the leading‐order corrections due to small transaction costs are obtained.     

Disagreement in a Multi‐Asset Market

This paper provides a simple framework to study the effect of disagreement in a multi‐asset market equilibrium by considering two agents who disagree about expected returns, variances, and correlation of returns of two risky assets. When agents' subjective beliefs are characterized by mean preserving spreads of a benchmark homogeneous belief, we show that the effect of the disagreement does not cancel out in general and the effect in a multi‐asset market can be very different from a single asset market. In particular, the market risk premium can increase and the risk‐free rate can decrease significantly even when the market is overoptimistic and overconfident.     

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     

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.     

MODEL‐INDEPENDENT NO‐ARBITRAGE CONDITIONS ON AMERICAN PUT OPTIONS

We consider the pricing of American put options in a model‐independent setting: that is, we do not assume that asset prices behave according to a given model, but aim to draw conclusions that hold in any model. We incorporate market information by supposing that the prices of European options are known. In this setting, we are able to provide conditions on the American put prices which are necessary for the absence of arbitrage. Moreover, if we further assume that there are finitely many European and American options traded, then we are able to show that these conditions are also sufficient. To show sufficiency, we construct a model under which both American and European options are correctly priced at all strikes simultaneously. In particular, we need to carefully consider the optimal stopping strategy in the construction of our process.     

THE RANGE OF TRADED OPTION PRICES

Suppose we are given a set of prices of European call options over a finite range of strike prices and exercise times, written on a financial asset with deterministic dividends which is traded in a frictionless market with no interest rate volatility. We ask: when is there an arbitrage opportunity? We give conditions for the prices to be consistent with an arbitrage-free model (in which case the model can be realized on a finite probability space). We also give conditions for there to exist an arbitrage opportunity which can be locked in at time zero. There is also a third boundary case in which prices are recognizably misspecified, but the ability to take advantage of an arbitrage opportunity depends upon knowledge of the null sets of the model.     

TUG‐OF‐WAR,MARKET MANIPULATION,AND OPTION PRICING

We develop an option pricing model based on a tug‐of‐war game. This two‐player zero‐sum stochastic differential game is formulated in the context of a multidimensional financial market. The issuer and the holder try to manipulate asset price processes in order to minimize and maximize the expected discounted reward. We prove that the game has a value and that the value function is the unique viscosity solution to a terminal value problem for a parabolic partial differential equation involving the nonlinear and completely degenerate infinity Laplace operator.     

Empirical tests of canonical nonparametric American option‐pricing methods

Alcock and Carmichael (2008, The Journal of Futures Markets, 28, 717–748) introduce a nonparametric method for pricing American‐style options, that is derived from the canonical valuation developed by Stutzer (1996, The Journal of Finance, 51, 1633–1652). Although the statistical properties of this nonparametric pricing methodology have been studied in a controlled simulation environment, no study has yet examined the empirical validity of this method. We introduce an extension to this method that incorporates information contained in a small number of observed option prices. We explore the applicability of both the original method and our extension using a large sample of OEX American index options traded on the S&P100 index. Although the Alcock and Carmichael method fails to outperform a traditional implied‐volatility‐based Black–Scholes valuation or a binomial tree approach, our extension generates significantly lower pricing errors and performs comparably well to the implied‐volatility Black–Scholes pricing, in particular for out‐of‐the‐money American put options. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 30:509–532, 2010     

Pricing Multiasset Cross‐Currency Options

This study develops a general pricing method for multiasset cross‐currency options, whose underlying asset consists of multiple different assets, and the evaluation currency is different from the ones used in the most liquid market of each asset; the examples include cross‐currency options, cross‐currency basket options, and cross‐currency average options. Moreover, in practice, fast calibration is necessary in the option markets relevant for the underlying assets and the currency, which is also achieved in this study. © 2012 Wiley Periodicals, Inc. Jrl Fut Mark 34:1–19, 2014     

PRICING OF HIGH‐DIMENSIONAL AMERICAN OPTIONS BY NEURAL NETWORKS

Pricing of American options in discrete time is considered, where the option is allowed to be based on several underlyings. It is assumed that the price processes of the underlyings are given Markov processes. We use the Monte Carlo approach to generate artificial sample paths of these price processes, and then we use the least squares neural networks regression estimates to estimate from this data the so‐called continuation values, which are defined as mean values of the American options for given values of the underlyings at time t subject to the constraint that the options are not exercised at time t. Results concerning consistency and rate of convergence of the estimates are presented, and the pricing of American options is illustrated by simulated data.     

ON THE DYBVIG‐INGERSOLL‐ROSS THEOREM

The Dybvig‐Ingersoll‐Ross (DIR) theorem states that, in arbitrage‐free term structure models, long‐term yields and forward rates can never fall. We present a refined version of the DIR theorem, where we identify the reciprocal of the maturity date as the maximal order that long‐term rates at earlier dates can dominate long‐term rates at later dates. The viability assumption imposed on the market model is weaker than those appearing previously in the literature.     

MULTIPLICATIVE APPROXIMATION OF WEALTH PROCESSES INVOLVING NO‐SHORT‐SALES STRATEGIES VIA SIMPLE TRADING

A financial market model with general semimartingale asset–price processes and where agents can only trade using no‐short‐sales strategies is considered. We show that wealth processes using continuous trading can be approximated very closely by wealth processes using simple combinations of buy‐and‐hold trading. This approximation is based on controlling the proportions of wealth invested in the assets. As an application, the utility maximization problem is considered and it is shown that optimal expected utilities and wealth processes resulting from continuous trading can be approximated arbitrarily well by the use of simple combinations of buy‐and‐hold strategies.     

Pricing of moving‐average‐type options with applications

Moving‐average‐type options are complex path‐dependent derivatives whose payoff depends on the moving average of stock prices. This article concentrates on two such options traded in practice: the moving‐average‐lookback option and the moving‐average‐reset option. Both options were issued in Taiwan in 1999, for example. The moving‐average‐lookback option is an option struck at the minimum moving average of the underlying asset's prices. This article presents efficient algorithms for pricing geometric and arithmetic moving‐average‐lookback options. Monte Carlo simulation confirmed that our algorithms converge quickly to the option value. The price difference between geometric averaging and arithmetic averaging is small. Because it takes much less time to price the geometric‐moving‐average version, it serves as a practical approximation to the arithmetic moving‐average version. When applied to the moving‐average‐lookback options traded on Taiwan's stock exchange, our algorithm gave almost the exact issue prices. The numerical delta and gamma of the options revealed subtle behavior and had implications for hedging. The moving‐average‐reset option was struck at a series of decreasing contract‐specified prices on the basis of moving averages. Similar results were obtained for such options with the same methodology. © 2003 Wiley Periodicals, Inc. Jrl Fut Mark 23:415–440, 2003