Pricing and hedging margrabe options with stochastic volatilities elisa alos thorsten rheinlander y february 23, 2017 abstract a margrabe or exchange option is an option to exchange one asset for another. In other words, perform all the computations in a relative price system. So the original option has become a call option on the first asset with its numeraire pricing with a strike of 1 unit of the riskless asset. The use of numeraires in multidimensional blackscholes. This particular asset is usually referred to as the numeraire.
Undoubtedly, the capital asset pricing model capm developed by sharpe 1964, lintner 1965, and mossin 1966 is the best known asset pricing model. They have an expression on earth that i believe applies to this situation. Most of the ideas in the book developed from long discussions with each of them, and trying to make sense of what each was saying in the language of the other. A generalization of the international asset pricing model 110 no foreigner is interested in the asset, and this of course has to take into account the covariances with and the expected. This led to modern portfolio theory mpt by harry markowitz in 1952 journal of finance and to the related capital asset pricing model capm by william sharpe in 1964 journal. With a view toward numeraire change yan zeng version 1. The use of numeraires in multidimensional blackscholes partial differential equations 3 we can reduce the spatial dimension by one. Hence, their numeraire asset has no market risk and pays an interest rate that equals zero, which is convenient for their analysis. Pricing kernels 3 pricing is reduced to riskless discounting and a distorted riskneutral conditional expectation. It is illegal to reproduce this article in any format. On the use of numeraires in option pricing it is illegal. Pdf changes of numeraire for pricing futures, forwards, and. Some of the proofs are not given in sections 2 and 3, as they tend to be quite long. The key message of the model is that the expected excess return on a risky.
A numeraire approach treats price as a number of units of one asset needed for an acquisition of a unit of another asset instead of expressing prices in dollar terms exclusively. Unlike much of the existing literature, stochastic finance. In either case, the chosen numeraire is the unique growthoptimal portfolio with respect to f, and p is the unique equivalent measure under which the discounted price process is a uniformly integrable martingale with respect to f. Change of numeraire in five formulas a numeraire stock, zcb, bankaccount. In a monetary economy, acting as the numeraire is one of the functions of money, to serve as a unit of account.
Asset pricing in production economies wharton finance. The presence of composition risk increases the mean and. Jermann finance department, the wharton school, university of pennsylvania, philadelphia, pa 19104, usa received 1 july 1994. Monika piazzesi, martin schneider, and selale tuzel may 2003 abstract this paper builds an equilibrium asset pricing model with housing consumption. Changes of numeraire for pricing futures, forwards, and options the self.
A numeraire is usually applied to a single good, which becomes the base value for the entire index or market. Frahm, 2015 pricing and valuation under the realworld measure to f, the market is sensitive to f. In pricing stock options, if the riskless rate is assumed to be constant, the money market account is a natural numeraire. Download limit exceeded you have exceeded your daily download allowance.
A numeraire is any tradeable asset with price process nt such that nt 0, for all times t. This paper develops a simple consumptionbased asset pricing model that reserves an explicit role for housing. In the context of inflationary cosmology, it is fair to say that the universe is the ultimate free lunch. Changes of numeraire for pricing futures, forwards, and. Introduction and overview asset market data are often ignored in evaluating macroeconomic mod els, and aggregate quantity data are often avoided in empirical investiga tions of asset market returns. Numeraire invariance and application to option pricing and. As we have seen in the previous lesson, proving that a market is arbitragefree may be very tedious, even under very simple circumstances. In 1989 papers by geman 1989 and jamshidian 1989 formalized the mathematics behind the numeraire approach. This pdf is a selection from an outofprint volume from the national.
A key concept in modern asset pricing theory is that of a numeraire. At jumps, the value of a dollar measured in units of the asset satis. Using stock price as numeraire in option pricing models with. Noarbitrage pricing and numeraire change springerlink. Using stock price as numeraire in option pricing models. Agents care about the composition of a consumption basket that contains shelter and other goods. While limited in scope, this approach offers precision in many applications.
For example, when pricing an option to exchange one asset for another, choosing one of the. The option selecting one asset among two assets with stochastic price is a typical example with several risk sources and the change of numeraire has advantage in pricing problems of such options. The arbitrage model was proposed as an alternative to the mean variance capital asset pricing model, introduced by sharpe, lintner, and treynor, that has become the major analytic tool for explaining phenomena observed in capital markets for risky assets. P that are arbitrarily close in the totalvariation distance to p. First, we can examine whether real world asset prices and investor portfolios conform to the.
In this lesson we will present the first fundamental theorem of asset pricing, a result that provides an alternative way to test the existence of arbitrage opportunities in a given market. The insight provided in 34, 49 is that dynamic asset pricing in the absence of arbitrage is captured by the existence of a positive with probability one martingale that can be used to represent prices in. In this article, we build on their results and, using the reinvested asset price as the numeraire, unify and extend the literature on option parity, or equivalence, results relating american call and put prices for asset and futures options. With power utility, the model predicts that excess stock returns are low, unpredictable and not very volatile. Aug 15, 2019 numeraire is an economic term that represents a unit of account. Recent work in asset pricing has tried to construct a macroeconomic model in which. Express all prices in the market, including that of the option, in terms of the chosen numeraire. Since the numeraire asset in the new price system is riskless by definition, we have reduced the number. We show here that many other probability measures can be defined in the same way to solve different assetpricing problems, in particular option pricing. Pricing and hedging margrabe options with stochastic volatilities.
Is the wellknown size effect or smallcap risk premium an academic fallacy. The payoff of the option, repriced under this change of numeraire, is max0, s 1 ts 2 t 1. This numeraire approach leads to simpler pricing options for complex products, such as barrier, lookback, quanto, and. Identifying a risky asset as numeraire has a correlation with the number of underlying assets to model. Changes of numeraire for pricing futures, forwards, and options. There are instances where a different numeraire is desirable. Pricing and hedging margrabe options with stochastic. This is a short version of the paper of exchange options 2007, concentrating on the principle of numeraire invariance. As this numeraire becomes worthless in default, certain technical di. Fundamental theorem of asset pricing in a nutshell. Numeraire invariance is a wellknown technique in option pricing and hedging theory. A generalization of the international asset pricing model 108 to be able to interpret also the second term in a as a covariance with some portfolio, the weights have to be rescaled.
It emphasizes application to unique pricing in arbitragefree model, the derivation of hedge ratios and the pde when price ratios are diffusions, explicit representations in the multivariate poisson model, and the role played by homogeneity. A representative agent consumes housing services and a numeraire nonhousing consumption good, both of which can be purchased in frictionless markets. There were informal markets for livestock over 3,000 years bce as evidenced by tablets in sumer for counting sheep in cuneiform. Pdf a generalization of the international asset pricing. The capital asset pricing model is an elegant theory with profound implications for asset pricing and investor behavior. Changes of numeraire, changes of probability measure and. Note the dividend rate q 1 of the first asset remains the same even with change of pricing. This numeraire approach leads to simpler pricing options for complex products, such as barrier, lookback. Not only can we deploy any nondividend paying traded asset in our pricing equation. The numeraire or numeraire is a basic standard by which value is computed. Estimating and evaluating asset pricing models 174 10 gmm in explicit discount factor models 177 10. Pdf a generalization of the international asset pricing model. In mathematical economics it is a tradable economic entity in terms of whose price the relative prices of all other tradables are expressed. There aint no such thing as a free lunch adam warlock and moondragon, in silver surferwarlock.
We show that the wealthconsumption ratio is the value of an annuity when the numeraire is changed from units of the consumption good to units of the consumption process. Abstract a survey of the fundamental theorem of asset pricing in mathematical. Contents 1 introduction 2 2 fundamental theorem of asset pricing. Evolution of stock pricing the need to determine a value in exchange or price began with the earliest trade in ancient times. Pdf changes of numeraire for pricing futures, forwards. Conference inriansf on mathematical finance, paris.
Workingwithdiscretetimemodels,leroy 1973, rubinstein1976, and lucas 1978 developed multiperiod extensions of the capm. The agent is endowed with a claim to future numeraire, as well as with a stock of. An overview of asset pricing models andreas krause university of bath school of management phone. But how useful is the model given the idealized world that underlies its derivation. Phases and patterns, annals of economics and finance, vol. It takes a convenient asset as the numeraire, as if it were the medium of exchange, and expresses all other asset and option prices in units of this numeraire. On the use of numeraires in option pricing it is illegal to.
Moreover, these probability measure changes are in fact. In a general stochastic volatility framework, by taking the second asset as a numeraire, we derive pricing as well as approximate pricing. Asset pricing problems are solved by judiciously choosing how much absolute and how much relative pricing one will do, depending on the assets in question and the purpose of the calculation. Changes of numeraire for pricing futures, forwards, and options a change of numeraire to the underlying asset price is associated with the new measure q, where dq qqdq dst s0. The deus ex machina of consumptionbased asset pricing models is that, for a given endowment process, household trading of financial assets is motivated by a desire to smooth consumption both over 1both the payoffs and the price are denominated in the numeraire consumption good. Download scholarly research papers on capital asset pricing models and their circularity. Numeraire in financial pricing determining an appropriate numeraire has foundation in several financial pricing models such as options and certain assets. Acknowledgments this book owes an enormous intellectual debt to lars hansen and gene fama. Numeraire is an economic term that represents a unit of account.
427 1156 1404 1416 1428 1289 229 991 1040 779 1173 1185 612 1138 1400 304 1048 1189 159 957 806 322 1130 695 88 852 84 44 1148 764 1097 834 346 855