No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
Option pricing with discrete rebalancing
Abstract
We consider option pricing when dynamic portfolios are discretely rebalanced. The portfolio adjustments only occur after fixed relative variation of the stock price. The stock price follows a marked point process and the market is incomplete. We first characterize the equivalent martingale measures. An explicit formula based on the minimal martingale measure is then provided together with the hedging strategy underlying portfolio adjustments. Under adequate conditions on the stock price dynamics, the minimal pricing formula converges to the Black-Scholes formula when the triggering price increment shrinks to zero. This is shown theoretically and numerically on two examples : a marked Poisson process and a jump process driven by a latent geometric Brownian motion. For the empirical application we use IBM intraday transaction data and compare option prices given by the marked Poisson model and the Black-Scholes model.
... There, the Authors approach the filtering problem (after time discretization in intervals of equal length) using a particle filter on the counting observations, based on a sampling importance resampling algorithm. Other works dealing with MPP but more concerned with option pricing and hedging can be found in Kirch and Runggaldier (2004), Prigent (2000) and Prigent et al. (2004). In Kirch and Runggaldier (2004), assuming a model in which the asset price follows a geometric Poisson process with unknown constant intensity, the optimal hedging strategy is constructed using stochastic control techniques. ...
... In Kirch and Runggaldier (2004), assuming a model in which the asset price follows a geometric Poisson process with unknown constant intensity, the optimal hedging strategy is constructed using stochastic control techniques. On the other hand, in Prigent (2000), in a very general context, the equivalent martingale measures are characterized by their Radon-Nykodim derivatives with respect to the natural probability, whereas in Prigent et al. (2004) the problem of option pricing is considered in the case in which the (dynamic) portfolios are adjusted only after fixed relative changes in the stock prices. Following a different modelling strategy, in our model the intensity process δ of the DSPP with marks is characterized as a function of time and of another underlying MPP. ...
... Thus, the problem of pricing a contingent claim, under the no arbitrage assumption, is reduced to taking expected values under the 'right' measure among all existing equivalent martingale measures. One possibility is to choose the so called minimal martingale measure Q introduced by Föllmer and Schweizer (1991) which arises very often in the financial literature (see Prigent et al. (2004) for a discussion and for further references). In our probabilistic setting, for the case of partial information, in which market agents are allowed to observe only the history of the stock price (that is, all past times and sizes of price changes, but not the history of the intensity process), we propose to use as a pricing measure the restriction to the filtration representing the available information of the measure Q derived in the case of complete information. ...
To model intraday stock price movements we propose a class of marked doubly stochastic Poisson processes, whose intensity process can be interpreted in terms of the effect of information release on market activity. Assuming a partial information setting in which market agents are restricted to observe only the price process, a filtering algorithm is applied to compute, by Monte Carlo approximation, contingent claim prices, when the dynamics of the price process is given under a martingale measure. In particular, conditions for the existence of the minimal martingale measure Q are derived, and properties of the model under Q are studied.
... Mello and Neuhaus [20] research the accumulated hedging error due to discrete rebalancing, extending the work by Figlewski [12] to imperfect markets. The idea of allowing hedging at the time when fixed relative changes in the stock price occur is explored in [25], where the price dynamic (in an incomplete market) is a marked point process. ...
It is well-known that using delta hedging to hedge financial options is not feasible in practice. Traders often rely on discrete-time hedging strategies based on fixed trading times or fixed trading prices (i.e., trades only occur if the underlying asset's price reaches some predetermined values). Motivated by this insight and with the aim of obtaining explicit solutions, we consider the seller of a perpetual American put option who can hedge her portfolio once until the underlying stock price leaves a certain range of values $(a,b)$. We determine optimal trading boundaries as functions of the initial stock holding, and an optimal hedging strategy for a bond/stock portfolio. Optimality here refers to the variance of the hedging error at the (random) time when the stock leaves the interval $(a,b)$. Our study leads to analytical expressions for both the optimal boundaries and the optimal stock holding, which can be evaluated numerically with no effort.
... Hence the resulting price process can be viewed as a marked point process. Option pricing in this context was studied by Prigent et al. (2002 Prigent et al. ( , 2004). In the setting of marked point processes, another observable quantity of interest is the durations, i.e., the waiting times between successive transactions of a given asset. ...
... In Prigent et al. (2004) the pricing of options on portfolios that are rebalanced after a fixed change in price occurred is considered. Krykova (2003) studied the valuation of path-dependent securities with low discrepancy methods. ...
ENGLISH ABSTRACT: Life insurance and pension funds offer a wide range of products that are invested in a mix of assets. These portfolios (II), underlying the products, are rebalanced back to predetermined fixed proportions on a regular basis. This is done by selling the better performing assets and buying the worse performing assets. Life insurance or pension fund contracts can offer the client a minimum payout guarantee on the contract by charging them an extra premium (a). This problem can be changed to that of the pricing of a put option with underlying . It forms a liability for the insurance firm, and therefore needs to be managed in terms of risks as well. This can be done by studying the option’s sensitivities. In this thesis the premium and sensitivities of this put option are calculated, using different Monte Carlo methods, in order to find the most efficient method. Using general Monte Carlo methods, a simplistic pricing method is found which is refined by applying mathematical techniques so that the computational time is reduced significantly. After considering Antithetic Variables, Control Variates and Latin Hypercube Sampling as variance reduction techniques, option prices as Control Variates prove to reduce the error of the refined method most efficiently. This is improved by considering different Quasi-Monte Carlo techniques, namely Halton, Faure, normal Sobol’ and other randomised Sobol’ sequences. Owen and Faure-Tezuke type randomised Sobol’ sequences improved the convergence of the estimator the most efficiently. Furthermore, the best methods between Pathwise Derivatives Estimates and Finite Difference Approximations for estimating sensitivities of this option are found. Therefore by using the refined pricing method with option prices as Control Variates together with Owen and Faure-Tezuke type randomised Sobol’ sequences as a Quasi-Monte Carlo method, more efficient methods to price this option (compared to simplistic Monte Carlo methods) are obtained. In addition, more efficient sensitivity estimators are obtained to help manage risks. AFRIKAANSE OPSOMMING: Lewensversekering en pensioenfondse bied die mark ’n wye reeks produkte wat belê word in ’n mengsel van bates. Hierdie portefeuljes (II), onderliggend aan die produkte, word op ’n gereelde basis terug herbalanseer volgens voorafbepaalde vaste proporsies. Dit word gedoen deur bates wat beter opbrengste gehad het te verkoop, en bates met swakker opbrengste aan te koop. Lewensversekeringof pensioenfondskontrakte kan ’n kliënt ’n verdere minimum uitbetaling aan die einde van die kontrak waarborg deur ’n ekstra premie (a) op die kontrak te vra. Die probleem kan verander word na die prysing van ’n verkoopopsie met onderliggende bate . Hierdie vorm deel van die versekeringsmaatskappy se laste en moet dus ook bestuur word in terme van sy risiko’s. Dit kan gedoen word deur die opsie se sensitiwiteite te bestudeer. In hierdie tesis word die premie en sensitiwiteite van die verkoopopsie met behulp van verskillende Monte Carlo metodes bereken, om sodoende die effektiefste metode te vind. Deur die gebruik van algemene Monte Carlo metodes word ’n simplistiese prysingsmetode, wat verfyn is met behulp van wiskundige tegnieke wat die berekeningstyd wesenlik verminder, gevind. Nadat Antitetiese Veranderlikes, Kontrole Variate en Latynse Hiperkubus Steekproefneming as variansiereduksietegnieke oorweeg is, word gevind dat die verfynde metode se fout die effektiefste verminder met behulp van opsiepryse as Kontrole Variate. Dit word verbeter deur verskillende Quasi-Monte Carlo tegnieke, naamlik Halton, Faure, normale Sobol’ en ander verewekansigde Sobol’ reekse, te vergelyk. Die Owen en Faure-Tezuke tipe verewekansigde Sobol’ reeks verbeter die konvergensie van die beramer die effektiefste. Verder is die beste metode tussen Baanafhanklike Afgeleide Beramers en Eindige Differensie Benaderings om die sensitiwiteit vir die opsie te bepaal, ook gevind. Deur dus die verfynde prysingsmetode met opsiepryse as Kontrole Variate, saam met Owen en Faure-Tezuke tipe verewekansigde Sobol’ reekse as ’n Quasi-Monte Carlo metode te gebruik, word meer effektiewe metodes om die opsie te prys, gevind (in vergelyking met simplistiese Monte Carlo metodes). Verder is meer effektiewe sensitiwiteitsberamers as voorheen gevind wat gebruik kan word om risiko’s te help bestuur. Thesis (MComm (Statistics and Actuarial Science)--University of Stellenbosch, 2010.
We give an explicit formula for the price of an European Option in presence of Poisson processes and easily verifiable sufficient conditions which ensure the uniqueness of the option price even in hypothesis of incomplete markets. In the last section of the paper some numerical examples are discussed. keywords: option price, Poisson processes, incomplete markets, equivalent martingale measure.
We describe a construction of a summation scheme with replacements of random variables. As the limit, we obtain a time nonhomogeneous
generalization of the Ornstein-Uhlenbeck process. This generalization is described by a transform of Lamperti type in which
an arbitrary continuous monotone function is used instead of the exponential one. Bibliography: 16 titles.
We consider pure-jump transaction-level models for asset prices in continuous
time, driven by point processes. In a bivariate model that admits
cointegration, we allow for time deformations to account for such effects as
intraday seasonal patterns in volatility, and non-trading periods that may be
different for the two assets. We also allow for asymmetries (leverage effects).
We obtain the asymptotic distribution of the log-price process. We also obtain
the asymptotic distribution of the ordinary least-squares estimator of the
cointegrating parameter based on data sampled from an equally-spaced
discretization of calendar time, in the case of weak fractional cointegration.
For this same case, we obtain the asymptotic distribution for a tapered
estimator under more
In this paper we examine the dependence of option prices in a general jump-diffusion model on the choice of martingale pricing measure. Since the model is incomplete there are many equivalent martingale measures. Each of these measures corresponds to a choice for the market price of diffusion risk and the market price of jump risk. Our main result is to show that for conves payoffs the option price is increasing in the the jump-risk parameter. We apply this result to deduce general inequalities comparing the prices of contingent claims under various martingale measures which have been propsed in the literature as candidate pricing measures. Our proods are based on couplings of stochastic processes. If there is only one possible jump size then we are able to utilize a second coupling to extend our results to include stochastic jump intensities.
We consider a nonparametric method to estimate copulas, i.e. functions linking joint distributions to their univariate margins. We derive the asymptotic properties of kernel estimators of copulas and their derivatives in the context of a multivariate stationary process satisfactory strong mixing conditions. Monte Carlo results are reported for a stationary vector autoregressive process of order one with Gaussian innovations. An empirical illustration containing a comparison with the independent, comotonic and Gaussian copulas is given for European and US stock index returns.
This paper investigates the relation between the term structure of rents and future spot rents. A rich database of office rental agreements for various maturities is used to estimate the term structure of rents, and from this structure implicit forward rents are extracted. The data pertain to commercial properties in the three largest Swedish cities for the period 1998-2002. A positive relation between forward and spot rents is found in some regions, but forward rents underestimate future rent levels. Another contribution of the paper lies in the area of rental index construction. We provide evidence that rental indices should not only be quality constant (i.e., control for characteristics), but should also be maturity constant. Copyright 2004 by the American Real Estate and Urban Economics Association
We analyze the joint convergence of sequences of discounted stock prices and Radon-Nicodym derivatives of the minimal martingale measure when interest rates are stochastic. Therefrom we deduce the convergence of option values in either complete or incomplete markets. We illustrate the general result by two main examples: a discrete time i.i.d. approximation of a Merton type pricing model for options on stocks and the trinomial tree of Hull and White for interest rate derivatives.
We propose new closed-form pricing formulas for interest rate options which guarantee perfect compatibility with volatility smiles. These cap pricing formulas are computed under variance optimal measures in the framework of the market model or the Gaussian model and achieve an exact calibration of observed market prices. They are presented in a general setting allowing to study model and numŽraire choice effects on the computed prices. NumŽraire dependence is particularly emphasized. A numerical example and an empirical application on market data are given to illustrate the practical use of the calibration procedure.
We introduce trading restrictions in the well known Black-Scholes model and Cox-Ross-Rubinstein model, in the sense that hedging is only allowed at some fixed trading dates. As a consequence, the financial market is incomplete in both modified models. Applying Schweizer's (and Schal's) variance-optimal criterion for pricing and hedging general claims, we first analyse the dynamic consistency of the strategies which minimize the variance of the total loss due to hedging a given claim. Then we establish some convergence results, when the number of trading dates is either kept fixed or increases to infinity.
We consider a very general diffusion model for asset prices which allows the description of stochastic and past-dependent volatilities. Since this model typically yields an incomplete market, we show that for the purpose of pricing options, a small investor should use the minimal equivalent martingale measure associated to the underlying stock price process. Then we present stochastic numerical methods permitting the explicit computation of option prices and hedging strategies, and we illustrate our approach by specific examples. Copyright 1992 Blackwell Publishers.
This paper discusses some properties of general asset prices in continuous time. We introduce the concept of a martingale density which is a generalization of an equivalent martingale measure, and we show that absence of arbitrage plus some technical conditions implies the existence of a martingale density. This is in turn already sufficient to derive a recent result of Back (1991) on local risk premia for asset returns. As an application, we obtain a simple condition, valid in arbitrary information structures, for the drift part of discounted security gains to be absolutely continuous with respect to the variance process of the martingale part.
In the setting of incomplete markets, this paper presents a general result of convergence for derivative assets prices. It is proved that the minimal martingale measure first introduced by Föllmer and Schweizer is a convenient tool for the stability under convergence. This extends previous well-known results when the markets are complete both in discrete time and continuous time. Taking into account the structure of stock prices, a mild assumption is made. It implies the joint convergence of the sequences of stock prices and of the Radon-Nikodym derivative of the minimal measure. The convergence of the derivatives prices follows.
This property is illustrated in the main classes of financial market models.
The information matrix (IM) test is shown to have a finite sample distribution which is poorly approximated by its asymptotic χ <sup>2</sup> distribution in models and sample sizes commonly encountered in applied econometric research. The quality of the χ <sup>2</sup> approximation depends upon the method chosen to compute the test. Failure to exploit restrictions on the covariance matrix of the test can lead to a test with appalling finite sample properties. Order O(n<sup>-1</sup>) approximations to the exact distribution of an efficient form of the IM test are reported. These are developed from asymptotic expansions of the Edgeworth and Cornish-Fisher types. They are compared with Monte Carlo estimates of the finite sample distribution of the test and are found to be superior to the usual χ <sup>2</sup> approximations in sample sizes of the magnitude found in applied micro-econometric work. The methods developed in the paper are applied to normal and exponential models and to normal regression models. Results are provided for the full IM test and for heteroskedasticity and nonnormality diagnostic tests which are special cases of the IM test. In general the quality of alternative approximations is sensitive to covariate design. However commonly used nonnormality tests are found to have distributions which, to order O(n<sup>-1</sup>), are invariant under changes in covariate design. This leads to simple design and parameter invariant size corrections for nonnormality tests.
If options are correctly priced in the market, it should not be possible to make sure profits by creating portfolios of long and short positions in options and their underlying stocks. Using this principle, a theoretical valuation formula for options is derived. Since almost all corporate liabilities can be viewed as combinations of options, the formula and the analysis that led to it are also applicable to corporate liabilities such as common stock, corporate bonds, and warrants. In particular, the formula can be used to derive the discount that should be applied to a corporate bond because of the possibility of default.
We consider the stochastic processes Xk+1 = [Gamma]k+1(Xk) + Wk+1 where {[Gamma]k} is a sequence of nonlinear random functions and {Wk} is a sequence of disturbances. Sufficient conditions for the existence of a unique invariant probability are obtained. Functional central limit theorem is proved for every Lipschitzian function on R.