One-state variable binomial models for European-/American-style geometric Asian options

Abstract

This paper is concerned with geometric Asian options whose pay-offs depend on the geometric average of the underlying asset prices. Following the Cox et al (1979 J. Financial Economics 7 229-63) arbitrage arguments, we develop one-state variable binomial models for the options on the basis of the idea of Cheuk and Vorst (1997 J. Int. Money Finance 16 173-87). The models are more efficient and faster than those lattice methods (for the options) proposed by Hull and White (1993 J. Derivatives 1 21-31), Ritchken et al (1993 Manage. Sci. 39 1202-13), Barraquand and Pudet (1996 Math. Finance 6 17-51) and Cho and Lee (1997 J. Financial Eng. 6 179-91). We also establish the equivalence of the models and certain difference schemes.     

A novel Monte Carlo approach to hybrid local volatility models

We present in a Monte Carlo simulation framework, a novel approach for the evaluation of hybrid local volatility [Risk, 1994, 7, 18–20], [Int. J. Theor. Appl. Finance, 1998, 1, 61–110] models. In particular, we consider the stochastic local volatility model—see e.g. Lipton et al. [Quant. Finance, 2014, 14, 1899–1922], Piterbarg [Risk, 2007, April, 84–89], Tataru and Fisher [Quantitative Development Group, Bloomberg Version 1, 2010], Lipton [Risk, 2002, 15, 61–66]—and the local volatility model incorporating stochastic interest rates—see e.g. Atlan [ArXiV preprint math/0604316, 2006], Piterbarg [Risk, 2006, 19, 66–71], Deelstra and Rayée [Appl. Math. Finance, 2012, 1–23], Ren et al. [Risk, 2007, 20, 138–143]. For both model classes a particular (conditional) expectation needs to be evaluated which cannot be extracted from the market and is expensive to compute. We establish accurate and ‘cheap to evaluate’ approximations for the expectations by means of the stochastic collocation method [SIAM J. Numer. Anal., 2007, 45, 1005–1034], [SIAM J. Sci. Comput., 2005, 27, 1118–1139], [Math. Models Methods Appl. Sci., 2012, 22, 1–33], [SIAM J. Numer. Anal., 2008, 46, 2309–2345], [J. Biomech. Eng., 2011, 133, 031001], which was recently applied in the financial context [Available at SSRN 2529691, 2014], [J. Comput. Finance, 2016, 20, 1–19], combined with standard regression techniques. Monte Carlo pricing experiments confirm that our method is highly accurate and fast.     

Nils Ekholm

Abstract

The problem of χ² tests of a linear hypothesis H₀ for ‘matched samples’ in attribute data has been discussed earlier by the author (Bennett, 1967, 1968). This note presents corresponding results for the hypothesis that the multinomial probabilities p satisfy (c ?1) functional restrictions: F ₁(p) = 0, ... , F _C?1(p) = 0. An explicit relationship between the usual ‘goodness-of-fit’ χ² and the modified minimum χ² (=χ*²) of Jeffreys (1938) and Neyman (1949) is demonstrated for this situation. An example of the test for the 2 × 2 × 2 contingency table is given and compared with the solution of Bartlett (1935).     

Book Reviews

Organized Uncertainty: Designing a World of Risk Management. Michael Power. Oxford University Press, 2007. xviii and 248pp. ISBN 978–0–9–925394–4. £24.99.

Intellectual Capital Reporting: Lessons from Hong Kong and Australia. J. Guthrie, R. Petty and F. Ricceri. The Institute of Chartered Accountants of Scotland, 2007, vii and 118pp. ISBN 978–1–904574–27–9. £15

The Routledge Companion to Fair Value and Financial Reporting. P. Walton (ed.). Routledge, 2007. xviii and 404 pp. ISBN 978–0–415–42356–4. £95.

UK Reporting of Intellectual Capital. Jeffrey Unerman, James Guthrie and Ludmila Striukova. ICAEW Centre for Business Performance, 2007. 68 pp. ISBN 978 1 84152 507 5. £20.     

CALL FOR PAPERS

Organized Uncertainty: Designing a World of Risk Management. Michael Power. Oxford University Press, 2007. xviii and 248pp. ISBN 978–0–9–925394–4. £24.99.

Intellectual Capital Reporting: Lessons from Hong Kong and Australia. J. Guthrie, R. Petty and F. Ricceri. The Institute of Chartered Accountants of Scotland, 2007, vii and 118pp. ISBN 978–1–904574–27–9. £15

The Routledge Companion to Fair Value and Financial Reporting. P. Walton (ed.). Routledge, 2007. xviii and 404 pp. ISBN 978–0–415–42356–4. £95.

UK Reporting of Intellectual Capital. Jeffrey Unerman, James Guthrie and Ludmila Striukova. ICAEW Centre for Business Performance, 2007. 68 pp. ISBN 978 1 84152 507 5. £20.     

Sequential Quasi-Credibility for Scale Dispersion Models

The sequential approach to credibility, developed by Landsman and Makov [(1999a) On stochastic approximation and credibility. Scand. Actuarial J. 1, 15-31; (1999b) Sequential credibility evaluation for symmetric location claim distributions. Insurance: Math. Econ. 24, 291-300] is extended to the scale dispersion family, which contains distributions often used in actuarial science: log-normal, Weibull, Half normal, Stable, Pareto, to mention only a few. For members of this family a sequential quasi-credibility formula is devised, which can also be used for heavy tailed claims. The results are illustrated by a study of log-normal claims.     

A Note on s. or n.s. cPp

Abstract

1. In a s. or n.s. cPp (stationary or non-stationary compound Poisson process) the probability for occurrence of m events, while the parameter (one-or more-dimensional) passes from zero to τ ₀ as measured on an absolute scale (the τ-scale), is defined as a mean of Poisson probabilities with intensities, which are distributed with distribution functions defining another random process, called the primary process with respect to the s. or n.s cPp. The stationarity (in the weak sence) and the non-stationarity of the primary process imply the same properties of the s. or n.s. cPp.     

Algebraic structure of vector fields in financial diffusion models and its applications

High-order discretization schemes of SDEs using free Lie algebra-valued random variables are introduced by Kusuoka [Adv. Math. Econ., 2004, 5, 69–83], [Adv. Math. Econ., 2013, 17, 71–120], Lyons–Victoir [Proc. R. Soc. Lond. Ser. A Math. Phys. Sci., 2004, 460, 169–198], Ninomiya–Victoir [Appl. Math. Finance, 2008, 15, 107–121] and Ninomiya–Ninomiya [Finance Stochast., 2009, 13, 415–443]. These schemes are called KLNV methods. They involve solving the flows of vector fields associated with SDEs and it is usually done by numerical methods. The authors have found a special Lie algebraic structure on the vector fields in the major financial diffusion models. Using this structure, we can solve the flows associated with vector fields analytically and efficiently. Numerical examples show that our method reduces the computation time drastically.     

The theory of confluent hypergeometric functions and its application to compound poisson processes

Abstract

A. Confluent Hypergeometric Functions

The theory of such functions will be recapitulated here following a recently published book (Slater, L. J., Confluent Hypergeometric Functions, Cambr. Univ. Press 1960).     

Robust portfolio estimation under skew-normal return processes

In this paper, we study issues related to the optimal portfolio estimators and the local asymptotic normality (LAN) of the return process under the assumption that the return process has an infinite moving average (MA) (∞) representation with skew-normal innovations. The paper consists of two parts. In the first part, we discuss the influence of the skewness parameter δ of the skew-normal distribution on the optimal portfolio estimators. Based on the asymptotic distribution of the portfolio estimator ? for a non-Gaussian dependent return process, we evaluate the influence of δ on the asymptotic variance V(δ) of ?. We also investigate the robustness of the estimators of a standard optimal portfolio via numerical computations. In the second part of the paper, we assume that the MA coefficients and the mean vector of the return process depend on a lower-dimensional set of parameters. Based on this assumption, we discuss the LAN property of the return's distribution when the innovations follow a skew-normal law. The influence of δ on the central sequence of LAN is evaluated both theoretically and numerically.     

Large sample validity of the binomial distribution for lives with unequal mortality rates

Abstract

Let us consider a group of n lives which are observed during some time or age interval. Suppose that the following conditions are satisfied: 1. The probability of death within the interval considered has the same value q for each person of the group.

2. These lives represent statistically independent observations (with respect to mortality).

     

A Note on Gaussian Estimation of the CKLS and CIR Models with Feedback Effects for Japan

In this note we extend the Gaussian estimation of two factor CKLS and CIR models recently considered in Nowman, K. B. (2001, Gaussian estimation and forecasting of multi-factor term structure models with an application to Japan and the United Kingdom, Asia Pacif. Financ. Markets 8, 23–34) to include feedback effects in the conditional mean as was originally formulated in general continuous time models by Bergstrom, A. R. (1966, Non-recursive models as discrete approximations to systems of stochastic differential equations, Econometrica 34, 173–182) with constant volatility. We use the exact discrete model of Bergstrom, A. R. (1966, Non-recursive models as discrete approximations to systems of stochastic differential equations, Econometrica 34, 173–182) to estimate the parameters which was first used by Brennan, M. J. and Schwartz, E. S. (1979, A continuous time approach to the pricing of bonds, J. Bank. Financ. 3, 133–155) to estimate their two factor interest model but incorporating the assumption of Nowman, K. B. (1997, Gaussian estimation of single-factor continuous time models of the term structure of interest rates, J. Financ. 52, 1695–1706; 2001, Gaussian estimation and forecasting of multi-factor term structure models with an application to Japan and the United Kingdom, Asia Pacif. Financ. Markets 8, 23–34). An application to monthly Japanese Euro currency rates indicates some evidence of feedback from the 1-year rate to the 1-month rate in both the CKLS and CIR models. We also find a low level-volatility effect supporting Nowman, K. B. (2001, Gaussian estimation and forecasting of multi-factor term structure models with an application to Japan and the United Kingdom, Asia Pacif. Financ. Markets 8, 23–34).     

Time-varying long-run mean of commodity prices and the modeling of futures term structures

The exploration of the mean-reversion of commodity prices is important for inventory management, inflation forecasting and contingent claim pricing. Bessembinder et al. [J. Finance, 1995, 50, 361–375] document the mean-reversion of commodity spot prices using futures term structure data; however, mean-reversion to a constant level is rejected in nearly all studies using historical spot price time series. This indicates that the spot prices revert to a stochastic long-run mean. Recognizing this, I propose a reduced-form model with the stochastic long-run mean as a separate factor. This model fits the futures dynamics better than do classical models such as the Gibson–Schwartz [J. Finance, 1990, 45, 959–976] model and the Casassus–Collin-Dufresne [J. Finance, 2005, 60, 2283–2331] model with a constant interest rate. An application for option pricing is also presented in this paper.     

Derivation of the Wishart distribution of the second order sample moments by straightforward integration of a multiple integral

1. Introduction.

In his textbook of statistics Kendall classifies the methods of deducing exact sampling distributions into four groups:

• (a) straightforward evaluation of the integral in question by ordinary analytical processes such as a convenient change of variable;

• (b) the use of geometrical terminology;

• (c) the use of characteristic functions; and

• (d) other analytical methods.

     

Positive asymmetric information in volatile environments: The black market dollar and sovereign bond yields in Venezuela

PurposeWe test the informational efficiency of Venezuelan USD sovereign bond yields when the black market exchange-rate premium (BMERP) changes.DesignWe use a non-parametric, asymmetric, Granger causality test to test our hypothesis.FindingsWe find that the bond market with less than or equal to 5 years of maturity seems to be efficient when good news is released on the BMERP. However, this market is not informationally efficient, and when combined with unbiased bad news regarding the BMERP, arbitrage opportunities are created.Originality/valueCapital controls that restrict free exchange-rate mechanisms create arbitrage opportunities with negative news as opposed to positive news.     

The distributions of the first and second order moments,the partial correlation coefficients and the multiple correlation coefficient in samples from a normal multivariate population

Abstract

1. When the frequency function of a statistical variable is known, one of the most important tasks of Theoretical Statistics is to find the frequency functions of some simple functions of this variable. The most important are the first and second order moments in a sample containing a certain number of values of the variable.     

An improved convolution algorithm for discretely sampled Asian options

We suggest an improved FFT pricing algorithm for discretely sampled Asian options with general independently distributed returns in the underlying. Our work complements the studies of Carverhill and Clewlow [Risk, 1990, 3(4), 25–29], Benhamou [J. Comput. Finance, 2002, 6(1), 49–68], and Fusai and Meucci [J. Bank. Finance, 2008, 32(10), 2076–2088], and, if we restrict our attention only to log-normally distributed returns, also Ve?e? [Risk, 2002, 15(6), 113–116]. While the existing convolution algorithms compute the density of the underlying state variable by moving forward on a suitably defined state space grid, our new algorithm uses backward price convolution, which resembles classical lattice pricing algorithms. For the first time in the literature we provide an analytical upper bound for the pricing error caused by the truncation of the state space grid and by the curtailment of the integration range. We highlight the benefits of the new scheme and benchmark its performance against existing finite difference, Monte Carlo, and forward density convolution algorithms.     

Fast Ninomiya–Victoir calibration of the double-mean-reverting model

Abstract

We consider the three-factor double mean reverting (DMR) option pricing model of Gatheral [Consistent Modelling of SPX and VIX Options, 2008], a model which can be successfully calibrated to both VIX options and SPX options simultaneously. One drawback of this model is that calibration may be slow because no closed form solution for European options exists. In this paper, we apply modified versions of the second-order Monte Carlo scheme of Ninomiya and Victoir [Appl. Math. Finance, 2008, 15, 107–121], and compare these to the Euler–Maruyama scheme with full truncation of Lord et al. [Quant. Finance, 2010, 10(2), 177–194], demonstrating on the one hand that fast calibration of the DMR model is practical, and on the other that suitably modified Ninomiya–Victoir schemes are applicable to the simulation of much more complicated time-homogeneous models than may have been thought previously.     

Application de la théorie collective du risque à la réassurance « Excess-Loss »

Abstract

Introduction.

Les méthodes de calcul de la prime de réassurance Excess-Loss utilisées jusqu'à présent ou bien avaient un caractère purement empirique, ou bien présumaient que les écarts entre le montant des sinistres constatés et le montant des sinistres attendus se répartissent suivant la loi de Laplace-Gauss.